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People working in or studying the theory of relations (Boolean matrices) or graph theory often use small examples and manipulate them with pencil and paper in order to prove or disprove some property. (Of course, this is no more feasible with bigger examples.) RELVIEW is an interactive computer system supporting such tasks. This report gives a description of the RELVIEW system (inclusive a user's manual and some implementation details) and informs also about the theoretical background.

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- Hajnal Andréka, Bjarni Jónsson, and István Németi.
Free algebras in discriminator varieties.
Algebra Universalis, 28:401447, 1991.
- Hajnal Andréka, Roger Duncan Maddux, and István Németi.
Splitting in relation algebras.
Proc. null Amer. null Math. null Soc., 111(4):10851094,
April 1991.
- Hajnal
Andréka, István Németi, and Ildikó Sain.
Abstract model theoretic approach to algebraic logic (an overview).
CCSOM Working Paper 92-92, Dept. null of Statistics and Methodology, PSCW,
Univ. null Amsterdam, 1992.
- Hajnal Andréka, István Németi, and Ildikó Sain.
Algebras of relations and algebraic logic. an introduction.
CCSOM Working Paper 93-91, Dept. null of Statistics and Methodology, PSCW,
Univ. null Amsterdam, 1993.
- Hajnal Andréka, István Németi, and Ildikó Sain.
Methodology of applying algebraic logic to logic.
Course Material Version, June 1993.
- Hajnal Andréka, Steven Givant, and István Németi.
The lattice of varieties of representable relation algebras.
Journal of Symbolic Logic, 59(2):631661, 1994.
- Hajnal Andréka.
Boolean reducts of relation and cylindric algebras and the cube problem.
Proc. null Amer. null Math. null Soc., 100(1):148153, May
1987.
- Hajnal Andréka.
On taking subalgebras of relativized relation algebras.
Algebra Universalis, 25:96100, 1988.
- Hajnal Andréka.
Representations of lattice-ordered semigroups with binary relations.
Algebra Universalis, 28:1225, 1991.
- Hajnal Andréka.
Weakly representable but not representable relation algebras.
Algebra Universalis, 32:3143, 1994.
- Irving H.
Anellis and Nathan Houser.
The 19
^{th}century roots of universal algebra and algebraic logic. In Andréka et al. [Andréka et al., 1991b], pages 136. - A. Apostolico,
M. Crochemore, Z. Galil, and U. Manber, editors.
Combinatorial Pattern Matching, Proceedings of the Third Annual
Symposium, Tucson, Arizona, April/May 1992, volume 644 of
LNCS. Springer-Verlag, 1992.
- M.A. Arbib and
E.G. Manes.
Partially-additive monoids, graph-growing and the algebraic semantics of
recursive calls.
In Claus et al. [Claus et al., 1978].
The way in which PFN (sets and partial functions) provides a setting for the semantics of deterministic programs [and REL (sets and relations) provides a setting for the semantics of nondeterministic programs] has led us to an axiomatic notion of a partially-additive monoid. We show that programs incorporating procedure calls may be represented by graph grammars, with one non-terminal and one production for each distinct procedure (including the program itself). Program execution may be construed as a process of interpretation of graphs obtained by repeated graph substitution. We show that the resultant interpretive semantics yields the same result ary of the canonical fixpoint for abstract recursion schemas introduced in an earlier paper.

- Ofer Arieli and
Arnon Avron.
Logical bilattices and inconsistent data.
In Proceedings, Ninth Annual IEEE Symposium on Logic in
Computer Science [IEE, 1994], pages 468476.
The notion of a

*bilattice*was first proposed by Ginsberg as a general framework for many applications. This notion was further investigated and applied for various goals by Fitting. In the present paper we develop*proof systems*, which correspond to bilattices in an essential way. We then show how to use those bilattices for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii concerning inconsistencies, which happen to suit well the framework of bilattices. The outcome is a paraconsistent logic with a lot of desirable properties. - W. W. Armstrong.
Dependency structures of database relationships.
In 1974 IFIP Congress, pages 580583. North-Holland, 1974.
- Andrea
Asperti and Giuseppe Longo.
Categories, Types, and Structures: An Introduction to Category Theory for
the Working Computer Scientist.
Foundations of Computing. MIT Press, Cambridge, MA, 1991.
- Jürgen Audretsch and Klaus Mainzer, editors.
Wieviele Leben hat Schrödingers Katze? Zur Physik und Philosophie der
Quantenmechanik.
BI-Wissenschaftsverlag, 1990.
- A. Avron.
The semantics and proof theory of linear logic.
Theoretical Computer Science, 57:161184, 1988.
- Arnon Avron.
Two types of multiple-conclusion systems.
Logic Journal of the IGPL, 6(5):695717, 1998.
- F. Baader and B. Hollunder.
How to prefer more specific defaults in terminological default logic.
DFKI Research Report RR-92-58, German Research Center for Artificial
Intelligence (DFKI), Kaiserslautern, Germany, 1992.
A short version will be published in the Proc. null of IJCAI'93.
In a recent paper we have proposed terminological default logic as a formalism which combines both means for structured representation of classes and objects, and for default inheritance of properties. The major drawback which terminological default logic inherits from general default logic is that it does not take precedence of more specific defaults over more general ones into account. This behaviour has already been criticized in the general context of default logic, but it is all the more problematic in the terminological case where the emphasis lies on the hierarchical organization of concepts. The present paper addresses the problem of modifying terminological default logic such that more specific defaults are preferred. It turns out that the existing approaches for expressing priorities between defaults do not seem to be appropriate for this purpose. Therefore we shall consider an alternative approach for dealing with prioritization in the framework of Reiter's default logic. The formalism is presented in the general setting of default logic where priorities are given by an arbitrary partial ordering on the defaults. We shall exhibit some interesting properties of the new formalism, compare it with existing approaches, and describe an algorithm for computing extensions.

- Franz Baader
and Klaus U. Schulz.
Unification in the union of disjoint equational theories: Combining decision
procedures.
Research Report RR-91-33, German Research Center for Artificial Intelligence
(DFKI), Saarbrücken, Germany, November 1991.
Most of the work on the combination of unification algorithms for the union of disjoint equational theories has been restricted to algorithms which compute finite complete sets of unifiers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of unification problems without computing unifiers. In this paper we describe a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called unification problems with constant restrictionsa slight generalization of unification problems with constantsis decidable for these theories. As a consequence of this new method, we can for example show that general A-unifiability, i.e., solvability of A-unification problems with free function symbols, is decidable. Here A stands for the equational theory of one associative function symbol. Our method can also be used to combine algorithms which compute finite complete sets of unifiers. Manfred Schmidt-Schauss's combination result, the until now most general result in this direction, can be obtained as a consequence of this fact. We also get the new result that unification in the union of disjoint equational theories is finitary, if general unificationi.e., unification of terms with additional free function symbolsis finitary in the single theories.

- F. Baader
and J.H. Siekmann.
Unification theory.
In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of
Logic in Artificial Intelligence and Logic Programming. Oxford
University Press, Oxford, UK, 1993.
Most knowledge-based systems in artificial intelligence (AI) with a commitment to a symbolic representation support one important operation: matching of descriptions. This operation, called

*unification*in the field of deduction systems, is the addition and multiplication of many AI systems, and is consequently often supported by special purpose hardware or by a fast instruction set. Unification theory provides the formal framework for investigations into the properties of this operation, which is in essence the solving of equations in an (equationally defined) free algebra. This paper gives an introduction into unification theory, and treats some of the important research topics in this area in more detail. - F. Baader, H.-J. Bürckert, B. Nebel, W. Nutt, and G. Smolka.
On the expressivity of feature logics with negation, functional uncertainty,
and sort equations.
Research Report RR-91-01, German Research Center for Artificial Intelligence
(DFKI), Stuhlsatzenhausweg 3, 6600 Saarbrücken 11, Germany, January 1991.
Feature logics are the logical basis for so-called unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with negation and the functional uncertainty construct needed for the description of long-distance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom.

- Franz Baader, Bernhard Hollunder, Bernhard Nebel,
Hans-Jürgen Profitlich, and Enrico Franconi.
An empirical analysis of optimization techniques for terminological
representation systems or: Making KRIS get a move on.
Research Report RR-93-03, German Research Center for Artificial Intelligence
(DFKI), Saarbrücken, Germany, January 1993.
A shorter version has been published in Proc. null KR'92.
We consider different methods of optimizing the classification process of terminological representation systems, and evaluate their effect on three different types of test data. Though these techniques can probably be found in many existing systems, until now there has been no coherent description of these techniques and their impact on the performance of a system. One goal of this paper is to make such a description available for future implementors of terminological systems. Building the optimizations that came off best into the KRIS system greatly enhanced its efficiency.

- Franz Baader.
Unification in varieties of completely regular semigroups.
In Schulz [Schulz, 1990], pages 210230.
All varieties of idempotent semigroups have been classified with respect to the unification types of their defining sets of identities. With the exception of eight finitary unifying theories, they are all of unification type zero. This yields countably many examples of theories of this type which are more ``natural'' than the first example constructed by Fages and Huet. The lattice of all varieties of idempotent semigroups is a sublattice of the lattice of all varieties of orthodox bands of groups, and this lattice is a sublattice of the lattice of all varieties of complex regular semigroups. The proof which was used to establish the result for the varieties of idempotent semigroups of type zero canwith some modificationsalso be applied to the larger lattice of all varieties of complete regular semigroups. This shows that type zero is not an exception, but rather common for varieties of semigroups. To establish the results for the eight exceptional finitary varieties of idempotent semigroups we have developed a method which under certain conditions allows to deduce the unification type of a join of varieties from the types of the varieties participating in this join. This method can also be employed for varieties of orthodox bands of abelian groups. Any variety of orthodox bands of abelian groups is the join of a variety of idempotent semigroups and a variety of abelian groups. It turns out that the unification type of such a join is just the type of the variety of idempotent semigroups taking part in this join. The emphasis of the paper is on describing the tools necessary for proving all the mentioned results.

- Franz Baader.
Unification theory.
In Schulz [Schulz, 1990], pages 151170.
The purpose of this paper is not to give an overview of the state of art in unification theory. It is intended to be a short introductin into the area of equational unification which should give the reader a feeling for what unification theory might be about. The basic notions such as complete and minimal compete sets of unifiers, and unification types of equational theories are introduced and illustrated by examples. The we shall describe the original motivations for considering unification (in the empty theory) in resolution theorem proving and term rewriting. Starting with Robinson's first unification algorithm it will be sketched how more efficient unification algorithms can be derived. We shall then explain the reasons which lead to the introduction of unification in non-empty theories into the above mentioned areas theorem proving and term rewriting. For theory unification it makes a difference whether single equations or systems of equations are considered. In addition, one has to be careful with regard to the signature over which the terms of the unification problems can be built. This leads to the distinction between elementary unification, unification with constants, and general unification (where arbitrary free function symbols may occur). Going from elementary unification to general unification is an instance of the so-called combination problem for equational theories which can be formulated as follows: Let E,F be equational theories over disjoint signatures. How can unification algorithms for E,F be combined to a unification algorithm for the theory E uni F.

- Leo
Bachmair and Harald Ganzinger.
Rewrite techniques for transitive relations.
In Proceedings, Ninth Annual IEEE Symposium on Logic in
Computer Science [IEE, 1994], pages 384393.
We propose inference systems for dealing with transitive relations in the contqext of resolution-type theorem proving. These inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods. We establish their refutational completeness and also prove their compatibility with the usual simplification techniques used in rewrite-based theorem provers. A key to the practicality of chaining techniques is the extent to which so-called variable chainings can be restricted. We demonstrate that rewrite techniques considerably restrict variable chaining, though we also show that they cannot be completely avoided for transitive relations in general. If the given relation satisfies additional properties, such as symmetry, further restrictions are possible. In particular, we discuss (partial) equivalence relations and congruence relations.

- Leo Bachmair, editor.
Rewriting Techniques and Applications, 11th International Conference,
RTA2000, Norwich, UK, July 2000, Proc., volume 1833 of
LNCS. Springer, 2000.
- R. J. R.
Back and J. von Wright.
Combining angels, demons and miracles in program specifications.
Theoretical Computer Science, 100:365383, 1992.
- R. J. R. Back.
On correct refinement of programs.
Journal of Computer and System Science, 23(1):4968, 1981.
- Fred Backer.
Representable relation algebras.
Berkeley, 1970.
Report for a seminar on relation algebras conducted by A. null Tarski,
mimeographed, Spring, 1970.
- Roland C. Backhouse and H. Doornbos.
Mathematical induction made calculational.
Computing Science Notes 94/16, Eindhoven Univ. null of Technology,
Dept. null of Mathematics and Computing Science, April 1994.
- Roland Backhouse and Maarten Fokkinga.
The
associativity of equivalence and the towers of hanoi problem.
2000.
Dijsktra and Scholten have argued that greater use should be made of the associativity of equivalence. This note shows how the property is used in specifying the rotation of the disks in the well-known Towers of Hanoi problem.

- Roland Backhouse and Paul Hoogendijk.
Final dialgebras:
From categories to allegories.
Theoretical Informatics, 33(4/5):401426, 1999.
The study of inductive and coinductive types (like finite lists and streams, respectively) is usually conducted within the framework of category theory, which to all intents and purposes is a theory of sets and functions between sets. Allegory theory, an extension of category theory due to Freyd, is better suited to modelling relations between sets as opposed to functions between sets. The question thus arises of how to extend the standard categorical results on the existence of final objects in categories (for example, coalgebras and products) to their existence in allegories. The motivation is to streamline current work on generic programming, in which the use of a relational theory rather than a functional theory has proved to be desirable. In this paper, we define the notion of a relational final dialgebra and prove, for an important class of dialgebras, that a relational final dialgebra exists in an allegory if and only a final dialgebra exists in the underlying category of maps. Instances subsumed by the class we consider include coalgebras and products. An important lemma expresses bisimulations in allegorical terms and proves this equivalent to Aczel and Mendler's categorical definition.

- Roland C.
Backhouse et al.
Fixed-point calculus.
Inform. null Process. null Lett., 53:131136, 1995.
- Roland Backhouse and Jaap van der Woude.
Domain operators and domain kinds.
ftp://ftp.win.tue.nl/pub/math.prog.construction, September 1993.
The notions of domain operator and domain kind are introduced. Several examples are presented. In particular, it is shown that the partial equivalence relations form a domain kind. The proof involves the construction of a Galois connection demonstrating that the partial equivalence relations form a complete lattice under the so-called domai ordering, thus providing another illustration of the importance of the early recognition of Galois connections.

- Roland C. Backhouse and Jaap van der Woude.
Demonic operators and monotype factors.
Mathematical Structures in Computer Science, 3(4):417433,
December 1993.
- Roland C. Backhouse, Peter de Bruin, Grant Malcolm,
Ed Voermans, and Jaap van der Woude.
A relational theory of types.
Dept. null of Computing Science, Rijksuniversiteit Groningen, and
Dept. null of Math. null and Computing Science, Technische Univ. null
Eindhoven, May 1990.
- [Backhouse et al., 1991a]
- Roland C. Backhouse, Peter J. de Bruin, Paul Hoogendijk, Grant Malcolm, Ed Voermans, and Jaap van der Woude. Polynomial relators. Computing Science Notes 91/10, Eindhoven Univ. null of Technology, Dept. null of Mathematics and Computing Science, May 1991.
- Roland C. Backhouse, Peter J. de Bruin, Grant Malcom, Ed Voermans, and Jaap van der Woude. Relational catamorphisms. In Möller [M{ö}ller, 1991a], pages 319371.
- Roland C. Backhouse, Ed Voermans, and Jaap van der Woude. A relational theory of types. In Proc. null EURICS Workshop on Calculational Theories of Program Structures, 1991.
- [Backhouse et al., 1992a]
- R.C. Backhouse, Peter J. de Bruin, P. Hoogendijk, G. Malcolm, T.S. Voermans, and J. van der Woude. Polynomial relators. In Nivat et al. [Nivat et al., 1992], pages 303362.
- Roland C. Backhouse, Paul Hoogendijk, Ed Voermans, and Jaap van der Woude. A relational theory of datatypes. Research report, Dept. null of Mathematics and Computer Science, Eindhoven Univ. null of Technology, The Netherlands, 1992.
- Roland C. Backhouse. An exploration of the BirdMeertens formalism. Computing Science Notes CS 8810, Univ. null of Groningen, Dept. null of Mathematics and Computing Science, 1988.
- Roland C. Backhouse. Pair algebras and galois connections. Information Processing Letters, 67(4):169176, August 1998.
- Roland Backhouse. Fixed point calculus. In Summer School and Workshop on Algebraic and Coalgebraic Methods in the Mathematics of Program Construction, Oxford, April 1114, 2000, 2000.
Fixed point calculus is about the solution of recursive equations defined by a monotonic endofunction on a partially ordered set. This tutorial discusses applications of fixed point calculus in the construction of computer programs, beginning with standard applications and progressing to recent research. The basic properties of least and greatest fixed points are presented. Well-foundedness and inductive properties of relations are expressed in terms of fixed points. A class of fixed point equations, called ``hylo'' equations, is introduced. A methodology of recursive program design based on the use of hylo equations is presented. Current research on generalisations of well-foundedness and inductive properties of relations, making these properties relative to a datatype, is introduced.

- Rolf Backofen and Gert Smolka. A complete and recursive feature theory. Research Report RR-92-30, German Research Center for Artificial Intelligence (DFKI), Stuhlsatzenhausweg 3, 6600 Saarbrücken 11, Germany, September 1992.
Various feature descriptions are being employed in logic programming languages and constrained-based grammar formalisms. The common notational primitive of these descriptions are functional attributes called features. The descriptions considered in this paper are the possibly quantified first-order formulae obtained from a signature of binary and unary predicates called features and sorts, respectively. We establish a first-order theory FT by means of three axiom schemes, show its completeness, and construct three elementarily equivalent models. One of the models consists of so-called feature graphs, a data structure common in computational linguistics. The other two models consist of so-called feature trees, a record-like data structure generalizing the trees corresponding to first-order terms. Our completeness proof exhibits a terminating simplification system deciding validity and satisfiability of possibly quantified feature descriptions.

- J.C.M. Baeten and W.P. Weijland. Process Algebra, volume 18 of Tracts in Theoretical Computer Science. Cambridge Univ. null Press, 1990.
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In this paper we discuss term-rewriting systems with rule priorities, which simply is a partial ordering on the rules. The procedural meaning of such an ordering then is, that the application of a rule of lower priority is allowed only if no rule of higher priority is applicable. The semantics of such a system is discussed. It turns out that the class of all bounded systems indeed has such a semantics

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Although the distinction between software and hardware is a posteriori, there is an a priori distinction that masquerades as the software-hardware distinction. This is the distinction between procedure interconnection, the semantics of flow chart diagrams, which is known to be described by the regular expression calculus; and system interconnection, the semantics of network diagrams, which is described by a certain logical calculus, dual to a calculus of regular expressions. This paper presents a proof of the duality in a special case, and gives the interpretation of the logical calculus for sequential machine interconnection. A minimal realization theorem for feedback systems is proved, which specializes to known open loop minimal realization theorems.

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- Rudolf Berghammer and Gunther Schmidt. The RELVIEW-system. In Choffrut and Jantzen [Choffrut and Jantzen, 1991], pages 535536.
People working in or studying the theory of relations or graph theory very often use more or less small examples of ``concrete'' relations and manipulate them with pencil and paper in order to prove or disprove some property. The RELVIEW system is a totally interactive and completely video-oriented computer system supporting such tasks.

- Rudolf Berghammer and Gunther Schmidt, editors. Programmiersprachen und Grundlagen der Programmierung, Kolloquium auf der Barbarahütte am Kreuzeck, Sep 1517, 1993, volume 93/09 of Tech. Rep. Fakultät für Informatik, Univ. null der Bundeswehr München, 1993.
- Rudolf Berghammer and Gunther Schmidt. Relational specifications. In C. Rauszer, editor, Proc. XXXVIII Banach Center Semester on Algebraic Methods in Logic and their Computer Science Applications, volume 28 of Banach Center Publications, pages 167190, Warszawa, 1993. Institute of Mathematics, Polish Academy of Sciences.
- Rudolf Berghammer and Gunther Schmidt. RELVIEW a computer system for the manipulation of relations. notes to a system demonstration. In Nivat et al. [Nivat et al., 1994], pages 403404.
- Rudolf Berghammer and Burghard von Karger. Towards a design calculus for CSP. Science of Computer Programming, 26:99115, 1996.
- Rudolf Berghammer and Burghard von Karger. Algorithms from relational specifications. In Brink et al. [Brink et al., 1997], chapter 9, pages 131149.
The calculus of relations turned into an important conceptual and methodological tool in computer science. The methods presented in this book include questions of relational databases, applications to program specification, resource-conscious linear logic, semantic and refinement consideration, nonclassical logics for reasoning about programs, tabular methods in software construction, algorithm development, linguistic problems, followed by a comprehensive bibliography. The reader gets an overview of the wide-ranging applicability of relational methods in computer science.

*``... While this is a multi-authored volume, the authors have done an excellent job of making it read like a single-authored work ... The book can be viewed as a set of snapshots of a family of research and researchers at one point in time. If you are interested in relational problems, I can think of no better introduction ...''* Computing Reviews- Rudolf Berghammer and Burghard von Karger. Relational semantics of functional programs. In Brink et al. [Brink et al., 1997], chapter 8, pages 115130.
The calculus of relations turned into an important conceptual and methodological tool in computer science. The methods presented in this book include questions of relational databases, applications to program specification, resource-conscious linear logic, semantic and refinement consideration, nonclassical logics for reasoning about programs, tabular methods in software construction, algorithm development, linguistic problems, followed by a comprehensive bibliography. The reader gets an overview of the wide-ranging applicability of relational methods in computer science.

*``... While this is a multi-authored volume, the authors have done an excellent job of making it read like a single-authored work ... The book can be viewed as a set of snapshots of a family of research and researchers at one point in time. If you are interested in relational problems, I can think of no better introduction ...''* Computing Reviews- Rudolf Berghammer and Hans Zierer. Relational algebraic semantics of functional programs. Technical Report TUM-INFO 8501, Technische Univ. null München, Fakultät für Informatik, 1985.
- Rudolf Berghammer and Hans Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123147, 1986.
- Rudolf Berghammer, Gunther Schmidt, and Hans Zierer. Symmetric quotients. Technical Report TUM-INFO 8620, Technische Universität München, Fakultät für Informatik, 1986. 18 p.
- Rudolf Berghammer, Herbert Ehler, and Hans Zierer. Development of graph algorithms by program transformation. In H. Göttler and H.J. Schneider, editors, Proc. 13th International Workshop on Graph-Theoretic Concepts in Computer Science, volume 314 of LNCS, pages 206218, Kloster Banz/Staffelstein, 1987. Springer.
- Rudolf Berghammer, Herbert Ehler, and Hans Zierer. Towards an algebraic specification of code generation. Technical Report TUM-I8707, Institut für Informatik, Technische Universität München, 1987.
- Rudolf Berghammer, Herbert Ehler, and Hans Zierer. Towards an algebraic specification of code generation. In F. Simon, editor, Proceedings Kolloquium ``Programmspezifikation'', Bericht Nr. 8711, Midlum, Föhr, 1987. Institut für Informatik und praktische Mathematik, Universität Kiel. gekürzte Version von [Berghammer et al., 1987b].
- Rudolf Berghammer, Herbert Ehler, and Hans Zierer. Towards an algebraic specification of code generation. Science of Computer Programming, 11:4563, 1988. gekürzte Version von [Berghammer et al., 1987b].
- Rudolf Berghammer, Gunther Schmidt, and Hans Zierer. Symmetric quotients and domain constructions. Inform. null Process. null Lett., 33(3):163168, 1989.
- Rudolf Berghammer, Herbert Ehler, and Bernhard Möller. On the refinement of non-deterministic recursive routines by transformations. In Broy and Jones [Broy and Jones, 1990], pages 5371.
- Rudolf Berghammer, Peter Kempf, Gunther Schmidt, and Thomas Ströhlein. Relational algebra and logic of programs. In Andréka et al. [Andréka et al., 1991a], pages 3758.
- Rudolf Berghammer, Birgit Elbl, and Ulf Schmerl. Proving correctness of programs in weak second-order logic. Technical Report 9206, Fakultät für Informatik, Universität der Bundeswehr München, 1992.
- Rudolf Berghammer, Thomas F. Gritzner, and Gunther Schmidt. Prototyping relational specifications using higher-order objects. In Heering et al. [Heering et al., 1993], pages 5675. also as Tech. Report. 9304, UniBwM, 1993, 33 pages.
- Rudolf Berghammer, Thomas F. Gritzner, and Gunther Schmidt. Prototyping relational specifications using higher-order objects. Technical Report 9304, Fakultät für Informatik, Universität der Bundeswehr München, 1993.
- Rudolf Berghammer, Armando M. Haeberer, Gunther Schmidt, and Paulo A. S. Veloso. Comparing two different approaches to products in abstract relation algebra. In Nivat et al. [Nivat et al., 1994], pages 167176.
- Rudolf Berghammer, Armando Martín Haeberer, Gunther Schmidt, and Paulo A.S. Veloso. A new class of partially evaluable fork algebras: Axiomatization and models. unpublished, 1995.
- Rudolf Berghammer. Eine Übertragung des Park'schen Lemmas auf die Abkömmlingsrelation. In W. Dosch, editor, Proc. null of Arbeitstreffen ``Logische und funktionale Programmierung Sprachen, Methoden, Implementierungen'', pages 6265, Hirschegg/Kleinwalsertal, 1989. Institut für Mathematik, Universität Augsburg. Report Nr. 214.
- Rudolf Berghammer. A mathematical basis for nondeterministic unfold/fold. In U. Furbach, M. Heisel, W. Reif, and W. Stephan, editors, Proceedings Workshop ``Verification, Konstruktion und Synthese von Programmen'', Bericht Nr. 10/89, pages 4648, Karlsruhe, apr 1989. Fakultät für Informatik, Universität Karlsruhe.
- Rudolf Berghammer. Transformational Programming with Non-deterministic and Higher-order Constructs. Habilitationsschrift, Fakultät für Informatik, Universität der Bundeswehr München, 1990. auch als Bericht Nr. null 9012.
- Rudolf Berghammer. Zur Beschreibung der ganzen Zahlen als termerzeugtes Modell einer Theorie der Prädikatenlogik erster Stufe. Technical Report 9012, Fakultät für Informatik, Universität der Bundeswehr München, 1990.
- Rudolf Berghammer. Codifying the differencing technique into formal transformation rules over CIP-L. In Broy [Broy, 1991], pages 406418.
- Rudolf Berghammer. Relational specification of data types and programs. Tech. null Report 9109, Fakultät für Informatik, Universität der Bundeswehr München, September 1991.
Abstract relation algebra is proposed as a practical means for specification of data types and programs. We define the concept of a relational specification by transferring some fundamental notions of the algebraic specification approach to the relational case. Furthermore, we demonstrate the usefulness of the relational approach and give an impression of relational calculations in the field of specifications by means of some examples. We treat generic constructions on direct products, the transformation of specifications, and non-determinism in more detail and show e.g., that relational specifications easily can deal with angelic and demonic non-determinism within a single context.

- Rudolf Berghammer. Computing the cut completion of a partially ordered set an example for the use of the RELVIEW-system. Technical Report 9205, Fakultät für Informatik, Universität der Bundeswehr München, July 1992.
The RELVIEW-system is a totally interactive and completely video-oriented computer system for the manipulation of concrete relations. This paper contains a brief description of the present version 2.0 of the RELVIEW-system and describes an application, viz. the computation of the cut completion of a partially ordered set. Also the main topics of future work on RELVIEW are sketched.

- Rudolf Berghammer. On the characterization of the integers: The hidden function problem revisited. In Wolfram-M. Lippe and Gudrun Stroot, editors, Proc. null Workshop ``Programmiersprachen Methoden, Semantik, Implementierungen'', Bericht 7/92-I, pages 8292, Landhaus Rothenberge, Germany, January 1992. Institut für Angewandte Mathematik und Informatik, Universität Münster.
- Rudolf Berghammer. Combining relational calculus and the dijkstra-gries method for deriving > relational programs. Information Sciences, 119(34):155171, December 1999.
We show how to derive imperative programs for relation-based discrete structures by combining relational calculus and the Dijkstra-Gries method. Three examples are given, viz. Warshall's algorithm for transitive closures, a breadth-first-search reachability algorithm, and an algorithm for spanning trees.

- Clifford H. Bergmann, Roger D. Maddux, and Don L.Pigozzi, editors. Algebraic Logic and Universal Algebra in Computer Science, Conference, Ames. Iowa, USA, June 1988, Proceedings, volume 425 of LNCS. Springer, 1990.
- J.A. Bergstra and Jan Willem Klop. Conditional rewrite rules: Confluence and termination. Journal of Computer and System Science, 32(3):323363, 1986.
Algebraic specifications of abstract data types can often be viewed as systems of rewrite rules. Here we consider rewrite rules with conditions, such as they arise, e.g., from algebraic specifications with positive conditional equations. The conditional term rewriting systems thus obtained which we will study, are based upon the well-known class of left-linear nonambiguous TRSS. A large part of the theory for such TRSS can be generalized to the conditional case. Our approach is nonhierarchical the conditions are to be evaluated in the same rewriting system. We prove confluence results and termination results for some well-known reduction strategies

- J. A. Bergstra and Gh. Stefanescu. Network algebra with demonic relation operators, 1995.
- C. Berline. Rétractions et interprétation interne du polymorphisme: Le problème de la rétraction universelle. RAIRO Inform. null Théor. null Appl., 26(1):5991, 1992.
Le but de cet exposé est de synthétiser en un seul article tous les résultats connus d'existence de ``rétractions universelles'' (= r.u.) dans des sous-classes intérressantes de l'ensemble R des rétractions d;un domaine de Scott. Ce problème, purement algébraique, est lié à la modélisation du polymorphisme. La solution donnée par Berardi fournit comme sous-produit des modéles non triviaux d'un &lgr;-calcul étendu qui n'est pas Church-Rosser. L'inter^et sést porté sur des sous-classes de R après qu'Ershov e^ut montré, en 1975, que R elle-m^eme n'avait pas d'objet universel dans D = P_&ohgr;. We give a condensed and uniform presentation of all known results (some of them unpublished) concerning the existence of ``universal'' retractions in interesting subclasses of the set of retractions of a Scott domain. This purely algebraic problem is linked to the modelisation of polymorphism. Berardi's solution provides, as a by-product, non trivial models to a non Church-Rosser extension of &lgr;-calculus.

- Paul Bernays. Über eine natürliche Erweiterung des Relationenkalküls. In A. Heyting, editor, Constructivity in Mathematics, Proc. null of the Colloq., 1957, pages 114, Amsterdam, 1959. North-Holland.
- G. Bernkopf. A history of infinite matrices. Arch. null Hist. null Exact Sci., 4:308358, 1968.
- Eike Best. Relational semantics of concurrent programs (with some applications). In D. Bjørner, editor, Formal Description of Programming Concept II, pages 431452. North-Holland, 1983.
- H. Bestougeff and G. Ligozat. Parameterized abstract objects for linguistic information processing. In Proc. null of the European Chapter of the Association for Computational Linguistics, pages 107115, Geneva, 1985.
- H. Bestougeff and G. Ligozat. Outils Logiques pour le Traitement du Temps: de la Linguistique à l'Intelligence Artificielle. Masson, Paris, 1989.
- Richard S. Bird and Oege de Moor. From dynamic programming to greedy algorithms. In B. Möller, H. Partsch, and S. Schuman, editors, Formal Program Development: Proc. of an IFIP TG2/WG 2.1 State of the Art Seminar, Rio de Janeiro, Jan. 1992, volume 755 of LNCS, pages 4361. Springer, 1992.
- Richard S. Bird and Oege de Moor. Solving optimisation problems with catamorphisms. In Bird et al. [Bird et al., 1992], pages 4566.
Efficient algorithms for solving optimization problems can often be expressed as homomorphisms on initial data types. Such homomorphisms, which correspond to the familiar sl fold operators in functional programming, are called

*catamorphisms*. In this paper, we report on an attempt to characterize those optimization problems whose efficient solution can be expressed as a catamorphism. Our results are a natural generalization of earlier work by Jeuring [6], [Jeuring-1990a], who considered the same problem in a slightly less abstract setting. The main result of this paper is to show how seemingly disparate results about subsequences, permutations, sequence partitions and subtrees can be stated as a single theorem.- Richard Bird and Oege de Moor. Relational program derivation and context-free language recognition. In Roscoe [Roscoe, 1994], chapter 2, pages 1735.
- Richard S. Bird and Oege De Moor. Algebra of Programming, volume 100 of International Series in Computer Science. Prentice Hall, 1997.
- Richard S. Bird, J. Gibbons, and G. Jones. Formal derivation of a pattern matching algorithm. Science of Computer Programming, 12:93104, 1989.
- Richard S. Bird, C. C. Morgan, and J. C. P. Woodcock, editors. Mathematics of Program Construction, Second International Conference Oxford, U.K., June/July 1992, volume 669 of LNCS. Springer, 1992.
- Richard Bird, Oege de Moor, and Paul Hoogendijk. Generic programming with relations and functors. unpublished draft, November 1993.
This paper explores the idea of generic programming in which programs are parameterised by data types. Part of the constructive theory of lists, specically the part dealing with properties of segments, is generalised in two ways: from lists to arbitrary inductive data types, and from functions to relations. The new theory is used to solve a generic problem about segments.

- Richard S. Bird. Using circular programs to eliminate multiple traversals of data. Acta Inform., 21, 1984.
- Richard S. Bird. Transformational programming and the paragraph problem. Science of Computer Programming, 6:159189, 1986.
- Richard S. Bird. A formal development of an efficient supercombinator compiler. Science of Computer Programming, 8:113137, 1987.
- Richard S. Bird. An introduction to the theory of lists. In M. Broy, editor, Logics of Programming and Calculi, volume F36 of NATO ASI Series, pages 342. Springer-Verlag, 1987.
- Richard S. Bird. Algebraic identities for program calculation. Comput. null J., 32(2):122126, April 1989.
- Richard S. Bird. A calculus of functions for program derivation. In David A. Turner, editor, Research Topics in Functional Programming, The UT Year of Programming Ser., chapter 11, pages 287308. Addison-Wesley, 1990.
- Richard S. Bird. The smallest upravel. Science of Computer Programming, 18:281292, 1992.
An

*unravel*of a sequence x is a bag of nonempty subsequences of x that when shuffled together can give back x. For example, the sequence ``accompany'' can be unravelled into three lists ``acm'', ``an'', and ``copy''. The order of these lists is not important but duplications do matter; for example, ``peptet'' can be unravelled into two copies of ``pet''. Thus, an unravel is essentially a*bag*of sequences and not a list or set. An unravel is called an*upravel*if all its component sequences are ascending. Since each of ``acm'', ``an'', and ``copy'' are ascending, they give an upravel of ``accompany''. Each nonempty sequence has at least one upravel, namely the upravel consisting of just singleton sequences. However, of all possible upravels we want to determine one with the least number of elements. The problem of the smallest upravel is one of the most instructive and challenging problems I have ever come across. ...- Garrett Birkhoff. Lattice-ordered groups. Ann. null of Math. null (2), 43:298331, 1942.
- Garrett Birkhoff. Lattice Theory, volume XXV of Amer. null Math. null Soc. null Colloq. null Publ. Amer. null Math. null Soc., Providence, R. null I., 1948.
- Garrett Birkhoff. Lattice Theory, volume XXV of Amer. null Math. null Soc. null Colloq. null Publ. Amer. null Math. null Soc., Providence, R. null I., 3rd edition, 1967.
- Balázs Biró and G. Serény. An explicit characterization of some non-representable cylindric algebras. Preprint no. 9, 1985, pp. null 9.
- Balázs Biró and Saharon Shelah. Isomorphic but not lower base-isomorphic cylindric set algebras. Journal of Symbolic Logic, 53:846853, 1988. Preprint no. 36, 1985, pp. null 20.
- Balázs Biró. Isomorphic but not base-isomorphic base-minimal cylindric set algebras. Algebra Universalis, 24:292300, 1987.
- Balázs Biró. Non-finite-axiomatizability results in algebraic logic. Journal of Symbolic Logic, 1987. Preprint, 1987, pp. null 19.
- J. Biskup. On the complementation rule for multivalued dependencies in database relations. Acta Inform., 10(3):297305, 1978.
- J. Biskup. Inference of multivalued dependencies in fixed and undetermined universes. Theoretical Computer Science, 10:93106, 1980.
- Dines Bjørner, Manfred Broy, and Igor Pottosin, editors. volume 735 of LNCS. Springer, 1994.
- Patrick Blackburn and Yde Venema. Dynamic squares. Logic Group Preprint Series 92, Dept. null of Philosophy, Utrecht Univ., 1993. To appear in J. null Philos. null Logic.
- Patrick Blackburn, Maarten de Rijke, and Yde Venema. The algebra of modal logic. CWI Report CS-R9463, CWI Amsterdam, 1994.
- Patrick Blackburn, Maarten de Rijke, and Yde Venema. Logic, language, and information. In Brink et al. [Brink et al., 1997], chapter 14, pages 211225.
The calculus of relations turned into an important conceptual and methodological tool in computer science. The methods presented in this book include questions of relational databases, applications to program specification, resource-conscious linear logic, semantic and refinement consideration, nonclassical logics for reasoning about programs, tabular methods in software construction, algorithm development, linguistic problems, followed by a comprehensive bibliography. The reader gets an overview of the wide-ranging applicability of relational methods in computer science.

*``... While this is a multi-authored volume, the authors have done an excellent job of making it read like a single-authored work ... The book can be viewed as a set of snapshots of a family of research and researchers at one point in time. If you are interested in relational problems, I can think of no better introduction ...''* Computing Reviews- S. L. Bloom, Z. Ésik, and Gheorghe c Stef u anescu. Notes on equational theories of relations. Technical report, Academiei Romane, Inst. null de Matematica, 1992.
- R.F. Blute, J.R.B. Cockett, R.A.G. Seely, and T.H. Trimble. Natural deduction and coherence for weakly distributive categories. To appear in Journal of Pure and Applied Algebra???
We define a two sided notion of proof nets, suitable for categories, like weakly distributove categories, which have the two-tensor structure (TIMES/PAR) of linear logic, but lack a NEGATION operator. These proof nets have a structure more closely parallel to that of traditional natural deduction than Girard's one-sided nets do. In particular, there is no cut, and cut elimination is replaced by normalization. We prove a sequentialization theorem for these nets and the corresponding sequent calculus, and deduce the coherence theorem for weakly distributive categories. We also extend these techniques to cover the case of non-symmetric (``planar'') tensors. We further extend the treatment of coherence to include the units for the tensors, giving a characterization of the Lambek equivalence relation on deductions (

*i.e. equality of morphisms*) in terms of the notion of empire. Finally, we derive a conservative extension result for the passage from weakly distributive categories to *-autonomous categories.- T. S. Blyth. Matrices over ordered algebraic structures. J. null London Math. null Soc., 39:427432, 1964.
- I. M Bochénski. History of Formal Logic. Chelsea, New York, 1970.
- Alexander Bockmayr. Algebraic and logical aspects of unification. In Schulz [Schulz, 1990], pages 171180.
During the last years unification theory has become an important subfield of automated reasoning and logic programming. The aim of the present paper is to relate unification theory to classical work on equation solving in algebra and mathematical logic. We show that many problems in unification theory have their counterpart in classical mathematics and illustrate by various examples how classical results can be used to answer unification-theoretic questions.

- Alexander Bockmayr. Model-theoretic aspects of unification. In Schulz [Schulz, 1990], pages 181196.
Unification is a fundamental operation in various areas of computer science, in particular in automated theorem proving and logic programming. In this paper we establih a relation between unification theory and classical model theory. We show how model-theoretic methods can be used to investigate a generalized form of unification, namely the problem whether, given an equational theory E and a system of equations S, there is an extension of the free algebra in E in which S is solvable.

- J. Bojanowski, M. Iglewski, Jan Madey, and A. Obaid. Functional approach to protocols specification. In Proc. null of the 14
^{th}Internat. null IFIP Sympos. null on Protocol Specification, Testing and Verification, PSTV'94, Vancouver, B.C., 7-10 June 1994, pages 371378, 1994.- Harold Boley. FIT-PROLOG: A functional/relational language comparison. Technical report, Univ. null Kaiserslautern, December 1983.
- Harold Boley. RELFUN: A functional/relational integration with valued clauses. Technical report, Universität Kaiserslautern, January 1986.
- George Boole. The Mathematical Analysis of Logic, Being an Essay Toward a Calculus of Deductive Reasoning. Macmillan, Cambridge, 1847.
- R. C. Bose and D. M. Mesner. On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. null Math. null Statist., 36:2138, 1959.
- Claudia Böttinger. On Scott's thesis for domains of information and well-quasi-orderings. Theoretical Computer Science, 70:151158, 1990.
- Michael Böttner and Wolf Thümmel, editors. Variable-free Semantics. Secolo, Osnabrück, 2000.
- Michael Böttner. State transition semantics. Theoret. null Linguist., 18:239286, 1992.
- Michael Böttner. Variable-free semantics for anaphora. J. null Philos. null Logic, 21:375390, 1992.
- Michael Böttner. Open problems in relational grammar. In P. Humphreys, editor, Patrick Suppes: Scientific Philosopher, volume 3, pages 1939. Kluwer, Dordrecht, 1994.
- Michael Böttner. A collective extension of relational grammar. J. null of the Interest Group in Pure and Applied Logics, 1996. to appear.
- Michael Böttner. Natural language. In Brink et al. [Brink et al., 1997], chapter 15, pages 226246.
- M. Böttner. A collective extension of relational grammar. Logic Journal of the IGPL, 6(2):175193, 1998.
Relational grammar was proposed in Suppes (1976) as a semantical grammar for natural language. Fragments considered so far are restricted to distributive notions. In this article, relational grammar is extended to collective notions.

- Michael Böttner. Visiting some relatives of peirce's. In Ali Jaoua, Peter Kempf, and Gunther Schmidt, editors, Using Relational Methods in Computer Science, Technical Report Nr. null 1998-03, pages 7183. Fakultät für Informatik, Universität der Bundeswehr München, July 1998.
The notion of a relational grammar is extended to ternary relations and illustrated by a fragment of English. Some of Peirce's terms for ternary relations are shown to be incorrect and corrected.

- Michael Böttner. Relationale Grammatik. Niemeyer, Tübingen, 1999.
- Michael Böttner. Meanings as state transitions. In Böttner and Thümmel [B{ö}ttner and Th{ü}mmel, 2000], pages 182198.
- N. Boudriga, F. Elloumi, and A. Mili. On the lattice of specifications: Applications to a specification methodology. Formal Aspects of Computing, 4:544571, 1992.
- Jonathan P. Bowen and Kevin C. Lano Peter T. Breuer. The REDO project: Final report. Technical Report PRG-TR-23-91, Programming Research Group, Oxford University Computing Laboratory, 1991.
This report gives an overview of the work performed by the Programming Research Group as part of the European collaborative ESPRIT II ``REDO'' project (no. 2487). This work covered the areas of reverse engineering: redocumentation and re-engineering; validation: post-hoc verification and generation of correct code from specifications; maintenance: new languages and methods to support maintenance. Research in areas of concurrent programming and decompilation were also performed.

- F. J. Brandenburg, G. Vidal-Naquet, and M. Wirsing, editors. volume 247 of LNCS. Springer, February 1987.
- S. Braun. Relationale Datenbanken mit multiplen Werten. In Manfred Broy, editor, Informatik und Mathematik. Festschrift zum 65. Geburtstag von F.L. Bauer, pages 115124. Springer-Verlag, 1991.
- D. A. Bredihin. Abstract characterization of some classes of algebras of binary relations. see Zbl 394.04001.
- D. A. Bredihin and Boris M. Schein. Representations of ordered semigroups and lattices by binary relations. Colloq. null Math., 39:112, 1978.
- Roland Brethauer. Ein Formelmanipulationssystem zur computergestützten Beweisführung in der Relationenalgebra. Master's thesis, Universität der Bundeswehr München, Fakultät für Informatik, December 1991. ID 43/91.
- Peter T. Breuer. An analysis/synthesis language with learning strategies. Technical Report PRG-TR-13-91, Programming Research Group, Oxford University Computing Laboratory, July 1991.
PARLEY is a declarative programming language based on the precept that `solution synthesis' from the solutions to subproblems and `problem analysis' into the set of subproblems ought to be the only components of a program description to concern the programmer. The programming style is introduced in conjunction with an operational semantics which lends itself to shared data and parallel processing models.

- Chris Brink and Ingrid Rewitzky. Modelling the algebra of weakest preconditions. South African Computer J., 6:1120, 1992.
- Chris Brink and Ingrid Rewitzky. Predicate transformers as power operations. Technical Report RR 137, Dept. null of Mathematics, Univ. null of Cape Town, 1992.
- Chris Brink and Ingrid Rewitzky. Predicate transformers as power operations. Formal Aspects of Computing, 7:169182, 1995.
- Chris Brink and Renate Schmidt. Subsumption computed algebraically. Comput. null Math. null Appl., 23:329342, 1992. Special Issue on semantic networks in Artificial Intelligence.
- Chris Brink and Gunther Schmidt, editors. Relational Methods in Computer Science, volume 80 of Dagstuhl-Seminar-Reports, Schloß Dagstuhl, 1994. Internat. null Begegnungs- und Forschungszentrum für Informatik, Dagstuhl. Dagstuhl-Seminar 9403, Jan 1721, 1994, 28 p.
- Chris Brink, J. J. C. Vermeulen, and J. P. G. Pretorius. Verisimilitude via vietoris. Technical Report RR 117, Dept. null of Mathematics, Univ. null of Cape Town, May 1991.
- Chris Brink, J. J. C. Vermeulen, and J. P. G. Pretorius. Verisimilitude via vietoris. J. null Logic Comput., 2:709718, 1992.
- Chris Brink, Katarina Britz, and Austin Melton. A note on fuzzy power relations. Fuzzy Sets and Systems, 54:115117, 1993.
- Chris Brink, Katarina Britz, and Renate Schmidt. Peirce algebras. Formal Aspects of Computing, 6:339358, 1994.
- Chris Brink, Dov Gabbay, and Hans Jürgen Ohlbach. Towards automating duality. Comput. null Math. null Appl., 29:7390, 1995.
- Chris Brink, Wolfram Kahl, and Gunther Schmidt, editors. Relational Methods in Computer Science. Advances in Computing. Springer, Wien, New York, 1997.
- Chris Brink. On Birkhoff's postulates for a relation algebra. J. null London Math. null Soc., 15:391394, 1977.
- Chris Brink. On Peirce's notation for the logic of relatives. Trans. null of the Charles S. null Peirce Society, 14:285304, 1978.
- Chris Brink. The algebra of relatives. Notre Dame J. null Formal Logic, 20:900908, 1979.
- Chris Brink. Two axiom systems for relation algebras. Notre Dame J. null Formal Logic, 20:909914, 1979.
- Chris Brink. Boolean modules. J. null Algebra, 71:291313, 1981.
- Chris Brink. On the application of relations. South African J. null of Philosophy, 7(2):105112, 1988.
- Chris Brink. Power structures. Algebra Universalis, 30:177216, 1993.
- S.D. Brookes and A.W. Roscoe. An improved failure model for communicating sequential processes. In Proc. null of the NFS-SERC Seminar on Concurrency, volume 197 of LNCS, pages 281305. Springer, 1984.
- Carolyn Brown and Doug Gurr. A representation theorem for quantales. Journal of Pure and Applied Algebra, 85:2742, 1993.
- Carolyn Brown and Graham Hutton. Categories, allegories and circuit design. In Proceedings, Ninth Annual IEEE Symposium on Logic in Computer Science [IEE, 1994], pages 372381.
Relational languages such as sc Ruby are used to derive hardware circuits from abstract specifications of their behaviour. Much reasoning is done informally in sc Ruby using pictorial representations of relational terms. We formalise this use of pictures in circuit design. We show that pictures naturally form a

*unitary pretabular allegory*. Homomorphisms of pictures correspond to adding new wires or circuit components. Two pictures are mutually homomorphic if and only if they represent equal allegorical terms. We prove soundness and completeness results which guarantee that deriving circuits using pictures does not lead to errors. We illustrate the use of pictures by deriving the ripple adder implementation from a high level, behavioural specification.- Carolyn Brown and Alan Jeffrey. Allegories of circuits. In Proc. Logic For Computer Science. Springer, 1994.
- Manfred Broy and C. B. Jones, editors. Programming Concepts and Methods, Proc. null of the IFIP WG 2.2/2.3, Working Conf. null on Programming Concepts and Methods. North-Holland, 1990.
- M. Broy, editor. Logic of Programming and Calculi of Discrete Design, volume F36 of NATO ASI Ser. null F. Springer, 1986.
- Manfred Broy, editor. Constructive Methods in Computing Science, volume 55 of NATO ASI Ser. null F. Springer, 1989.
- Manfred Broy, editor. Informatik und Mathematik. Proc. null Colloq. null ``Informatik im Kreuzungspunkt von Numerischer Mathematik, Rechnerentwurf, Programmierung, Algebra und Logik'' 12.-14.6.1989, for Prof. null Dr. null Dr. null h.c. null mult. null F. null L. null Bauer for his 65
^{th}birthday. Springer, 1991.- Kim B. Bruce, Albert R. Meyer, and John C. Mitchell. The semantics of second-order lambda calculus. Inform. null and Comput., 85(1):76134, 1990.
In the second-order (polymorphic) typed lambda calculus, lambda abstraction over type variables leads to terms denoting polymorphic functions. Straightforward cardinality considerations show that a naive set-theoretic interpretation of the calculus is impossible. We give two definitions of semantic models for this language and prove them equivalent. Our syntactical ``environment model'' definition and a more algebraic ``combinatory model'' definition for the polymorphic calculus correspond to analogous model definitions for untyped lambda calculus. Soundness and completeness theorems are proved using the environment model definition. We verify that some specific interpretations of the calculus proposed in the literature indeed yield models in our sense.

- R. H. Bruck and H. J. Ryser. The nonexistence of certain finite projective planes. Canad. null J. null Math., 1:8893, 1949.
- Roberto Bruni, Fabio Gadducci, and Ugo Montanari. Normal forms for partitions and relations. In Fiadeiro [Fiadeiro, 1999], pages 3147.
In recent times there has been a growing interest towards algebraic structures which are able to express formalisms different from the well-known, tree-like presentation of terms. Many of the approaches adopted for such descriptions reveal a common, specific interest towards their application in the ``distributed and concurrent systems'' field, but an exhaustive comparison between them is very difficult because their presentations can be quite different. This work is a first step towards a unified view, which is able to recast all those formalisms into a more general one, where they can be easily compared. We introduce a general schema for describing a characteristic normal form for many interesting algebraic formalisms, and show that those normal forms can be thought of as arrows of suitable concrete monoidal categories, whose operations preserve the normal form itself.

- Roberto Bruni, Fabio Gadducci, and Ugo Montanari. Normal forms for partitions and relations. Theoretical Computer Science, 2000. to appear.
Recent years have seen a growing interest towards algebraic structures that are able to express formalisms different from the standard, tree-like presentation of terms. Many of these approaches reveal a specific interest towards the application to the ``distributed and concurrent systems'' field, but an exhaustive comparison between them is difficult because their presentations can be quite dissimilar. This work is a first step towards a unified view, which is able to recast all those formalisms into a more general one, where they can be easily compared. We introduce a general schema for describing a characteristic normal form for many algebraic formalisms, and show that those normal forms can be thought of as arrows of suitable concrete monoidal categories.

- Jacqueline Brunning. Peirce's Development of the Algebra of Relations. PhD thesis, Univ. null of Toronto, Toronto, 1980.
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Terminological Knowledge Representation Systems (TKRS) are tools for designing and using knowledge bases that make use of terminological languages (or concept languages). We analyze from a theoretical point of view a TKRS whose capabilities go beyond the ones of presently available TKRS. The new features studied, all of practical interest, can be summarized in three main points. First, we consider a highly expressive terminological language, called ALCNR, including general complements of concepts, number restrictions and role conjunction. Second, we allow to express inclusion statements between general concepts, and terminological cycles as a particular case. Third, we prove the decidability of a number of desirable TKRS-deduction services (like satisfiability-, subsumption- and instance checking) through a sound, complete and terminating calculus for reasoning in ALCNR-knowledge bases. Our calculus extends the general technique of constraint systems and can be easily turned into a procedure using exponential space. As a byproduct of the proof, we get also the result that inclusion statements in ALCNR can be simulated by terminological cycles, if descriptive semantics is adopted.

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^{th}Hungarian Computer Science Conf., Györ, Hungary 1985, 373383.A relational logic is given for proving database dependencies represented by means of binary relations.

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A categorical calculus of relations is used to derive a unified setting for higher order logic and polymorphic lambda calculus.

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We combine the principles of the Floyd-Warshall-Kleene algorithm, enriched categories, and Birkhoff arithmetic, to yield a useful class of algebras of transitive vertex-labeled spaces. The motivating application is a uniform theory of abstract or parametrized time in which to any given notion of time there corresponds an algebra of concurrent behaviors and their operations, always the same operations but interpreted automatically and appropriately for that notion of time. An interesting side application is a language for succinctly naming a wide range of datatypes.

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Relational programming was originally proposed by MacLennan [4,5,6]. He advocated a language based on binary relations and operators for combining and manipulating relations. Such operators form a

*relational algebra* a set of combining forms for relations which generalise a language like FP from functions to relations. MacLennan designed a relational language and built an interpreter for it [1] but his implementation was achieved at the expense of an explicit compromise to relational abstraction. He split relational programming into two worlds an intensional world (relations represented by many-to-one functions) and an extensional world (relations represented by association lists). The relational operators were similarly segregated into two groups one applicable to intensional, the other to extensional relations. This representation divide considerably inhibited freedom of programmer expression.- E. Cerny, F. Corella, M. Langevin, X. Song, S. Tahar, and Z. Zhou. Automated verification with abstract state machines using multiway decision diagrams. In Kropf [Kropf, 1997], pages 79113.
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^{th}Internat. null Conf., 1982, volume 1004 of Lect. null Notes in Math., pages 91103, Puebla, Mexico, 1983.- Stephen D. Comer. A new foundation for the theory of relations. Notre Dame J. null Formal Logic, 24:181187, 1983.
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Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. Starting from a functorial presentation of multi-algebras based on

*gs-monoidal theories*, we argue that specifications for multi-algebras should be based on the notion of*term graphs*instead of on standard terms. We consider the simplest case of (term graph) equational specification, showing that it enjoys an unrestricted form of substitutivity. We discuss the expressive power of equational specification for multi-algebras, and we sketch possible extensions of the calculus.- Marc-Michel Corsini, Alain Griffault, and Antoine Rauzy. Yet another application for toupie: Verification of mutual exculion algorithms. In Voronkov [Voronkov, 1993], pages 8697.
Toupie is a finite domain u -calculus model checker that uses extended decision diagrams to represent relations and formulae. In recent papers, we have demonstracted that such a language can model and solve difficult problems, such as AI Puzzles, Abstract Interpretation of Logic Programs with very good running times. Hereafter we show how, in Toupie, one can handle transition systems and check properties of Mutual Exclusion Algorithms.

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^{th}Annual IEEE Symposium on Foundations of Computer Science, pages 137146, Houston, Texas, Oct. 2527 1976.- B. Courcelle and G. Rozenberg, editors. Selected Papers of the International Workshop on Computing by Graph Transformation, Bordeaux, France, March 2123, 1991. Elsevier, 1993. Theoretical Computer Science
**109**(12).- Bruno Courcelle. On using context-free graph grammars for analyzing recursive definitions. In Fuchi and Kott [Fuchi and Kott, 1987b], pages 83122.
Certain recursive definitions can be represented by context-free graph grammars. The grammar associated with a recursive definition generates the set of its

*computation graphs*. Properties of recursive definitions expressible as monadic second-order logical properties of their computation graphs, are decidable. Applications to recursive applicative program schemes and to recursive queries in relational data bases are given.- Bruno Courcelle. Graph rewriting: An algebraic and logic approach. In van Leeuwen [van Leeuwen, 1990], chapter 5, pages 193242.
- Bruno Courcelle. The monadic second-order logic of graphs vii: Graphs as relational structures. Theoretical Computer Science, 101(1):333, 1992.
Relational structures form a unique framework in which various types of graphs and hypergraphs can be fromalized and studied. We define operations on structures that are compatible with monadic second-order logic, and that are powerful enough to represent context- free graph and hypergraph grammars of various types, namely, the so-called hyperedge replacement, C-edNCE, and separated handle rewriting grammars. Several results concerning monadic second-order properties of the generated sets are obtained in a uniform way. We also give a logical characterization of the equational sets of structures that generalies the ones obtained by Engelfriet and Courcelle for hyperedge replacement and C- edNCE sets of graphs.

- Patrick Cousot and Radhia Cousot. Comparing the galois connection and widening/narrowing approaches to abstract interpretation. In Bruynooghe and Wirsing [Bruynooghe and Wirsing, 1992], pages 269295. invited lecture.
The use of infinite abstract domains with widening and narrowing for accelerating the convergence of abstract interpretations is shown to be more powerful than the Galois connection approach restricted to finite lattices (or lattices satisfying the chain condition).

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- Aubert Daigneault, editor. Studies in Algebraic Logic, volume 9 of MAA Studies in Mathematics. The Mathematical Assoc. null of America, Washington, D.C., 1974.
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We present a methodology that allows temporal constraints to be imposed on the behaviour of term-rewriting systems and in particular allows the evaluation order of pure functional programs to be constrained. This permits the use of these languages in applications such as operating systems and real-time control, where control of evaluation order is necessary to achieve a correct implementation. This control is achieved, in the declarative spirit, by utilising a temporal logic that allows the user to specify, at a high level, the temporal conditions his program must satisfy. The temporal logic is a meta-language which talks about events occurring in the execution of the associated program. The program together with the temporal constraints is then automatically transformed to produce a single program that is guaranteed to behave correctly on any implementation. These techniques are of particular interest when developing programs for parallel asynchronous machines such as ALICE [Cripps et al, 1987] that can exhibit genuinely non-deterministic evaluation (even of deterministic programs). We detail the temporal specification language, the transformations used and their implementation and give an example showing the use of the methodology with an illustration of its execution on the parallel graph reduction machine ALICE.

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Efficient algorithms for solving optimization problems can often be expressed as homomorphisms on initial data types. Such homomorphisms, which correspond to the familiar sl fold operators in functional programming, are called

*catamorphisms*. In this paper, we report on an attempt to characterize those optimization problems whose efficient solution can be expressed as a catamorphism. Our results are a natural generalization of earlier work by Jeuring [6], [Jeuring-1990a], who considered the same problem in a slightly less abstract setting. The main result of this paper is to show how seemingly disparate results about subsequences, permutations, sequence partitions and subtrees can be stated as a single theorem.- Oege de Moor and D.S. Swierstra. Virtual data structures. Presented at IFIP WG 2.1 state of the art summer school, Itacuruçá Island, Brazil, Jan. 10-23, 1992. to appear., 1992.
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Functional programming formalisms have the advantage of a very rich type structure: the presence of higher-order functions allows the expression of algebraic identities with a minimum of bound variables. A category theorist might explain this phenomenon by saying that the category of sets and total functions has a very rich type structure, in that it allows all major categorical constructions like limits, colimits and exponentials. This property is not shared by the category of sets and relations, and as a consequence programming calculi based on relations lack the abundant type structure of their functional counterparts. Motivated by this observation, we examine a categorical construction of relations where relations are defined in terms of total functions. This construction makes it possible to extend the popular higher-order operators of functional programming to relations. It is also possible to tell which algebraic properties of these functional operators are still valid when they are lifted to relations. As an application of the calculus obtained in this manner, we consider the derivation of dynamic programming algorithms.

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We discuss a binary operator on relations, which has division like properties. We review the mathematical properties of this operator, then investigate its relevance to program construction.

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- Randall R. Dipert. Review. Hist. null Philos. null Logic, 4:19, 1984. of Peirce: Studies in Logic by Members of the Johns Hopkins University.
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In this note a concise formulation of mathematical induction is presented. This formulation is in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form lends itself very well for use in calculational proofs. As a non-trivial example a proof of a generalization of Newman's lemma is given.

- Henk Doornbos and Roland Backhouse. Induction and recursion on datatypes. In Möller [M{ö}ller, 1995], pages 242256.
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We give a criterion for the union of well-founded (i.e., noetherian) relations to be well-founded, generalizing results of Geser and of Bachmair-Dershowitz. The proof is written in a calculational style and is conducted entirely in regular algebra.

- Henk Doornbos, Netty van Gasteren, and Roland Backhouse. Programs and datatypes. In Brink et al. [Brink et al., 1997], chapter 10, pages 150165.
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We consider equational theories of binary relations, in a language expressing composition, convers, and lattice operations. We treat the equaations valid in the standard model of sets and also define a hierarchy of equational axiomatisations stratifying the standard theory. By working directly with a presentation of relation-expressions as

*graphs*we are able to define a notion of reduction which is confluent and strongly normalising, in sharp contrast to traditional treatments based on first-order terms. As consequences we obtain normal forms, decidability of the decision problem for equality for each theory. in particular we show a non-deterministic polynomial-time upper bound for the complexity of the decision problems.- D. R. Dowty. Word Meaning and Montague Grammar. Reidel, 1979.
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- Ivo Düntsch, Gunther Schmidt, and Michael Winter. A necessary relation algebra for mereotopology. Studia Logica, 1999. in print.
- Juan E. Durán. Some classes containing a fork algebra equivalent variety involving projections. Logic Journal of the IGPL, 6(2):203226, 1998.
Some varieties that are extensions of relational algebras with two constants that play the role of projections are studied. The classes have as a subvariety the abstract fork algebra (

**AFA**) equivalent variety involving projections. They are obtained by weakening some laws valid in**AFA**. Some applications of the varieties in the literature and in the specification of abstract data types are exhibited. For each of the classes obtained, an answer is given to the question: ``Is the relational reduct of the class representable?''. For the subvarieties formed with the models that have a repressentable relational reduct, a repressentation theorem is proved. For them the finitization problem is studied. Next the varieties presented are compared by means of the inclusion order. For each class the problem of characterizing finite models is considered. Simple models in the varieties are studied. Finally, the existence of equivalent classes with a binary operation like fork is studied.- Barry Dwyer. Relational programming in libra. In Ali Jaoua, Peter Kempf, and Gunther Schmidt, editors, Using Relational Methods in Computer Science, Technical Report Nr. null 1998-03, pages 3558. Fakultät für Informatik, Universität der Bundeswehr München, July 1998.
Libra is a general-purpose programming language based on the algebra of binary relations. It attempts to unify functional and logic programming, retaining the advantages of both, and avoiding some of the problems. It has all the features needed of a programming language, and a straightforward semantic interpretation. Since program specifications are easily expressed as relations, it offers a simple path from a specification to a program and from the program to its proof of correctness. The algebra of binary relations has several operators whose effects are like those of familiar procedural language constructs, for example, relational composition is analogous to sequential execution. The Libra language is illustrated by its application to a simple programming exercise. Some conclusions are drawn.

- Roy Dyckhoff, editor. Extensions of Logic Programming, 4th International Workshop, ELP '93, St. Andrews, U.K., March/April 1993, Proceedings, volume 798 of LNAI. Springer-Verlag, 1994.
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**109**(12).Parallel and distributed derivations are introduced and studied in the single-pushout approach, which models rewriting by pushout constructions in appropriate categories of partial morphisms. We present a categorical framework for this approach in an axiomatic way. Models of this categorical framework are among others: graphs, hypergraphs, relational structures, and algebraic specifications with suitable partial morphisms. Several new results concerning parallelism and distributed parallelism are presented which are even new in the example categories.

- Hartmut Ehrig and Fernando Orejas, editors. Recent Trends in Data Type Specification, 9th Workshop on Specification of Abstract Data Types, Joint with the 4th COMPASS Workshop, Caldes de Malavella, Spain, October 1992, Selected Papers, volume 785 of LNCS. Springer-Verlag, 1994.
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- Andrea Formisano, Eugenio G. Omodeo, and Marta Simeoni. A graphical approach to map reasoning. unpublished draft (as of 2000-10-17), October 2000.
Map reasoning is concerned with relations over an unspecified domain of discourse. Two limitations w.r.t. first-order reasoning are: only dyadic relations are taken into account; all map formulas are equations, having the same expressive power as first-order sentences in three variables. The map formalism inherits from the Peirce-Schröder tradition, through contributions of Tarski and many others. Algebraic manipulation of map expressions (equations in particular) is much less natural than developing inferences in first-order logic; it may in fact appear to be overly machine-oriented for direct hand-based exploitation. The situation radically changes when one resorts to a convenient representation of map expressions based on labeled graphs. The paper provides details of this representation, which abstracts w.r.t. inessential features of expressions. Formal techniques illustrating three uses of the graph representation of map expressions are discussed: one technique deals with translating first-order specifications into map algebra; another one, with inferring equalities within map calculus with the aid of convenient diagram-rewriting rules; a third one with checking, in the specialized framework of set theory, the definability of particular set operations. Examples of use of these techniques are produced; moreover, a possible approach to mechanization of graphical map-reasoning is outlined.

- Andrea Formisano, Eugenio G. Omodeo, and Marco Temperini. Goals and benchmarking for automated map reasoning. J. null Symbolic Comput., 29(2):259297, 2000.
Tarski-Givant's map calculus is briefly reviewed, and a plan of research is outlined aimed at investigating applications of this ground equational formalism in the theorem-proving field. The main goal is to create synergy between first-order predicate calculus and the map calculus. Techniques for translating isolated sentences, as well as entire theories, from first-order logic into map calculus are designed, or in some cases simply brought nearer through the exercise of specifying properties of a few familiar structures (natural numbers, nested lists, finite sets, lattices). It is also highlighted to what extent a state-of-the-art theorem-prover for first-order logic, namely Otter, can be exploited not only to emulate, but also to reason about, map calculus. Issues regarding ``safe'' forms of map reasoning are singled out, in sight of possible generalizations to the database area.

- Martin Fränzle, Bernhard von Stengel, and Arne Wittmüss. A generalized notion of semantic independence. Inform. null Process. null Lett., 53:59, 1995.
- Marc Frappier, Ali Mili, and Jules Desharnais. Program construction by parts. Theoretical Computer Science, 199?
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We propose a method which integrates program modification to the refinement calculus style of program development. Given a program developed through stepwise refinement of a specification, we propose an approach to specify modifications and to derive a new program from the existing refinement steps. This approach is based on the refinement lattice operator meet. A modification to a specification is represented by taking the meet of the old specification and the new feature to add. A solution to the new specification is constructed by

*coercing*the new feature to match the structure of the existing refinement steps. The method fosters reuse of refinement steps and their proofs. We also show that program construction is streamlined by using coercion.- Marc Frappier. A Relational Basis for Program Construction by Parts. PhD thesis, University of Ottawa, Computer Science Department, 150 Louis Pasteur, Ottawa, ON, K1N 6N5, Canada, 1995.
- Gottlob Frege. Kritische Beleuchtung einiger Punkte in E. null Schröders Vorlesungen über die Algebra der Logik. Archiv für systematische Philosophie, 1:433456, 1895. Engl. null translation in Frege [Frege1952].
- Gottlob Frege. Translations from the Philosophical Writings of Gottlob Frege. Blackwell, Oxford, 1952. Peter Geach and Max Black, editors.
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^{th}Latinamerican Sympos. null on Mathematical Logic, Colombia, 1995.- Marcelo F. Frias and S.E. Gordillo. Semantical optimization of queries in deductive objectoriented databases. In Proc. null of ADBIS'95, Moscow, pages 5572. Springer, 1995.
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The representation theorem for fork algebras was always misunderstood regarding its applications in program construction. Its application was always described as ``the portability of properties of the problem domain into the abstract calculus of fork algebras''. In this paper we show that the results provided by the representation theorem are by far more important. We show that not only the heuristic power coming from concrete binary relations is captured inside the abstract calculus, but also design strategies for program development can be successfully expressed. This result makes fork algebras a programming calculus by far more powerful than it was previously thought.

- Kazuhiro Fuchi and Laurent Kott, editors. Programming of Future Generation Computers II: Proc. null of the 2
^{nd}Franco-Japanese Sympos. null on Programming of Future Generation Computers. North-Holland, 911 November 1987.- Kazuhiro Fuchi and Laurent Kott, editors. Programming of Future Generation Computers II: Proceedings of the second Franco-Japanese Symposium on Programming of Future Generation Computers. North-Holland, 9-11 November 1987.
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This paper presents a set of rules for the transformation of GHC (Guarded Horn Clauses) programs based on unfolding. The proposed set of rules, called UR-set, is shown to preserve freedom from deadlock and to preserve the set of solutions to be derived. UR-set is expected to give a basis for various program transformations, especially partial evaluation of GHC programs.

- Hitoshi Furusawa and Wolfram Kahl. A study on symmetric quotients. Technical Report 1998-06, Fakultät für Informatik, Universität der Bundeswehr München, December 1998.
Symmetric quotients, introduced in the context of heterogeneous relation algebras, have proven useful for applications comprising for example program semantics and databases. Recently, the increased interest in fuzzy relations has fostered a lot of work concerning relation-like structures with weaker axiomatisations. In this paper, we study symmetric quotients in such settings and provide many new proofs for properties previously only shown in the strong theory of heterogeneous relation algebras. Thus we hope to make both the weaker axiomatisations and the many applications of symmetric quotients more accessible to people working on problems in some specific part of the wide spectrum of relation categories.

- Hitoshi Furusawa. Algebraic Formalisations of Fuzzy Relations and Their Representation Theorems. PhD thesis, Department of Informatics, Kyushu University, March 1998.
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In this paper we present an algebraic construction of the category of monotonic predicate transformaers from the category of relations which is similar to the standard algebraic construction of the integers from the natural numbers. The same construction yields the category of relations from the category of total functions. This provides a mechanism through which the rich type structure of the category of total functions can be promoted to successively weaker ones in the categories of relations and predicate transformers. In addition, it has exposed two complete rules for the refinement and composition of specifications in Morgan's refinement calculus.

- D. Gardy and G. Louchard. Dynamic analysis of the sizes of relations. In Mayr and Puech [Mayr and Puech, 1995], pages 433444.
We present a dynamic modelization of a database when submitted to a sequence of queries and updates, that allows us to study the evolution of the sizes of relations. While the problem of estimating the sizes of derived relations at a given time (``static'' case) has been the subject of several studies, to the best of our knowledge the evolution of the relation sizes under queries and updates (``dynamic'' cases) has not been studied so far. We consider the size of a relation as a random variable, and we study its probability distribution when the database is submitted to a sequence of insertions, deletions and queries. We show that it behaves asymptotically as a Gaussian process, whose expectation and covariance are proportional to the time. This approach also allows us to analyze the maximum of the size of the derived relation.

- Emmanuelle Garel and Jean-Pierre Olivier. The opoid generated by transitive closure and interior and symmetric closure and interior, a charaterization using generators and relations. ?, October 1994.
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This is not the first book on rough set analysis and certainly not the first book on knowledge discovery algorithms, but it is the first attempt to do this in a non-invasive way. The term "non-invasive" in connection with knowledge discovery or data analysis is new and needs some introductory remarks. We have worked from about 1993 on topics of knowledge discovery and/or data analysis (both topics are sometimes hard to distinguish), and we felt that most of the common work on this topics was based on at least discussable assumptions. We regarded the invention of Rough Set Data Analysis (RSDA) as one of the big events in those days, because, at the start, RSDA was clearly structured, simple, and straightforward from basic principles to effective data analysis. It is our conviction that a model builder who uses a structural and/or statistical system should be clear about the basic assumptions of the model. Furthermore, it seems to be a wise strategy to use models with only a few (pre-)assumptions about the data. If both characteristics are fulfilled, we call a modelling process non-invasive. This idea is not really new, because the non-parametric statistics approach based on the motto of R.A. Fisher ``Let the data speak for themselves'' can be transferred to the context of knowledge discovery. It is no wonder that e.g. the randomisation procedure (one of the flagships of non-parametric statistics) is part of the non-invasive knowledge discovery approach. In this book we present an overview of the work we have done in the past seven years on the foundations and details of data analysis. During this time, we have learned to look at data analysis from many different angles, and we have tried not to be biased for or against any particular method, although our ideas take a prominent part of this book. In addition, we have included many citations of papers on RSDA in knowledge discovery by other research groups as well to somewhat alleviate the emphasis on our own work. We hope that the presentation is neither too rough nor too fuzzy, so that the reader can discover some knowledge in this book

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The syntax of a formal language is effectively given. This is not immediately so for the semantics. This paper deals with the simple but sufficiently powerful applicative language (&lgr;-calculus) and studies effectiveness properties of its semantics. In particular it analyses the effectiveness of the interpretation of &lgr;-terms as well as different notions of computability over models.

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Recently, J.Y. Girard discoverd that usual logic connectors such as zimp (implication) could be broken up into more elementary

*linear*connectors. This provided a new*linear*logic [Girard86] where hypothesis are (in some sense) used*once*and*only once*. The most surprising is that all the power of the usual logic can be recovered by means of recursive logical operators (connector ``of course''). There are two versions of the*linear logic*: the*intuitionistic*one and the*classical*one. It seems that the second provides a appropriate formalism for*parallelism*and*communication*. This approach is entirely new and requires further development. Here we restrict out attention to the*intuitionistic*version and to the consequences os the*linear*constraint to the computation process. medskip We give two equivalent presentations of the (propositional part of)*linear*logic: a sequent calculus and a (categorical) combinator system. Then we introduce*inductive*and*projective*connectors, in particular the connector ! (read ``of course''). It plays a fundamental role in the encoding of usual intuitionistic logic into*linear*logic. There is a*cut elimination theorem*for the sequent calculus that corresponds to an*evaluation mechanism*for the combinator system. We present a very simple (abstract) machine that performs*linear*computations with the following features: begin itemize item A*very natural*lazy evaluation mechanism. item No need of*garbage collector*. end itemize Finally, we discuss the relevance of*linear*logic to implement functional languages.- Jean-Yves Girard, Paul Taylor, and Yves Lafont. Proofs and Types, volume 7 of Cambridge Tracts Theoret. null Comput. null Sci. Cambridge Univ. null Press, 1989.
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In the context of the executable specification language

*OBJ3*, an order-sorted completion procedure is implemented, providing automatically convergent specifications from user-given ones. This feature is of first importance to insure unambiguity and termination of the rewriting execution process. We describe here how we specified a modular completion design in terms of inference rules and control language, using*OBJ3*itself. On another hand, the specific problems encountered to integrate a completion process in an already reduction-oriented environment are pointed out.- R. Goldblatt. Logics of Time and Computation. CSLI Publications, Stanford, 1987.
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We represent concurrent processes as Boolean propositions or gates, cast in the role of acceptors of concurrent behavior. This properly extends other mainstream representations of concurrent behavior such as event structures, yet is defined more simply. It admits an intrinsic notion of duality that permits processes to be viewed as either schedules or automata. Its algebraic structure is essentially that of linear logic, with its morphisms being consequence-preserving renamings of propositions, and with its operations forming the core of a natural concurrent programming language.

- Vineet Gupta. Chu Spaces: A Model of Concurrency. PhD thesis, Stanford University, September 1994.
A Chu space is a binary relation between two sets. In this thesis we show that Chu spaces form a non-interleaving model of concurrency which extends event structures while endowing them with an algebraic structure whose natural logic is linear logic. We provide several equivalent definitions of Chu spaces, including two pictorial representations. Chu spaces represent processes as automata or schedules, and Chu duality gives a simple way of converting between schedules and automata. We show that Chu spaces can represent various concurrency concepts like conflict, temporal precedence and internal and external choice, and they distinguish between causing and enabling events. We present a process algebra for Chu spaces including the standard combinators like parallel composition, sequential composition, choice, interaction, restriction, and show that the various operational identities between these hold for Chu spaces. The solution of recursive domain equations is possible for most of these operations, giving us an expressive specification and programming language. We define a history preserving equivalence between Chu spaces, and show that it preserves the causal structure of a process.

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Much of the earlier development of abstract interpretation, and its application to imparative programming languages, has concerned techniques for finding fixed points in large (often infinite) lattices. The standard approach in the abstract interpretation of functional languages has been to work with small, finite lattices and this supposedly circumvents the need for such techniques. However, practical experience has shown that, in the presence of higher order functions, the lattices soon become too large (although still finite) for the fixed-point finding problem to be tractable. This paper develops some approximation techniques which were first proposed by Hunt and shows how these techniques relate to the earlier use of widening and narrowing operations by the Cousots.

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Recursivity is well known to be a crucial and important concept in programming theory. The simplest scheme of recursion in the context of logic programming is the binary Horn clause P(l

_{1},...,l_{n}) gets P(r_{1},...,r_{n}) . The decidability of the satisfiability problem of programs consisting of such a rule, a fact and a goal called*smallest binary program* has been a goal of research for some time. In this paper the undecidability of the smallest binary program is shown by a simple reduction of the Post Correspondence Problem.- G. Hansoul. A duality for boolean algebras with operators. Algebra Universalis, 17:3449, 1983.
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In computer science, one is interested mainly in finite objects. Insofar as

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*ATLAS*: A typed language for algebraic specification. In Heering et al. [Heering et al., 1993], pages 146168.We introduce an implementation of rewriting for type and combinator terms called

*ATLAS*. This system implements the algebraic and term rewriting theory for abstract types and combinators developed in [Meinke-1991,Meinke-1992b]. The system is intended to support the execution of equational specifications of abstract types and combinators. The type checking algorithms of the system also allow it to function as a framework for defining logics and proof checking. We present a short tutorial introduction to*ATLAS*by means of examples taken from first and higher order algebraic specifications and logics.- Peter Heath. On the Syllogism, and Other Logical Writings. Routledge and Kegan Paul, 1966.
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Usually, modal correspondences are investigated between monomodal and classical predicate logic formulas (van Benthem[76]). We want to propose a more algebraic view, concentrating on correspondences between general multimodal formulas (seen in the context of modal algebras), and relation algebraic formulas (which may be classified as restricted predicate logic formulas). Correspondences will be obtained quite systematically by means of relation algebraic principles of extensionality. For first order expressible correspondences, these will provide for the quantifier elimination, as well as they give the translation of all modal expressible relation algebraic primitives. This leads to a simple method of translation of certain relation algebraic formulas into their multimodal counterparts, which then have to be proved equivalent to the usual monomodal correspondences. Further, we discuss how (by the use of binary operators) the modal operator principle could be extended to capture the entire relation algebra by means of modal correspondences.

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The family of terminological representation systems has its roots in the representation system KL-ONE. Since the development of this system more than a dozen similar representation systems have been developed by various research groups. These systems vary along a number of dimensions. In this paper, we present the results of an empirical analysis of six such systems. Surprisingly, the systems turned out to be quite diverse leading to problems when transporting knowledge bases from one system to another. Additionally, the runtime performance between different systems and knowledge bases varied more than we expected. Finally, our empirical runtime performance results give an idea of what runtime performance to expect from such representation systems. These findings complement previously reported analytical results about the computational complexity of reasoning in such systems.

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Versions of constraint rewriting for completion of rewrite systems in the presence of associative commutative operators with identities have been proposed, in which constraints are used to limit the applicability of rewrite rules. We extend these approaches such that the initially given equations can contain constraints, and such that a suitable version of unification modulo associativity, commutativity and identity can be interleaved with the process of completion.

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A mathematical model for communicating sequential processes is given, and a number of its interesting and useful properties are stated. The possibilities of non-determinism are fully taken into account.

- C. A. R. Hoare, I. J. Hayes, Jifeng He, C. C. Morgan, A. W. Roscoe, J. W. Sanders, I. H. Sørensen, J. M. Spivey, and B. A. Sufrin. Laws of programming. Comm. null ACM, 30(8):672686, August 1987. Corrigenda in
**30**, 9, p. null 770.A complete set of algebraic laws is given for Dijkstra's nondeterministic sequential programming language. Iteration and recursion are explained in terms of Scott's domain theory as fixed points of continuous functionals. A calculus analogous to weakest preconditions is suggested as an aid to deriving programs from their specifications.

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This paper suggests that input and output are basic primitives of programming and that parallel composition of communicating sequential processes is a fundamental program structuring method. When combined with a development of Dijkstra's guarded command, these concepts are surprisingly versatile. Their use is illustrated by sample solutions of a variety of familiar programming exercises.

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A compiler is specified by a description of how each construct of the soure language is translated into a sequence of object code instructions. The meaning of the onject code can be defined by an interpreter written in the source language itself. A proof that the compiler is correct must show that interpretation of the object code is at least as good (for any relevant purpose) as the corresponding source program. The proof is conducted using standard techniques of data refinement. All the calculations are based on algebraic laws governing the source language. The theorems are expressed in a form close to a logic program, which may be used as a compiler prototype, or as a check on the results of a particular compilation. It is suggested that this formal framework provides appropriate interfaces for compiler implementors, and hardware designers, as well as users of the language.

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This paper shows that the nice properties of logic programs extend to definite clause specifications over arbitrary constraint languages. The notion of a constraint language sees a constraint as a piece of syntax with unknown internal structure that constrains the values variables can take in interpretations. Examples of constraint languages are Predicate Logic and its sublanguages as well as attributive concept description languages developed for knowledge representation. Our framework generalizes the constraint logic programming scheme of Jaffar and Lassez to make it applicable to knowledge representation: the constraint language is not required to be a sublanguage of predicate logic and may come with more than one interpretation, and the interpretations of the constraint language are not required to be solution compact. We present a semantic type discipline for our generalized definite clause specifications and establish a notion of well-typedness that is decidable provided the underlying constraint language is decidable. Finally, we give a type inference rule for computing most general well-typed weakenings of specifications.

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In this paper we demonstrate that the basic rules and calculational techniques used in two extensively documented program derivation methods can be expressed, and, indeed, can be generalised within a relational theory of datatypes. The two methods to which we refer are the so-called ``Bird-Meertens formalism'' (see [22]) and the ``Dijkstra-Feijen calculus'' (see [15]). The current paper forms an abridged, though representative, version of a complete account of the algebraic properties of the Boom hierarchy of types [19,18]. Missing is an account of extensionality and the so-called cross-product.

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A

*collecting interpretation of expressions*is an interpretation of a program that allows one to answer questions of the sort: ``What are the possible values to which an expression might evaluate during program execution?'' Answering such questions in a denotational framework is akin to traditional data flow analysis and, when used in the context of abstract interpretation, allows one to infer properties that approximate the run-time behaviour of expression evaluation. Exact collection interpretations of expressions are developed for three abstract functional languages: a strict first-order language, a nonstrict first-order language, and a nonstrict higher order language ( the full untyped lambda calculus with constants). It is argued that the method is simple (in particular, no powerdomains are needed), natural (it captures the intuitive operational behaviour of a cache), yet more expressive than existing methods (it is the first exact collecting interpretation for either nonstrict or higher order languages). Correctness of the interpretations with respect to the standard semantics is shown via a generalization of the notion of strictness. It is further shown how to form abstractionsof these exact interpretations, using as an example a collecting strictness analysis which yields compile-time information not previously captured by conventional strictness analyses.- P. Hudak, editor. 17th Annual ACM Symposium on Principles of Programming Languages, San Francisco, California, January 1990. acm press.
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A representation changer is a function that converts a concrete representation of an abstract value into a different concrete representation of that value. Many useful functions can be recognised as representation changers; examples include compilers, and arithmetic functions such as addition and multiplication. Functions that can be specified as the right inverse of other functions are special cases of representation changers. In recent years, a number of authors have used a relational calculus to derive representation changers from their specifications. In this paper we show that the generality of relations is not essential, and representation changers can be derived within the more basic setting of functional programming. We illustrate our point by deriving a carry-save adder and a base-converter, two functions which have previously been derived relationally.

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This paper presents the Centaur system through a tutorial describing the creation of an environment for a small language of mathematical expressions called Exp. With Centaur, the user may interactively generate programming language environments, including structured editors, debuggers, interpreters, and other tools. In this tutorial, all phases of language specification are covered: the design of the abstract and concrete syntax of Exp in Metal and Sdf, the pretty printing rules in Ppml, and the semantics of an Exp interpreter in Typol. The tools generated by Centaur based on these specifications are enhanced by a user interface built with Centaur graphic primitives.

- Bart Jacobs, Eugenio Moggi, and Thomas Streicher. Relating models of impredicative type theories. In Pitt et al. [Pitt et al., 1991], pages 197218.
The object of study of this paper is the categorical semantics of three impredicative type theories, viz. Higher Order &lgr;-calculus F&ohgr;, the Calculus of Constructions and Higher Order ML. The latter is particularly interesting because it is a two-level type theory with type dependency at both levels. Having described appropriate categorical structures for these calculi, we establish translations back and forth between all of them. Most of the research in the paper concerns the theory of fibrations and comprehension categories.

- Bart Jacobs. Comprehension categories and the semantics of type dependency. Theoretical Computer Science, 107:169207, 1993.
A

*comprehension category*is defined as a functor cal P:**E**tfun**B**^{ tfun }satisfying (a) cod ø cal P is a fibration, and (b) f is cartesian in**E**implies that cal P f is a pullback in**B**. This notion captures many structures which are used to describe type dependency (like display-map categories (Taylor (1986), Hyland and Pitts (1989) and Lamarche (1988)), categories with attributes (Cartmell (1978) and Moggi (1991)), D-categories (Ehrhard (1988)) and comprehensive fibrations (Pavlovic (1990))). It also captures comprehension as occurring in topos theory and as described by Lawvere's (1970) hyperdoctrines. This paper is meant as an introduction to these comprehension categories. A comprehension category will be called*closed*if it has appropriate dependent products and sums. A few examples of closed comprehension categorieswill be described here; more of them may be found in Jacobs (1991); applications occur in Jacobs (1991) and Jacobs et al. (1991).- Ryszard Janicki, David Lorge Parnas, and Jeffery Zucker. Tabular representations in relational documents. In Brink et al. [Brink et al., 1997], chapter 12, pages 184196.
- R. Janicki. Towards a formal semantics of Parnas tables. In Proc. null of the 17
^{th}Internat. null Conf. null on Software Engineering, Seattle, WA, pages 231240, 1995.- Ali Jaoua and M. Beaudry. Difunctional relations: A formal tool for program design. Rapport de recherche no 55, Département de Mathématique et d'Informatique, Univ. null de Sherbrooke, Québec, Canada, 1989.
- Ali Jaoua and Gunther Schmidt. Relational methods in computer science: Introduction. Information Sciences, 119(34):131133, December 1999.
- Ali Jaoua, Nadir Belkhiter, Jules Desharnais, and Ridha Khédri. Augmentation automatique de la fiabilité d'un logiciel. ICO Québec, 3(3):332337, 1991.
- Ali Jaoua, N. Boudriga, J. L. Durieux, and A. Mili. Regularity of relations: A measure of uniformity. Theoretical Computer Science, 79:323339, 1991.
- Ali Jaoua, N. Belkhiter, J. Desharnais, and T. Moukam. Propriétés des dépendances difonctionelles dans les bases de données relationnelles. INFOR. null Information Systems and Operational Research, 30(3):297316, August 1992.
- Ali Jaoua, H. Ounalli, and N. Belkhiter. Automatic entity extraction from an sl n-ary relation: Towards a general law for information decomposition. In Joint Conf. null on Information Sciences (JCIS), pages 9295, Pinehurst, Duke Univ., NC, November 1994.
- Ali Jaoua, Nadir Belkhiter, Habib Ounalli, and Théeodore Moukam. Databases. In Brink et al. [Brink et al., 1997], chapter 13, pages 197210.
- Ali Jaoua, Peter Kempf, and Gunther Schmidt, editors. Using Relational Methods in Computer Science. Fakultät für Informatik, Universität der Bundeswehr München, July 1998. Tech. null Rep. null Nr. null 1998-03, iv+83 pp.
- Ali Jaoua. Recouvrement avant de programmes sous les hypothèses de spécifications déterministes et non-déterministes. Diss. null de Doctorat d'Etat dès sciences, Univ. null de Toulouse, 1987.
- Ali Jaoua, editor. Third International Seminar on The Use of Relational Methods in Computer Science, 610 January 1997, Hammamet, Tunisia, Participant's Proceedings. University of Tunis II, Faculty of Sciences of Tunis, Department of Computer Science, 1997.
- J. Jaspars and E. Krahmer. Unified dynamics. Technical Report CS-R95, CWI, Amsterdam, 1995.
- Johan Jeuring. Algorithms from theorems. In Broy and Jones [Broy and Jones, 1990], pages 247266.
- Peter Jipsen and Erzsébet Lukács. Representability of finite simple relation algebras with many identity atoms. In Andréka et al. [Andréka et al., 1991b].
- Peter Jipsen, Chris Brink, and Gunther Schmidt. Background material. In Brink et al. [Brink et al., 1997], chapter 1, pages 121.
- Peter Jipsen. Infinite ra's that have no finite nontrivial subalgebras, 1989. Preprint, April 17, 1989.
- Peter Jipsen. Computer-aided investigations of relation algebras. PhD thesis, Vanderbilt University, May 1992.
- W. E. Johnson. The logical calculus, I, general principles. Mind, 1, New Series:330, 1892.
- W. E. Johnson. The logical calculus, II. Mind, 1, New Series:235250, 1892.
- W. E. Johnson. The logical calculus, III. Mind, 1, New Series:340357, 1892.
- J. Johnson. Nonfinitizability of classes of representable polyadic algebras. Journal of Symbolic Logic, 34:344352, 1969.
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- Geraint Jones and Mary Sheeran. Relations and refinement in circuit design. Technical Report PRG-TR-13-90, Programming Research Group, Oxford University Computing Laboratory, 1990.
A language of relations and combining forms is presented in which to describe both the behaviour of circuits and the specifications which they must meet. We illustrate a design method that starts by selecting representations for the values on which a circuit operates, and derive the circuit from these representations by a process of refinement entirely within the language.

- Geraint Jones and Mary Sheeran. The study of butterflies. Technical Report PRG-TR-14-90, Programming Research Group, Oxford University Computing Laboratory, 1990.
Butterfly networks arise in many signal processing circuits and in parallel algorithms for many sorts of message-passing computers. This paper attempts to explain why this should be, and what butterfly networks are, using a new and elegant formulation based on a language of relations. Most of the material covered by this paper has appeared in a less tractable form in earlier papers [7,8]. The novelty here is in the simplicity and elegance of presentation, which derives from an appropriate choice of high-level structures. These structures are represented by functions which are used to compose circuits from components, and are chosen to have simple mathematical properties. This presentation makes it easier to explain how the design comes about, showing that butterflies are natural implementations of divide-and-conquer algorithms.

- Mark P. Jones. Computing with lattices: an application of type classes. Technical Report PRG-TR-11-90, Programming Research Group, Oxford University Computing Laboratory, 1990.
This report presents a simple framework for performing calculations with the elements of (finite) lattices. A particular feature of this work is the use of type classes to enable the use of overloaded function sysmbols within a strongly typed language. Previous applications of type classes have been in areas that are of most interest to language implementors. This report suggests that type classes might also be useful as a general tool in the development of clear and modular programs.

- Mark P. Jones. Towards a theory of qualified types. Technical Report PRG-TR-6-91, Programming Research Group, Oxford University Computing Laboratory, April 1991. preliminary version of [Jones, 1992].
In a language with a polymorphic type system, a term of type all t.f(t) can be treated (possibly after suitable instantiation) as having any of the types in the set: $ { f(a) | a is a type }. A natural extension of such systems supports a more restricted form of polymorphism in which, rather than simply taking on all possible values, the type a is constrained to satisfy a specified predicate &pgr;(a)$.

- Mark P. Jones. A theory of qualified types. In Krieg-Brückner [Krieg-Br{ü}ckner, 1992], pages 287306. also as TechReport [Jones, 1991] (A-0088).
In a language with a polymorphic type system, a term of type all t.f(t) can be treated (possibly after suitable instantiation) as having any of the types in the set: $ { f(t) | t is a type }. It is natural to consider a more restricted form of polymorphism in which the value taken by t may be constrained to a particular subset of types. In this situation we write all t.&pgr;(t) Rightarrow f(t), where &pgr;(t) is a predicate of types, for the type of an object which can be treated (after suitable instantiation) as having any of the types in the set: { f(t) | t is a type such that &pgr;(t) holds }. A term with a restricted polymorphic type of this kind is often said to be

*overloaded*, having different interpretations for different argument types. This paper presents a general theory of overloading based on the use of*qualified types*, which are types of the form &pgr; Rightarrow &sgr; denoting those instances of type &sgr; which satisfy the predicat &pgr;$. The main benefits of using qualified types are: begin itemize item A general approach which includes a range of familiar type systems as special cases. Results and tools developed for the general system are immediately applicable to each particular application. item A precise treatment of the relationship between implicit and explicit overloading. This is particularly useful for describing the implementation of systems supporting qualified types. item The ability to include local constraints as part of the type of an object. This enables the definition and use of polymorphic overloaded values within a program.- Bjarni Jónsson. The theory of binary relations, a first draft. Preprint, 1984, pp. null 65.
- Bjarni Jónsson and Alfred Tarski. Boolean algebras with operators. Bull. null Amer. null Math. null Soc., 54:7980, 1948. Abstract 88.
- Bjarni Jónsson and Alfred Tarski. Representation problems for relation algebras. Bull. null Amer. null Math. null Soc., 54:80 and 1192, 1948. Abstract 89.
- Bjarni Jónsson and Alfred Tarski. Boolean algebras with operators, Part I. Amer. null J. null Math., 73:891939, 1951.
- Bjarni Jónsson and Alfred Tarski. Boolean algebras with operators, Part II. Amer. null J. null Math., 74:127167, 1952.
- Bjarni Jónsson. Representation of modular lattices and of relation algebras. Trans. null Amer. null Math. null Soc., 92:449464, 1959.
- Bjarni Jónsson. Defining relations for full semigroups of finite transformations. Michigan Math. null J., 9:7785, 1962.
- Bjarni Jónsson. Extensions of relational structures. In The Theory of Models, pages 146157. North-Holland, AmsterdamLondon, 1972.
- Bjarni Jónsson. Varieties of relation algebras. Algebra Universalis, 15:273298, 1982.
- Bjarni Jónsson. Maximal algebras of binary relations. In Contributions to Group Theory: Papers Published in Honor of Roger Lyndon on his 65
^{th}Birthday, pages 299307. Amer. null Math. null Soc., Providence, 1984. Contemporary Mathematics 33, edited by Kenneth I. Appel, John G. Ratcliffe, and Paul E. Schupp QA171.C683 1984.- Bjarni Jónsson. On binary relations. In G. Hutchinson, editor, Proc. null of the NIH Conf. on Universal Algebra and Lattice Theory, pages 25, Bethesda, Maryland, 1986. National Inst. null of Health, Laboratory of Computer Research and Technology.
- Bjarni Jónsson. Relation algebras and Schröder categories. Discrete Math., 70:2745, 1988.
- Bjarni Jónsson. The theory of binary relations. In Andréka et al. [Andréka et al., 1991a], pages 245292.
- Bjarni Jónsson. A survey of boolean algebras with operators. In Algebras and Orders, volume 389, pages 239286. Kluwer, Dordrecht, 1993. ed. null by Ivo G. null Rosenberg and Gert Sabidussi North American Treaty Organization, Advanced Science Institutes Series, Series C: Mathematical and Physical Sciences.
- Mark B. Josephs and David Redmond-Pyle. Entity-relationship models expressed in Z: a synthesis of structured and formal methods. Technical Report PRG-TR-20-91, Programming Research Group, Oxford University Computing Laboratory, July 1991.
Structured methods are widely used in systems analysis and design for commercial data processing applications. One of the most important features of these methods is the use of entity-relationship diagrams as a data medelling technique. This paper contributes to the understanding of such methods by taking a typical one, LBMS systems Engineering, and providing a systematic translation of its diagrams into Z. We also demonstrate how the expressiveness and precision of structured methods can be enhanced by specifying in Z further constraints on the data model and the effect of transactions on the system state. In structured methods, the former is usually done by informal comments and the latte by pseudo-code, if at all. This work also has important consequences as far as the widespread adaption of formal methods is concerned. It provides a style of writing Z specifications that could easily be adopted by someone already familiar with entity-relationship modelling, and does so in a way that standardizes the use of schemas as much as possible.

- Wolfram Kahl and Claudia Hattensperger. Second-order syntax in HOPS and in RALF. In Buth et al. [Buth et al., 1998], pages 140164. ISBN: 3-8265-3806-4.
HOPS and RALF are two interactive symbol manipulation systems one for functional programming and program transformation, the other for proving relation algebraic formulae that both implement interactive application of second-order rewriting rules, HOPS on term graphs and RALF on conventional terms and formulae. Both systems support a larger class of second-order rewriting rules than commonly found in other systems. In this paper we provide a homogeneous underpinning to second-order syntax and rewriting for easing the transition from terms to term graphs and vice versa, so that aspects that are easier to understand in one view also further understanding in the other view, altogether making a case for bringing second-order syntax more directly to the user interface than usual in most systems today.

- Wolfram Kahl and Gunther Schmidt. Exploring (finite) Relation Algebras using Tools written in Haskell. Technical Report 2000-02, Fakultät für Informatik, Universität der Bundeswehr München, October 2000.
During the last few years, relational methods have been gaining more and more acceptance and impact in computer science. Besides applications of concrete relations, also non-standard models of the relation algebraic axioms are important in fields as far apart as artificial intelligence and distributed computing. Also weaker structures have been considered, such as Dedekind categories in connection with fuzzy reasoning, and different kinds of allegories. indent In this report we present a library of Haskell modules that allows to explore relation algebras and several weaker structures by providing different means to construct and test such algebras. indent The kernel of our library is strictly conformant to the Haskell 98 standard, and can therefore be expected to be usable on future Haskell systems, too. For ease of use, we additionally provide a more elegant interface using non-standard extensions.

- Wolfram Kahl. Can functional programming be liberated from the applicative style? In Bjørn Pehrson and Imre Simon, editors, Technology and Foundations, Information Processing '94, Proceedings of the IFIP 13th World Computer Congress, Hamburg, Germany, 28 August 2 September 1994, Volume I, volume A-51 of IFIP Transactions, pages 330335. IFIP, North-Holland, 1994.
Modern functional languages all support higher-order functions. Nevertheless actual applications are mostly written in an applicative style. We show how working with a language based on slightly shifted concepts and embedded into a powerful environment can bring about change.

- Wolfram Kahl. Kategorien von Termgraphen mit gebundenen Variablen. Technischer Bericht 9503, Fakultät für Informatik, Universität der Bundeswehr München, September 1995.
With the aim of extending algebraic term graph rewriting to the expressiveness of Combinatory Reduction Systems, we first introduce a novel definition of term graphs with primitive notions of variable binding and variable identity, and with metavariables with successors. hbox After discussing identification and sharing in these graphs, we introduce intervals and segments to serve as images of metavariables, including those with successors. Building on this we are able to establish a hierarchy of structure-preserving mappings between our term graphs, including at its top a concept of homomorphism avoiding ``capture of variables'' and catering for multiple instances of metavariables. The individual categories this gives rise to have different uses, namely in term graph rewriting, and the appendix provides an overview over a novel approach to algebraic term graph rewriting (fully presented in [Kahl, 1996]) together with the two crucial proofs concerning the viability of the constructions involved.

- Wolfram Kahl. Algebraische Termgraphersetzung mit gebundenen Variablen. Reihe Informatik. Herbert Utz Verlag Wissenschaft, München, 1996. ISBN 3-931327-60-4; also Doctoral Diss. null at Univ. null der Bundeswehr München, Fakultät für Informatik.
This thesis presents a first algebraic approach to term graph rewriting encompassing the treatment of bound variables. Building on a novel definition of term graphs with primitive notions of variable binding and variable identity, we present a concept of homomorphism avoiding ``capture of variables'' and catering for multiple instances of metavariables. Rewriting of these term graphs within the algebraic approach requires a new extension that is interesting in itself, the

*fibered approach*to rewriting. As one result we obtain the first algebraic characterisation of graph reduction. Summing up, we have extended the algebraic approach to term graph rewriting, that so far only covered conventional term rewriting systems, to the expressive power of combinatory reduction systems and even slightly more general second-order rewriting systems. Thus we lay a theoretical foundation for implementations of functional programming languages, program transformation systems and other symbolic computation systems.- Wolfram Kahl. Algebraic graph derivations for graphical calculi. In d'Amore et al. [d'Amore et al., 1997], pages 224238.
Relational formalisations can be very concise and precise and can allow short, calculational proofs under certain circumstances. [...] In situations corresponding to the simultaneous use of many variables in predicate logic, however, either a style using predicate logic with point variables has to be adopted or impractical and clumsy manipulations of tuples have to be employed inside relation calculus. In the application of relational formalisation to term graphs with bound variables [...] we have been forced to employ both methods extensively, and, independently of other approaches, have been driven to develop a

*graphical calculus*for making complex relation algebraic proofs more accessible. It turns out that, although our approach shares many common points with those presented in the literature [...], it still is more general and more flexible than those approaches since we draw heavily on additional background in algebraic graph rewriting- Wolfram Kahl. A fibred approach to rewriting how the duality between adding and deleting cooperates with the difference between matching and rewriting. Technical Report 9702, Fakultät für Informatik, Universität der Bundeswehr München, May 1997.
We present a new approach to rewriting obtained by enhancing and unifying existing variants inside the algebraic (or better categorical) approach to (graph) rewriting. Our approach is motivated by second-order term graph rewriting and stresses on one hand the two-step nature of rule application consisting of deleting and adding items and on the other hand the heterogeneous nature of the rewriting setup where rule steps should be clearly distinguished from matching of rule sides into redexes. Complementing the existing opfibration approach with a dual fibration step turns out to yield a natural and flexible approach with useful new applications. The resulting fibred approach takes advantage of the heterogeneous setting and appropriately reflects the duality between deleting and adding in the course of rewriting, in contrast with the double-pushout approach which simplifies this duality into a symmetry. An important contribution is the universal characterisation of the host object, which has to be found as a pushout-complement in the double pushout approach. The fibred approach is presented in abstract and independent from any concrete application categories in the manner of High-Level Replacement Systems. Our original motivation for the development of the fibred approach comes from term graphs with bound variables where all other approaches failed; in this paper we present an unusual view on term rewriting as running example.

- Wolfram Kahl. The
**H**igher**O**bject**P**rogramming**S**ystem User Manual for HOPS . Fakultät für Informatik, Universität der Bundeswehr München, 1998. electronically available via URL: url http://ist.unibw-muenchen.de/kahl/HOPS/hopsmanual.ps.gz.- Wolfram Kahl. Internally typed second-order term graphs. In Juraj Hromkovic and Ondrej Sykora, editors, Graph Theoretic Concepts in Computer Science, 24th International Workshop, WG '98, Smolenice Castle, Slovak Republic, June 1998, Proceedings, volume 1517 of LNCS, pages 149163. Springer, 1998.
We present a typing concept for second-order term graphs that does not consider the types as an external add-on, but as an integral part of the term graph structure. This allows a homogeneous treatment of term-graph representations of many kinds of typing systems, including second-order &lgr;-calculi and systems of dependent types. Applications can be found in interactive systems and as typed intermediate representation for example in compilers.

- Wolfram Kahl. Relational treatment of term graphs with bound variables. Logic Journal of the IGPL, 6(2):259303, March 1998.
We show how and why it makes sense to use a relational formalisation instead of the usual functional one in the treatment of term graphs. Special attention is paid to term graphs with bound variables, that have, to our knowledge, never been formalised with such a generality before. Besides the novel treatment of term graphs themselves, we present an innovative relational homomorphism concept that for the first time allows to consider terms, resp. null term trees as a special case of term graphs and still have the full power of (second-order) substitution available.

- Wolfram Kahl. Beyond pretty-printing: Galley concepts in document formatting combinators. In Gopal Gupta, editor, Practical Aspects of Declarative Languages, First International Workshop, PADL '99, San Antonio, Texas, USA, January 1999, Proceedings, volume 1551 of LNCS, pages 7690. Springer, 1999.
Galleys have been introduced by Jeff Kingston as one of the key concepts underlying his advanced document formatting system Lout. Although Lout is built on a lazy functional programming language, galley concepts are implemented as part of that language and defined only informally. In this paper we present a first formalisation of document formatting combinators using galley concepts in the purely functional programming language Haskell.

- Wolfram Kahl. Explicit graphs and computer aided notation. Semiotica, 125(1/3):143154, 1999.
In many cases, linear notation systems can be seen to encode underlying,

*implicit*graphs. This paper focusses on the way that making these graphs*explicit*is useful for human understanding, and on the use of computers to make handling of notations based on explicit graphs feasible, efficient and productive.- Wolfram Kahl. Relational matching for graphical calculi of relations. Information Sciences, 119(34):253273, December 1999.
In this paper we extend an earlier approach to graphical relation calculi towards relational matching, thus allowing proofs with fewer auxiliary steps and concentrating more on the essential proof ideas. For facilitating the formal argument we introduce hierarchical relational diagrams as an intermediate structure and employ more of the algebraic graph rewriting repertoire for defining relational rewriting of these hierarchical diagrams.

- Wolfram Kahl. The term graph programming system HOPS. In Rudolf Berghammer and Yassine Lakhnech, editors, Tool Support for System Specification, Development and Verification, Advances in Computing Science, pages 136149, Wien, March 1999. Springer-Verlag. ISBN: 3-211-83282-3.
[...] The bf Higher bf Object bf Programming bf System HOPS, which has been developed by a group led by Gunther Schmidt since the mid-eighties [...] is a graphically interactive term graph programming system designed for transformational program development. In HOPS, only syntactically correct and well-typed programs can be constructed. The choice of the language is only constrained by certain restrictions of the term graph formalism and of the typing system. [...] The design of this system relies on recent advances in the theories of untyped and typed second-order term graphs.

- Wolfram Kahl. Stratified term graphs. In Ehrig and Taentzer [Ehrig and Taentzer, 2000], pages 115122. Report Nr. 2000-2.
We propose

*stratified term graphs*as enrichment of conventional term graph structures with*synchronisation borders*that intuitively capture constraints such as that values along these borders should be available simultaneously in distributed implementations. This additional structure requires a weakening of the algebraic characterisation, and we propose*coherent unsharp ps-semigroup categories*as generalisation of gs-monoidal categories. These capture exactly the essence of stratified term graphs with at least one root.- Wolfram Kahl. Unsharp demonic products and stratified term graphs. In Desharnais [Desharnais, 2000], pages 165174.
Investigating the interplay between demonic operators and direct products in abstract relation algebras against the background of gs-monoidal categories, we discover that the direct product gives rise to a gs-monoidal category with demonic composition, while a new concept of

*demonic product*gives rise to a structure that fails to be gs-monoidal mostly through the lack of functoriality of the demonic product. However, this lack is interesting on its own account, since it is an example of what has been studied as*unsharpness*in the context of direct products in relation algebras, and there it can only occur if not all products exist. We show how an intuitive understanding of our demonic products coincides with the intuition behind the unsharpness research, and, generalising the approach of using term graphs as syntax for gs-monoidal categories, discuss the generalisation of*stratified term graphs*to be used for the unsharp variant.- Gilles Kahn, editor. Functional Programming Languages and Computer Architecture, volume 274 of LNCS, Portland, Oregon, USA, September 1987. Springer.
- J. Kalicki and D. Scott. Equational completeness of abstract algebras. Indag. null Math., 17:650659, 1955.
- Fairouz Kamareddine. A system at the cross-roads of functional and logic programming. Science of Computer Programming, 19:239279, 1992.
The type-free &lgr;-calculus is powerful enough to contain all the polymorphic and higher-order nature of functional programming and furthermore types could be constructed inside it. However, mixing the type-free &lgr;-calculus with logic is not very straightforward (see Aczel [1] and Scott [15]). In this paper, a system that combines polymorphism and higher-order functions with logic is presented. The system is suitable for both the functional and logical paradigms of programming as from the functional paradigm's point of view, the system enables one to have all the polymorphism and higher order that exist in functional languages and much more. In fact even the fixed point operator Y which is defined as &lgr; f. (&lgr; x.f(x x))(&lgr; x.f(x x)) can be type checked to ((&agr; tfun &agr;) tfun &agr;) where &agr; is a variable type. (&lgr; x. x x)(&lgr; x. x x) can be type-checked too, something not allowed in functional languages. From the point of view of theorem proving, the system is expressive enough to allow self-referential sentences and those sentences that lead to Russel's and Curry's paradoxes. However, the paradoxes do not hold due to the notion of circular types which contain the type of propositions. In fact both sentences &lgr; x. lnot x x and &lgr; x.x x tfun bottom are ill-typed according to the system, because their resulting types are circular. Hence the application of either sentence to itself will not result in a proposition. The system is implementes in Milner's ML and can be seen as extending ML in two important ways. First, it extends the part related to the functional paradigm in taht it can type terms that could not be typed in ML, namely the terms that contain self-application such as the Y term above. Second, our system extends ML by adding logic to it in a consistent way.

- H. Kamel. Relational algebra. Bull. null Amer. null Math. null Soc., 58:391, 1952.
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^{th}Amsterdam Colloq., pages 377391, Amsterdam, 1994. ILLC.- Max I. Kanovich. Efficient program synthesis: Semantics, logic, complexity. In Ito and Meyer [Ito and Meyer, 1991], pages 615632.
The problem of program synthesis is considered. begin itemize item [1.] A computational semantics is introduced for relational knowledge bases. Our semantics naturally arises from practical experience of databases and knowledge bases. item [2.] It is stated that the corresponding logic coincides exactly with the intuitionistic one. item [3.] Our methods of proof of the general theorems turn out to be very useful for designing new efficient algorithms.In particular, one can construct a program synthesizer that runs in linear space.As a corollary, we can explain why there exist programs that solve PSPACE-complete problems ``in a reasonable time'' despite of their theoretical exponential uniform lower bound. end itemize

- Stéphane Kaplan and Mitsuhiro Okada, editors. Conditional and Typed Rewriting Systems, 2
^{nd}Internat. null CTRS Workshop, June 1990, Montreal, Canada, volume 516 of LNCS, Montreal, Canada, June 1990. Springer.- Yasuo Kawahara and Yoshihiro Mizoguchi. Categorical assertion semantics in topoi. Adv. null Software Sci. null Tech., 4:137150, 1992.
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In this paper we present a basic notion of relational structures which includes simple graphs, labelled graphs and hypergraphs, and introduce a notion of partial morphisms between them. An existence theorem of pushouts in the category of relational structures and their partial morphisms is proved under a certain functorial condition, and it enables us to discuss single pushout rewritings of relational structures.

- Yasuo Kawahara, Hitoshi Furusawa, and Masao Mori. Categorical representation theorems of fuzzy relations. In Proceedings of 4th International Workshop on Rough Sets, Fuzzy Sets, and Machine Discovery (RSFD '96), pages 190197, 1996.
- Yasuo Kawahara, Hitoshi Furusawa, and Masao Mori. Categorical representation theorems of fuzzy relations. Information Sciences, 119(34):235251, December 1999.
This paper provides a notion of Zadeh categories as a categorical structure formed by fuzzy relations with sup-min composition, and proves two representation theorems for Dedekind categories (relation categories) with a unit object analogous to one-point set, and for Zadeh categories without unit objects.

- Yasuo Kawahara. Matrix calculus in I-categories and an axiomatic characterization of relations in a regular category. Mem. null Fac. null Sci. null Kyushu Univ. null Ser. null A, 27(2):249273, 1973.
- Yasuo Kawahara. Notes on the universality of relational functors. Mem. null Fac. null Sci. null Kyushu Univ. null Ser. null A, 27(2):275289, 1973.
- Yasuo Kawahara. On the class of regular epimorphisms. Comm. null Algebra, 3(9):851857, 1973.
- Yasuo Kawahara. Relations in categories with pullbacks. Mem. null Fac. null Sci. null Kyushu Univ. null Ser. null A, 27(1):149173, 1973.
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- Yasuo Kawahara. Pushout-complements and basic concepts of grammars in toposes. Theoretical Computer Science, 77:267289, 1990.
An existence theorem of pushout-complements is given in an elementary topos by using category theory of binary relations, called relational calculus, and it is also shown more explicitly in the category of directed graphs, which is a typical example of toposes, as an application. Moreover an embedding theorem and Church-Rosser theorem on grammars (derivations) in a topos are proved.

- Yasuo Kawahara. Relational set theory. volume 953 of LNCS, pages 4458. Springer, 1995.
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A fixed point semantics for nondeterministic data flow is introduced which refines and extends work of Park (1983). It can be seen also as an extension to the general case of Kahn's (1974) successful fixed point semantics for deterministic data flow. An associativity result for network construction is proved which shows that anomalies such as those of Brock and Ackerman do not arise in this semantics. The semantics is shown to be extensional, in the natural sense that nondeterministic processes which induce identical output relations in all contexts are equal.

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We model processes by means of a mathematical entity that we call a relational process. This model describes a process as an open system from which the description of the process as a closed system can be easily obtained. Also, it represents not only the actions of the process but also the resources needed to accomplish its behaviour. Using this model, we first define two operators. Each of these represents an extreme perception of concurrency. One, the interleaved parallel composition operator, reduces concurrency to interleaving and the other, the maximal totally synchronous parallel composition operator, reduces concurrency to a totally synchronous behaviour. Second, by combining these operators, we define the maximal true-concurrency composition operator, which is an operator expressing true concurrency. When many processes interfere on the same resource in order to modify it, each in its way, the two maximal operators express this situation by letting the final value of the variable modelling this resource be indeterminate. So, they allow the detection of interferences between processes. We present some of the properties of these operators.

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The reader is introduced to Morgan's Refinement Calculus notation, by means of a simple case study. This example is taken all the way from an abstract specification down to a program in Dijkstra's language of guarded commands, using the laws of the refinement calculus to justify each step. This program is then transliterated into Pascal

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We present an algorithm for lambda expression reduction that avoids any copying that could later cause duplication of work. It is optimal in the sense defined by Lévy. The basis of the algorithm is a graphical representation of the kinds of commonality that can arise from substitutions; the idea can be adapted to represent other kinds of expressions besides lambda expressions. The algorithm is also well suited to parallel implementations, consisting of a fixed set of local graph rewrite rules.

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Rewriting with non-symmetric relations can be considered as a computational model of many specification languages based on non-symmetric relations. ...

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This short note is to refresh an old problem that several researchers have tried to tackle unsuccessfully. This problem originated by Wadsworth's PhD dissertation can be fortunately easy stated: ``how to evaluate efficiently lambda expressions?''. By efficient, we mean optimal which is maybe not the same. And optimal means without duplication of contraction of redexes.

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This thesis examines a tree automata approach to tree matching problems. This approach is motivated by the finite automata approach which has been very succesful in designing string matching algorithms. In particular, we show how the KMP algorithm can be generalized to give tree matching algoritms which preprocess the pattern tree. We also define structures for trees which are analogous to suffix tries and DAWGs for strings and show how they can be used for tree pattern matching. Additionally, we explore some other approaches to tree matching problems.

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`linear-request@cs.stanford.edu`.- Wolfram-M. Lippe and Gudrun Stroot, editors. Programmiersprachen Methoden, Semantik, Implementierungen, Landhaus Rothenberge, Germany, January 1992. Univ. null Münster, Inst. null für Angewandte Mathematik und Informatik.
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We study the use of relation calculi for compilation and execution of Horn Clause programs with an extended notion of input and output. We consider various other extensions to the Prolog core.

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Formal countermodels may be used to justify the unprovability of formulae in the Heyting calculus (the best accepted formal system for constructive reasoning), on the grounds that unprovable formaulae are not constructively valid. We argue that the

*intuitive*impact of such countermodels becomes more transparent and convincing as we move from Kripke/Beth models based on possible worlds, to Läuchli realizability models. We introduce a new semantics for constructive reasoning, called*relational realizability*, which strengthens further the intuitive impact of Läuchli realizability. But, none of htese model theories provides countermodels with the compelling impact of classical truth-table countermodels for classically unprovable formulae. We outline a proof that the Heyting calculus is sound for relational realizability, and conjecture that there is a constructive choice-free proof of completeness. In this respect, relational realizability improves the metamathematical constructivity of Läuchli realizability (which uses choice in two crucial ways to prove completeness) in the same sort of way Berth semantics improves Kripke semantics.- 9
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In this paper we consider ordered h-ary trees, that is, trees whose nodes have exactly h sons; and ranked trees, where the number of sons depends on the node label. We define the subtree distance between two ordered h-trees T

_{1}, T_{2}as the number of subtrees to be inserted or deleted in T_{1}to obtain T_{2}, and consider the problem of finding all the occurrences with bounded distance k, of an h-ary tree P as a subtree of another h-ary tree T. This problem is solved in time O(h size P + max(h,k) size T). We then study the classical problem of finding all the occurrences of a ranked tree P in another tree T, where the two trees are labelled, and a special label v in the leaves of P stands for any subtree in T. An extension of the previous algorithm allows to solve this problem in time O( size P + k size T), where k is the number of labels v in P. We also discuss some natural variants of the two problems.- R. D. Luce. A note on boolean matrix theory. Proc. null Amer. null Math. null Soc., 3:382388, 1952.
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- Roger Duncan Maddux. The Equational Theory of CA
_{3}is Undecidable. Journal of Symbolic Logic, 45:311316, 1980.- Roger Duncan Maddux. Embedding modular lattices into relation algebras. Algebra Universalis, 12:244246, 1981.
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- Roger Duncan Maddux. Pair-dense relation algebras. Draft paper, Iowa State Univ., Ames, 1987.
- Roger Duncan Maddux. Canonical relativized cylindric set algebras. Proc. null Amer. null Math. null Soc., 107(2):465478, October 1989.
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- Roger Duncan Maddux. Nonfinite axiomatizability results for cylindric and relation algebras. Journal of Symbolic Logic, 54(3):951974, September 1989.
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- Roger Duncan Maddux. The neat embedding problem and the number of variables required in proofs. Proc. null Amer. null Math. null Soc., 112:195202, 1991.
- Roger Duncan Maddux. The origin of relation algebras in the development and axiomatization of the calculus of relations. Studia Logica, 50(3/4):421455, 1991.
- Roger Duncan Maddux. Pair-dense relation algebras. Trans. null Amer. null Math. null Soc., 328:83131, 1991.
- Roger Duncan Maddux. Relation algebras of every dimension. Journal of Symbolic Logic, 57(4):12131229, December 1992.
- Roger Duncan Maddux. A working relational model: The derivation of the Dijkstra-Scholten predicate transformer semantics from tarski's axioms for the Peirce-Schröder calculus of relations. Technical report, Dept. null of Mathematics, Iowa State Univ., Ames, Iowa 50011, USA, September 1992. Superseded by [Maddux1993a].
- Roger Duncan Maddux. A working relational model: The derivation of the Dijkstra-Scholten predicate transformer semantics from Tarski's axioms for the Peirce-Schröder calculus of relations. South African Computer J., 9:92130, 1993.
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- Roger Duncan Maddux. Relation algebras for reasoning about time and space. In Nivat et al. [Nivat et al., 1994], pages 2744.
- Roger Duncan Maddux. Undecidable semiassociative relation algebras. Journal of Symbolic Logic, 59:398418, 1994.
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Transformations of hypergraphs by applying rewriting rules are considered. An idea of progrmming such transformations and a suitable language with a denotational semantics is presented. It is shown that in this language one can program sequential and parallel processes of rewritimg as particular cases.

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The construction os structure-preserving maps, ``homomorphisms'', is described for an arbitrary data type, and a ``promotion'' theorem is derived for proving equalities of homomorphisms. Examples are given for finite lists, tree structures and types defined by mutual induction; the construction is then dualised to data types with infinite objects, such as infinite lists. The promotion theorem allows the development of concise, calculational proofs: several examples are given of its application to program transformation.

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^{th}Internat. null Joint Conf. null on Artificial Intelligence, Karlsruhe, W. null Germany, August 1983 (IJCAI), pages 343345, 1983.- Cristian Mallol, Jean-Pierre Olivier, and Dany Serrato. Groupoids, idempotents and pointwise inverses in relational categories. Journal of Pure and Applied Algebra, 36:2351, 1985.
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second edition, edited by J. M. Martin

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- Maarten Marx. Relation algebras can tile. Information Sciences, 119(34):173191, December 1999.
Undecidability of the equational theory of the class RA of relation algebras can easily be proved using the undecidability of the word-problem for semigroups. With some effort and ingenuity, one can push this proof through for the larger class SA. We provide another "cause" for undecidability which works for even larger classes than SA. The reason is that we can encode the tiling problem. In doing so we will meet very simple BAO-varieties with undecidable equational theories which might be useful in other undecidability proofs. Our work is part of the research project which tries to establish the border between undecidability and decidability in relational type algebras, cf. [15,16,12,1] and the references therein. The ultimate goal of this research is to come up with versions of relational algebra which are still suitable for modern dynamic applications but whose equational theory is decidable or even tractable.

- Markus A. Marzetta. A quantifier-free type inference system. Technical Report 91-014, Bern University, 1991.
Several kinds of logical system have been introduced in order to establish properties of programs like termination and correctness: second order typed lambda calculus due to Girard and Reynolds', Martin-Löf's intuitionistic type theories, the calculus of constructions due to Huet and Coquand, etc. These systems are mostly characterised by a great proof-theoretical strength, which allows to prove totality/ter-mination for a large class of functions/programs, perhaps using essentially impredicative methods, but also increases the difficulty of finding such a proof (automatically). Starting from this observation and from work of Mitchell and Harper on the programming language ML, Feferman has developed constructive (type) theories ranging in strength from primitive recursive arithmetic it (PRA) up to fairly strong subsystems of analysis. Following these lines we present quantifier-free type inference systems by means of which types can be assigned to &lgr;-expressions. Formally these are deduction systems for sequents of the form a

_{1}mkern -2mu mathrel : mkern -2mu A_{1}, allowbreak ..., allowbreak a_{n}mkern -2mu mathrel : mkern -2mu A_{n}allowbreak mathrel supset t_{1}mkern -2mu mathrel : mkern -2mu B and a_{1}mkern -2mu mathrel : mkern -2mu A_{1}, allowbreak ..., allowbreak a_{n}mkern -2mu mathrel : mkern -2mu A_{n}allowbreak mathrel supset t_{1}=t_{2}mkern -2mu mathrel : mkern -2mu B where the a_{i}are free variables, the t_{i}are individual terms of explicitly typed lambda calculus and A_{i},B are type terms built up from the basic type hbox rlap sf I hskip 0.15em sf N, equational types and (optionally) sum and product types. Special interest is put on the analysis of the proof-theoretical strength of these systems. Our basic system &tgr;*pr*^{-}is shown to be proof-theoretically equivalent to*PRA*. The extension &tgr;*pr*includes new type constructions, dependent sum and product, but still has the same strength. A stronger system &tgr;*ha*can indeed be obtained from &tgr;*pr*by varying the type constructions for which induction/recursion is allowed.- D. W. Matula. An algorithm for subtree identification. SIAM Rev., 10:273274, 1968. Abstract.
Sketch of Reyners algorithm

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Of the various approaches to program correctness, that of ``Transformational Programming'' appears to be the most helpful in constructing correct programs. The essence of the method is to start with an obviously correct but possibly hopelessly inefficient algorithm, and to improve it by successively applying correctness-preserving transformations. The manipulations involved are akin to those used in mathematics. Two important impediments to this method are the verbosity of algorithmic notations, making the process cumbersome, and the semantic baroqueness of many primitives, making it hard to verify the validity of transformations. Computer Science can profit here from the lessons taught by the history of Mathematics. Another major step, comparable to one made long ago in Mathematics, is not to insist on the ``executability'' of algorithmic descriptions. This makes it possible to treat initial high-level specifications in the same framework as the final programs. Just as Mathematics evolved from ``Transformational Arithmetic'', Transformational Programming may come of age as ``Algorithmics''.

- Lambert Meertens. Constructing a calculus of programs. In van de Snepscheut [van de Snepscheut, 1989], pages 6690.
- Lambert Meertens. Paramorphisms. Formal Aspects of Computing, 4(5):413424, 1992.
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We develop a calculus for lazy functional programming based on recursion operators associated with data type definitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example functions in Bird and Wadler's ``Introduction to Functional Programming'' can be expressed using these operators.

- Karl Meinke and L.J. Steggles. Specification and verification in higher-order algebra: a case study of convolution. In Heering et al. [Heering et al., 1993], pages 189222.
We present a case study of higher order algebraic methods applied to the specification of convolution as a second order transformation on streams. Two systolic synchronous concurrent algorithms (SCAs) for convolution are formally specified and verified using higher order equational logic. We then study the metamathematics of these verification proofs by means of non-standard models.

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A theory of general logics is outlined as a basis for an axiomatic notion of ``logic programming language''. It is shown that a wide variety of logical programming languages are instances of the general notion. The problem of designing multiparadigm logic programming languages that overcome the present limitations faced by relational and functional logical languages in dealing with state change and reactive systems is approached by a method based on the use of the axiomatic notion of logic programming language and of mappings between logics to guide the search for a logic in which the desired multiparadigm integration can be attained. Following this method, rewriting logic is proposed as a logic in which the functional, relational, and concurrent object-oriented paradigms can be unified in a simple and rigorous way. Two languages based on this logic, Maude and MaudeLog, are briefly described and illustrated with examples.

- Emily Michael. Peirce's early study of the logic of relations, 18651867. Trans. null of the Charles S. null Peirce Society, 10:6375, 1974.
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- Emily Michael. A note on Peirce on Boole's algebra of logic. Notre Dame J. null Formal Logic, 20:636638, 1979.
- Renato Migliorato. Isomorphisms of finite hypergroupoids. In Barlotti et al. [Barlotti et al., 1988], pages 301310.
- Szabolcs Mikulás, Ildikó Sain, and András Simon. Complexity of the equational theory of relational algebras with projection elements. Bull. null of the Sect. null of Logic, Univ. null of Lódz, 21(3):103111, 1992.
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In this paper we define computationally well-behaved versions of classical first-order logic and prove that the validity problem is decidable.

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^{th}Internat. null Conf. null on Software Engineering (ICSE 7), pages 499509, Orlando, FL, March 1984.- Ali Mili and Jules Desharnais. Toward the automatic symbolic execution of while statements. In Proc. null 17
^{th}Hawaii Internat. null Conf. null on System Sciences, pages 378382, Honolulu, HI, January 1984.- Fatma Mili and Ali Mili. Heuristics for constructing while loops. Science of Computer Programming, 18:67106, 1992.
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When writing programs to manipulate structures such as algebraic expressions, logical formulas, proofs, and programs, it is highly desirable to take the linear, human-oriented, concrete syntax of these structures and parse them into a more computation oriented syntax. For a wide variety of manipulations, concrete syntax contains too much useless information (

*e.g.*, keywords and white space) while important information is not explicitely represented (*e.g.*, function-argument relations and the scope of operators). In parse trees, much of the semantically useless information is removed while other relationships, such as between function and argument, are made more explicit. Unfortunately, parse trees do not adequately address important notions of object-level syntax, such as bound and free object-variables, scopes, alphabetic changes of bound variables, and object-level substitution. I will argue here that the*abstract syntax*of such objects should be organized around &agr;-equivalence classes of &lgr;-terms instead of parse trees. Incorporating this notion of abstract syntax into programming languages is an interesting challenge. This paper briefly describes a logic programming language that directly supports this notion of syntax. An example specifications in this programming language is presented to illustrate its approach to handling object-level syntax. A model-theoretic semantics for this logic programming language is also presented.- H. D. Mills, V. R. Basili, J. D. Gannon, and R. G. Hamlet. Principles of Computer Programming. A Mathematical Approach. Allyn and Bacon, 1987.
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- John C. Mitchell and Albert R. Meyer. Second-order logical relations. In Parikh [Parikh, 1985], pages 225236. extended abstract.
Logical relations are a generalization of homomorphisms between models of typed lambda calculus. We define logical relations for second-order typed lambda calculus and use these relations to give a semantic characterization of second-order lambda definability. Logical relations are also used to state and prove a general representation independence theorem. Representation independence implies that the meanings of expressions do not depend on whether true is represented by 1 and false by 0, as long as all the functions that manipulate truth values are represented correctly.

- O. H. Mitchell. On a new algebra of logic. In Peirce [Peirce, 1883], pages 72125.
- Yoshihiro Mizoguchi and Yasuo Kawahara. Relational graph rewritings. Theoretical Computer Science, 141:311328, 1995.
- Yoshihiro Mizoguchi. Properties of graphs preserved by relational graph rewritings. Information Sciences, 119(34):289299, December 1999.
We formulate graphs and graph rewritings using binary relations and call them relational graphs and relational graph rewritings. In this framework, rewriting is defined using a pushout in a category of relational graphs. It is known that an important theorem of rewriting systems called critical pair's lemma can be proved using simple and clear categorical properties. In this paper, we construct treelike graphs and Raoult Graphs by some relational conditions. We give a sufficient condition for rewriting rules and matchings which guarantees the closedness of those graph rewritings. These results show that the critical pair's lemma also holds under some conditions for a graph rewriting system in which graphs are restricted to treelike graphs or Raoult Graphs.

- Axel Möbus. Relationale Algebren. PhD thesis, Univ. Düsseldorf, Mathematisch-Naturwissenschaftliche Fakultät, 1981.
- Eugenio Moggi. A modular approach to denotational semantics. In Pitt et al. [Pitt et al., 1991], pages 138139.
We propose an incremental approach to the denotational semantics of complex programming languages based on the idea of

*monad transformer*.- R. Mohr and T. C. Henderson. Arc and path consistency revisited. Artificial Intelligence, 28:225233, 1986.
- Bernhard Möller and Martin Russling. Shorter paths to graph algorithms. In Bird et al. [Bird et al., 1992], pages 250268.
We illustrate the use of formal languages and relations in compact formal derivations of some graph algorithms.

- Bernd Möller. On the algebraic specification of infinite objectsordered and continous models of algebraic types. Acta Inform., 22:537578, 1985.
- Bernd Möller, editor. Constructing Programs From Specifications, Proc. null of the IFIP TC2 Working Conf. null on Constructing Programs From Specifications. IFIP WG 2.1, North-Holland, 1991.
- Bernhard Möller. Relations as program development language. In Möller [M{ö}ller, 1991a], pages 319371.
- Bernd Möller. Ordered and continuous models of higher-order algebraic specifications. In Heering et al. [Heering et al., 1993], pages 223255.
We investigate the existence of continuous and fixpoint models of higher-order specifications. Particular attention is paid to the question of extensionality. We use ordered specifications, a particular case of Horn specifications. The main tool for obtaining continuous models is the ideal completion. Unfortunately, it may destroy extensionality. This problem is inherent: we show that there is no completion method which is guaranteed to preserve extensionality. To restore it, generally a quotient has to be taken. It is shown that under certain conditions this preserves the existence of least fixpoints. Examples of the specification method include the essential concepts of Backus' FP and Hoare's CSP.

- B. Möller. Algebraic calculation of graph and sorting algorithms. In Bjørner et al. [Bj{ø}rner et al., 1994], pages 98127.
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- Bernhard Möller, editor. Mathematics of Program Construction, Third International Conference, MPC '95, Kloster Irsee, Germany, July 1995, volume 947 of LNCS. Springer, 1995.
- Bernhard Möller. Calculating with acyclic and cyclic lists. Information Sciences, 119(34):135154, December 1999.
We use a relational model of pointer structures to calculate a number of standard algorithms on singly linked lists, both acyclic and cyclic. This shows that our techniques are not just useful for tree-like structures, but apply to general pointer structures as well.

- J. Donald Monk. Relation algebras and cylindric algebras. Notices Amer. null Math. null Soc., 8:358, 1961.
- J. Donald Monk. Studies in cylindric algebra. PhD thesis, Univ. null of California, Berkeley, Berkeley, 1961. Doctoral Diss.
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- J. Donald Monk. Model-theoretical methods and results in the theory of cylindric algebras. In Addison [Addison, 1965], pages 238250.
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- J. Donald Monk. Completions of boolean algebras with operators. Math. null Nachr., 46:4755, 1970.
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- Richard Montague. Formal Philosophy. Yale Univ. null Press, New Haven, 1974.
- Ugo Montanari and Francesca Rossi. Fundamental properties of networks of constraints: A new formulation. pages 426449. no Journal!!!!!
- Ugo Montanari and Francesca Rossi. Graph rewriting, constraint solving and tiles for coordinating distributed systems, 199?
- Ugo Montanari and Francesca Rossi. Modeling process coordination via tiles, graphs, and constraints, 199?
- Ugo Montanari and Francesca Rossi. Perfect relaxation in constraint logic programming. In Beaumont and G. [Beaumont and G., 1991], pages 223237.
- Ugo Montanari and Francesca Rossi. Contextual occurrence nets and concurrent constraint programming. In Schneider and Ehrig [Schneider and Ehrig, 1993], pages 280295.
This paper proposes a new semantics for concurrent constraint programs. The meaning of each program is defined as a contextual net, which is just a usual net where context conditions, besides pre- and post-conditions, are allowed. Context conditions are just items which have to be present in order for an event to take place, but which are not affected by the event. They are very useful for describing situations where different events share a common resource and want to read it simultaneously. In fact, such events are concurrent in the net. The causal dependency relation of the net induces a partial order among objects in the same computation, while its mutual exclusion relation provides a way of expressing nondeterministic information. Such information can be of great help to a scheduler while trying to find an efficient execution of the program, or also to a compile-time optimizer.

- Ugo Montanari and Francesca Rossi. Graph rewriting for a partial ordering semantics of concurrent constraint programming. In Courcelle and Rozenberg [Courcelle and Rozenberg, 1993], pages 225256. Theoretical Computer Science
**109**(12).The concurrent constraint logic programming framework extends both logic programming and concurrent logic programming in that a program consists of the concurrent execution of agents which add and check constraints on a shared set of variables, and whose behavior is described by a set of clauses. This formulation is very general and can be seen as a concurrent logic programming shell which is parametrized w.r.t. the underlying constraint system. Graphs and graph grammars can be conveniently used to describe such a framework and the modelling is so elegant and expressive that they provide what we believe is the most natural abstract machine for concurrent constraint programming.

- Ugo Montanari. Networks of constraints: Fundamental properties and applications to picture porcessing. Inform. null Sci., 7:95132, 1974.
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- Gregory H. Moore. A house divided against itself: The emergence of first-order logic as the basis for mathematics. In Studies in the History of Mathematics, volume 26 of MAA Studies in Mathematics, pages 98136. The Mathematical Assoc. null of America, 1987.
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- J. M. Morris. A theoretical basis for stepwise refinement and the programming calculus. Science of Computer Programming, 9:287306, 1987.
- M.A. Moshier. Featureless HPSG, 1995. Unpublished manuscript.
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- Andy Mück. CAMEL: An extension of the categorical abstract machine to compile functional/logical programs. In Bruynooghe and Wirsing [Bruynooghe and Wirsing, 1992], pages 341354.
In this paper we present a clean implementation technique for functional/logic (or algebraic) programming languages. First we define an intermediate language to which a functional / logic program is compiled. In order to implement this intermediate language, we extend the Categorical Abstract Machine (CAM) by an additional data structure to handle logical variables and by a few instructions covering unification and backtracking. Finally, we show how the intermediate language is compiled into the instruction set or our Categorical Abstract Machine extension.

- J. J. Murphy. On the addition and multiplication of logical relatives. Memoirs of the Manchester Literary and Philosophical Society, ser. null 3, 7(27):201224, 1882.
- Bernhard Nebel and Christer Bäckström. On the computational complexity of temporal projection and some related problems. Research Report RR-91-34, German Research Center for Artificial Intelligence (DFKI), Stuhlsatzenhausweg 3, 6600 Saarbrücken 11, Germany, October 1991. Also published as Research Report LiTH-IDA-R-91-34, Department of Computer and Information Science, Linköping University, Linköping, Sweden.
One kind of temporal reasoning is temporal projectionthe computation of the consequences for a set of events. This problem is related to a number of other temporal reasoning tasks such as story understanding, plan validation, and planning. We show that one particular simple case of temporal projection on partially ordered events turns out to be harder than previously conjectured. However, given the restrictions of this problem, planning and story understanding are easy. Additionally, we show that plan validation, one of the intended applications of temporal projection, is tractable for an even larger class of plans. The incomplete decision procedure for the temporal projection problem that has been proposed by other authors, however, fails to be complete in the case where we have shown plan validation to be tractable.

- Bernhard Nebel and Hans-Jürgen Bürckert. Reasoning about temporal relations: A maximal tractable subclass of Allen's interval algebra. Research Report RR-93-11, German Research Center for Artificial Intelligence (DFKI), Saarbrücken, Germany, March 1993.
We introduce a new subclass of Allen's interval algebra we call "ORD-Horn subclass," which is a strict superset of the "pointisable subclass." We prove that reasoning in the ORD-Horn subclass is a polynomial-time problem and show that the path-consistency method is sufficient for deciding satisfiability. Further, using an extensive machine-generated case analysis, we show that the ORD-Horn subclass is a maximal tractable subclass of the full algebra (assuming P <> NP). In fact, it is the unique greatest tractable subclass amongst the subclasses that contain all basic relations.

- Bernhard Nebel and Jana Koehler. Plan modification versus plan generation: A complexity-theoretic perspective. Research Report RR-92-48, German Research Center for Artificial Intelligence (DFKI), Saarbrücken, Germany, November 1992.
The ability of a planner to modify a plan is considered as a valuable tool for improving efficiency of planning by avoiding the repetition of the same planning effort. From a computational complexity point of view, however, it is by no means obvious that modifying a plan is computationally as easy as planning from scratch if the modification has to follow the principle of ``conservatism,'' i.e., to reuse as much of the old plan as possible. Indeed, considering propositional STRIPS planning, it turns out that conservative plan modification is as hard as planning and can sometimes be harder than plan generation. Furthermore, this holds even if we consider modification problems where the old and the new goal specification are similar. We put these results into perspective and discuss the relationship to existing plan modification systems. Although sometimes claimed otherwise, these systems do not address the modification problem, but use a non-conservative form of plan modification as a heuristic technique.

- Bernhard Nebel and Gert Smolka. Attributive description formalisms and the rest of the world. Research Report RR-91-15, German Research Center for Artificial Intelligence (DFKI), Stuhlsatzenhausweg 3, 6600 Saarbrücken 11, Germany, 1991. Published in: O. Herzog and C.-R. Rollinger, Text Understanding in LILOG, Springer-Verlag, Berlin, Heidelberg, New York, 1991, 439452.
Research in knowledge representation has led to the development of so-called terminological logics, the purpose of which is to support the representation of the conceptual and terminological part of Artificial Intelligence applications. Independently, in computational linguistics, so-called feature logics have been developed which are aimed at representing the semantic and syntactic information natural language sentences convey. Since both of these logics rely mainly on attributes as the primary notational primitives for representing knowledge, they can be jointly characterized as attributive description formalisms. Although the intended applications for terminological logics and feature logics are not identical, and the computational services of systems based on the respective formalisms are quite different for this reason, the logical foundations turn out to be very similar as we pointed out elsewhere. In this paper, we will show how attributive description formalisms relate to ``the rest of the world.'' Recently, a number of formal results in the area of attributive description formalisms have been obtained by exploiting other research fields, such as formal language theory, automata theory, and modal logics. This connection between these different fields of formal research will be highlighted in the sequel.

- Bernhard Nebel. Belief revision and default reasoning: Syntax-based approaches. Research Report RR-91-11, German Research Center for Artificial Intelligence (DFKI), Stuhlsatzenhausweg 3, 6600 Saarbrücken 11, Germany, April 1991. A shorter version of this paper was published in: J. A. Allen, R. Fikes, and E. Sandewall (eds.),
*Principles of Knowledge Representation and Reasoning: Proceedings of the Second International Conference*, Morgan Kaufmann, San Mateo, CA, 1991, 417428.Belief revision leads to temporal nonmonotonicity, i.e., the set of beliefs does not grow monotonically with time. Default reasoning leads to logical nonmonotonicity, i.e., the set of consequences does not grow monotonically with the set of premises. The connection between these forms of nonmonotonicity will be studied in this paper focusing on syntax-based approaches. It is shown that a general form of syntax-based belief revision corresponds to a special kind of partial meet revision in the sense of the theory of epistemic change, which in turn is expressively equivalent to some variants of logics for default reasoning. Additionally, the computational complexity of the membership problem in revised belief sets and of the equivalent problem of derivability in default logics is analyzed, which turns out to be located at the lower end of the polynomial hierarchy.

- G. Nelson. A generalization of Dijkstra's calculus. ACM Transactions on Programming Languages and Systems (TOPLAS), 11(4):517561, 1989.
- István Németi. Logic with 3 variables has Gödel's Incompleteness PropertyThus Free Cylindric Algebras are not Atomic. Hungarian Acad. null Sci., Math. null Inst., Budapest, 1985. Preprint No. null 49/85.
- István Németi. Free Algebras and Decidability in Algebraic Logic. Hungarian Acad. null Sci., Budapest, 1986. Doctoral Diss.
- István Németi. A non-representable cylindric algebra with pairing functions. Algebra Universalis, 22:117119, 1986.
- István Németi. Decidability of relation algebras with weakened associativity. Proc. null Amer. null Math. null Soc., 100(2):340344, June 1987.
- István Németi. On varieties of cylindric algebras with applications to logic. Ann. null Pure Appl. null Logic, 36:235277, 1987.
- István Németi. Algebraizations of quantifier logics, an introductory overview. 10
^{th}Version, October 1991.- István Németi. On cylindric algebraic model theory. ?, 199x.
- M. Nesi, V. de Paiva, and E. Ritter. Rewriting properties of combinators for intuitionistic linear logic. In Heering et al. [Heering et al., 1993], pages 256275.
In this paper we investigate the possibility of developing a (semi-)automatic rewriting tool for manipulating and reasoning about combinators for intuitionistic linear logic. In particular, we develop a canonical (i.e. confluent and terminating) term rewriting system associated to a theory of categorical combinators for (rudimentary) linear logic. In order to do that, we make use of the Knuth-Bendix completion algorithm to transform the equational theory for the combinators into an equivalent canonical rewrite system. This means that a set of categorical combinators for linear logic has first to be derived, and then the resulting system of combinators can be checked for rewriting properties using rewriting techniques.

- Peter M. Neumann. Finite permutation groups, edge-colored graphs and matrices. In Topics in Group Theory and Computation. Academic Press, 1977. edited by Michael P. J. Curran.
- W. H. Newton-Smith. The Structure of Time. Routledge Kegan Paul, 1980.
- Kan Ching Ng and Alfred Tarski. Relation algebras with transitive closure. Notices Amer. null Math. null Soc., 24:A29, 1977.
- Kan Ching Ng. The Cantor-Bernstein theorem and related results in a relation algebraic setting. Notices Amer. null Math. null Soc., 24:A30, A304, 1977.
- Kan Ching Ng. Relation Algebras with Transitive Closure. PhD thesis, Univ. null of California, Berkeley, Berkeley, 1984. Doctoral Diss.
- Thanh Tung Nguyen. Multi-Valued Function Theory for Computer Programming. PhD thesis, Univ. Cath. de Louvain, Belgium, 1988.
- Thanh Tung Nguyen. A relational model of demonic nondeterministic programs. Internat. null J. null Found. null Comput. null Sci., 2(2):101131, 1991.
- Thanh Tung Nguyen. The connection between predicate logic and demonic relation calculus. Technical Report CRIN 92-R-187, Centre de Recherche en Informatique de Nancy, November 1992.
- Thanh Tung Nguyen. Duality between relations and predicate transformers. Technical Report SIGRAPA/INFO/RR.95-01, SIGRAPA, Kraainen, Belgium, May 1995.
- J. M. Nicolas. Mutual dependencies and some results on undecomposable relations. In 4
^{th}Internat. null Conf. null on Very Large Data Bases, pages 360367, Berlin, September 1978.- Joachim Niehren and Gert Smolka. A confluent relational calculus for higher-order programming with constraints. In Louannaud [Louannaud, 1994], pages 89104.
We present the &rgr;-calculus, a relational calculus parameterized with a logical constraint system. The rho-calculus provides for higher-order relational programming with first-order constraints, and subsumes higher-order functional programming as a special case. It captures important aspects of the concurrent constraint programming langauge Oz. ...

- Joachim Niehren, Andreas Podelski, and Ralf Treinen. Equational and membership constraints for infinite trees. Research Report RR-93-14, German Research Center for Artificial Intelligence (DFKI), April 1993.
We present a new constraint system with equational and membership constraints over infinite trees. It provides for complete and correct satisfiability and entailment tests and is therefore suitable for the use in concurrent constraint programming systems which are based on cyclic data structures. Our set defining devices are greatest

*fixpoint solutions*of regular systems of equations with a deterministic form of union. As the main technical particularity of the algorithms we present a novel memorization technique. We believe that both satisfiability and entailment tests can be implemented in an efficient and incremental manner.- Flemming Nielson and Hanne Riis Nielson. Layered predicates. In de Bakker et al. [de Bakker et al., 1992], pages 425456.
We review the concept of logical relations and how they interact with structural induction; furthermore we give examples of their use, and of particular interest is the combination with the PER-idea (partial equivalence relations). This is then generalized to Kripke-logical relations; the major application is to show that in combination with the PER-idea this solves the problem of establishing a substitution property in a manner conducive to structural induction. Finally we introduce the concept of Kripke-layered predicates; this allows a modular definition of predicates and supports a methodology of ``proof in stages'' where each stage focuses on only one aspect and thus is more manageable. All of these techniques have been tested and refined in ``realistic applications'' that have been documented elsewhere.

- Maurice Nivat, Charles Rattray, Teodore Rus, and Giuseppe Scollo, editors. Proc. null 2
^{nd}Internat. null Conf. null on Algebraic Methodology and Software Technology, AMAST '91, Enschede, June 2125. Springer, 1992.- Maurice Nivat, Charles Rattray, Teodore Rus, and Giuseppe Scollo, editors. Proc. null 3
^{rd}Internat. null Conf. null Algebraic Methodology and Software Technology, Enschede, June 2125, Workshops in Computing. Springer, 1994.- Maurice Nivat. On the interpretation of recursive polyadic program schemes. In Convegni del Feb. null e dell` Apr. del 1973, volume 15 of Sympos. null Math., pages 255281, London, 1975. Istituto Nazionale di Alta Matematica, Academic Press.
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J. C. Reynolds suggested that Strachey's intuitive concept of ``parametric''(i.e., uniform) polymorphism is closely linked to R-representation independece, and used logical relations to formalize this principle in languages with type variables and user-defined types. Here, we use relational parametricity to address long-standing problems with the semantics of local-variable declarations, by showing that interactions between local and nonlocal entities satisfy certain relational criteria. The new model is based on a cartesian closed category of ``relation-preserving'' functors and natural transformations which is induced by a suitable category of ``possible worlds'' with relations assigned to its objects and morphisms. The semantic interpretation supports straightforward validations of all the test equivalences that have been proposed in the literature, and encompasses standard methods of reasoning about data representations; however, it is not known whether it is fully abstract.

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A unifier is a substitution that makes two terms syntactically equal. In this paper, we discuss a more semantical unifier: an equational unifier, which is a substitution that makes two terms equal modulo a congruence relation. As a result we will give a general procedure that enumerates a complete set of equational unifiers for a given pair of terms under a given congruence.

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The standard Algebraic Theory of Graph Grammars is based on the notion of ``color-preserving'' graph morphisms and on a ``double pushout'' construction to represent gluing of graphs. In this paper, we impose a simple structure on the sets of colors to allow variables in both graphs and productions. Instantiations are performed by graph morphisms. Using relative unification, we define the composition of rules and prove the Concurrency Theorem in this more general framework. By restricting our attention to rooted directed acyclic graphs, we can represent standard Term Rewriting with First order substitutions. One of the motivations for this study is the attempt to provide a description of the static behavior of Rule-Based Expert Systems.

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^{th}GI Jahrestagung, volume 257 of Informatik-Fachberichte, page 12, Stuttgart, 1990. Springer. 10 October1990.- David Lorge Parnas. Tabular representation of relations. Technical Report CRL Report 260, McMaster Univ., Communications Research Laboratory, TRIO (Telecommunications Research Inst. null of Ontario), October 1992.
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An interactive theorem prover,

*Isabelle*, is under development. In LCF, each inference rule is representedby one function for forwards proof and another (a*tactic*) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. Resolution gives both forwards and backwards proofs, supporting a large class of logics. Isabell has been used to prove theorems in Martin-Löf's constructive type theory. Quantifiers pose several difficulties: substitution, bound variables, Skloemization. Isabelle's representation of logical syntax is the typed &lgr;-calculus, requiring higher-order unification. It may have potential for logic programming. Depth-first subgoaling along inference rules constitutes a higher-order PROLOG.- L. C. Paulson. Logic and Computation (Interactive Proof with Cambridge LCF), volume 2 of Cambridge Tracts Theoret. null Comput. null Sci. Cambridge Univ. null Press, 1987.
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RML is a programming language intended for the implementation of Natural Semantics specifications. The basic procedural elements are

*relations*: many-to-many mappings defined by a number of*axioms*or*inference rules*. It has control flow, logical variables and (explicit) unification as in Prolog; from ML it borrows a ploymorphic type system, data structures, and pattern matching; a facility for separately-compilable modules also exists. A simple prototype compiler, based on translating RML to Continuation-Passing Style and then to C, has been implemented. Benchmarks indicate that this compiler generates code that is*several orders of magnitude*faster than Typol, and two times faster than standard Prolog compilers.- Mikael Petterson. Compiling Natural Semantics, volume 1549 of LNCS. Springer, 1999.
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^{th}Annual IEEE Sympos. null on Foundations of Computer Science, pages 109121, October 1976.- Terrence W. Pratt. Definition of programming language semantics using grammars for hierarchical graphs. In Claus et al. [Claus et al., 1978], pages 389400.
Directed graphs are a useful formal structure in the modeling and definition of programming language semantics. Graph grammars are a valuable tool for defining sets of directed graphs in this setting. The goal of this paper is to describe this application of graph grammars and to provide the underlying formal definitions for the particular graph structures and graph grammars used. Translation of programming languages into graph structures may also be defined using graph grammars paired with ordinary BNF (context-free) grammars. This closely related application is also described. The graph grammar form used here and the theory underlying its application were developed about 1970 (see [1]). Graph grammars have been used regularly since then in the definition of a number of programming languages, including ALGOL60, PASCAL, LISP, and HAL/S.

- Vaughan Pratt. Models of program logics. In Proc. null of the 20
^{th}IEEE Sympos. null on Foundations of Computer Science, pages 115122, 1979.- Vaughan Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15(1):3371, February 1986.
Concurrency has been expressed variously in terms of formal languages (typically via the shuffle operator), partial orders, and temporal logic, inter alia. In this paper we extract from these three approaches a single hybrid approach having a rich language that mixes algebra and logic and having a natural class of models of concurrent processes. The heart of the approach is a notion of partial string derived from the view of a string as a linearly ordered multiset by relaxing the linearity constraint, thereby permitting partially ordered multisets or pomsets. Just as sets of strings form languages, so do sets of pomsets form processes. We introduce a number of operations useful for specifying concurrent processes and demonstrate their utility on some basic examples. Although none of the operations is particularly oriented to nets it is nevertheless possible to use them to express processes constructed as a net of subprocesses, and more generally as a system consisting of components. The general benefits of the approach are that it is conceptually straightforward, involves fewer artificial constructs than many competing models of concurrency, yet is applicable to a considerably wider range of types of systems, including systems with buses and ethernets, analog systems, and real-time systems.

- Vaughan Pratt. Dynamic algebras as a well-behaved fragment of relation algebras. In Bergmann et al. [Bergmann et al., 1990], pages 77110.
- Vaughan Pratt. Dynamic algebras as a well-behaved fragment of relational algebras. Technical Report CS-TR-90-1309, Stanford University, Department of Computer Science, March 1990.
The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras.

- Vaughan Pratt. Modeling concurrency with geometry. In POPL '91 [POPL '91, 1991], pages 311322.
The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer one over the other? We identify dimension as the culprit: 1-dimensional automata are skeletons permitting only interleaving concurrency, whereas

*true n-fold concurrency resides in transitions of dimension n*. The truly concurrent automaton dual to a schedule is not a skeletal distributive lattice but a solid one! We introduce true nondeterminism and define it as monoidal homotopy; from this perspective nondeterminism in ordinary automata arises from forking and joining creating nontrivial homotopy. The automaton dual to a poset schedule is simply connected whereas that dual to an event structure schedule need not be, according to monoidal homotopy though not to group homotopy. We conclude with a formal definition of higher dimensional automaton as n-complex or n-category, whose two essential axioms are associativity of concatenation within dimension and an interchange principle between dimensions.- Vaughan Pratt. Origins of the calculus of binary relations. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 248254, Santa Cruz, California, 2225 June 1992. IEEE Computer Society Press.
The calculus of binary relations was introduced by De Morgan in 1860, and was subsequently greatly developed by Peirce and Schroeder. Half a century later Tarski, J'onsson, Lyndon, and Monk further developed the calculus from the perspective of modern model theory.

- V.R. Pratt. Arithmetic + logic + geometry = concurrency. In Proc. First Latin American Symposium on Theoretical Informatics, Sao Paulo, Brazil, volume 583 of LNCS, pages 430447. Springer-Verlag, April 1992.
We relate the arithmetic of concurrent schedules to the higher-dimensional cellular geometry of concurrent automata using the logic of their Birkhoff-Stone duality. This collects and unifies ideas from several of the author's previous papers.

- V.R. Pratt. The duality of time and information. In Proc. of CONCUR'92, Stonybrook, New York, pages 237253. Springer, August 1992.
The states of a computing system bear information and change time, while its events bear time and change information. We develop a primitive algebraic model of this duality of time and information for rigid local computation, or straightline code, in the absence of choice and concurrency, where time and information are linearly ordered. This shows the duality of computation to be more fundamental than the logic of computation for which choice is disjunction and concurrency conjunction. To accommodate flexible distributed computing systems we then bring in choice and concurrency and pass to partially ordered time and information, the formal basis for this extension being Birkhoff-Stone dualtiy. A degree of freedom in how this is done permits a perfectly symmetric logic of computation amounting to Girard's full linear logic, which we view as the natural logic of computation when equal importance is attached to choice and concurrency. We conclude with an assessment of the prospects for extending the duality to other organizations of time and information besides partial orders in order to accommodate real time, nonmonotonic logic, and automata that can forget, and speculate on the philosophical significance of the duality.

- Vaughan R. Pratt. The second calculus of binary relations. In Andrzej M. Borzyszkowski and Stefan Sokolowski, editors, Mathematical Foundations of Computer Science 1993, 18th International Symposium, volume 711 of LNCS, pages 142155, Gdansk, Poland, 30 August 3 September 1993. Springer.
We view the Chu space interpretation of linear logic as an alternative interpretation of the language of the Peirce calculus of binary relations. Chu spaces amount to K-valued binary relations, which for K=2

^{n}we show generalize n-ary relational structures. We also exhibit a four-stage unique factorization system for Chu transforms that illuminates their operation.- V.R. Pratt. Linear logic for generalized quantum mechanics. In Proc. Workshop on Physics and Computation (PhysComp'92, Dallas), pages 166180. IEEE, 1993.
Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard's linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of linear automata and their dual linear schedules. In this extension the uncertainty tradeoff emerges via the ``structure veil.'' When VLSI shrinks to where quantum effects are felt, their computer-aided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics.

- Vaughan Pratt. Chu realizes all small concrete categories, July 1994. Stanford University.
The category Chu is concretely universal for much of concrete mathematics; in particular it concretely represents or realizes all categories of relational structures and their homomorphisms, as well as all topological such. This note extends these results to all small concrete categories, equivalently all small subcategories of Set. The category C is realized in Chu(Set,K) where K is the disjoint union of the underlying sets of objects of C. Each object is realized as the normal Chu space (A,X) where X consists of all functions from A in C astricted to K.

- Vaughan Pratt. Shorter proof of universality of Chu spaces, August 1994. Stanford University.
We give a shorter proof of the result in section 5 of our MFPS'93 paper [Pratt, 1993a], that every k-ary relational structure is realizable as a Chu space.

- V.R. Pratt. Chu spaces: Automata with quantum aspects. In Proc. Workshop on Physics and Computation (PhysComp'94, Dallas), pages 186195. IEEE, 1994.
Chu spaces are a recently developed model of concurrent computation extending automata theory to express branching time and true concurrency. They exhibit in a primitive form the quantum mechanical phenomena of complementarity and uncertainty. The complementarity arises as the duality of information and time, automata and schedules, and states and events. Uncertainty arises when we define a measurement to be a morphism and notice that increasing structure in the observed object reduces clarity of observation. For Chu spaces this uncertainty can be calculated in an attractively simple way directly from its dimensions.

- V.R. Pratt. Chu spaces: Complementarity and uncertainty in rational mechanics. Technical report, Budapest, 1994. Course notes, TEMPUS summer school, 35pp.
Notes for five lectures given at the Tempus summer school, Budapest, July 1994. Topics covered: Introduction to Chu spaces. Behavior: from event structures to rational mechanics. Algebra: from linear logic to process algebra. Relational structures. Heisenberg uncertainty in Chu spaces.

- V.R. Pratt. Time and information in sequential and concurrent computation. In Proc. Theory and Practice of Parallel Programming (TPPP'94) Sendai, Japan, pages 124, November 1994.
Time can be understood as dual to information in extant models of both sequential and concurrent computation. The basis for this duality is phase space, coordinatized by time and information, whose axes are oriented respectively horizontally and vertically. We fit various basic phenomena of computation, and of behavior in general, to the phase space perspective. The extant two-dimensional logics of sequential behavior, the van Glabbeek map of branching time and true concurrency, event-state duality and schedule-automaton duality, and Chu spaces, all fit the phase space perspective well, in every case confirming our choice of orientation.

- V.R. Pratt. Chu spaces and their interpretation as concurrent objects. In J. van Leeuwen, editor, Computer Science Today: Recent Trends and Developments, volume 1000 of LNCS, pages 392405. Springer, 1995.
A Chu space is a binary relation =| from a set A to an antiset X defined as a set which transforms via converse functions. Chu spaces admit a great many interpretations by virtue of realizing all small concrete categories and most large ones arising in mathematical and computational practice. Of particular interest for computer science is their interpretation as computational processes, which takes A to be a schedule of events distributed in time, X to be an automaton of states forming an information system in the sense of Scott, and the pairs (a,x) in the =| relation to be the individual transcriptions of the making of history. The traditional homogeneous binary relations of transition on X and precedence on A are recovered as respectively the right and left residuals of the heterogeneous binary relation =| with itself. The natural algebra of Chu spaces is that of linear logic, made a process algebra by the process interpretation.

- V.R. Pratt. Rational mechanics and natural mathematics. In TAPSOFT'95, volume 915 of LNCS, pages 108122. Springer, 1995.
Chu spaces have found applications in computer science, mathematics, and physics. They enjoy a useful categorical duality analogous to that of lattice theory and projective geometry. As natural mathematics Chu spaces borrow ideas from the natural sciences, particularly physics, while as rational mechanics they cast Hamiltonian mechanics in terms of the interaction of body and mind. This paper addresses the chief stumbling block for Descartes' 17th-century philosophy of mind-body dualism, how can the fundamentally dissimilar mental and physical planes causally interact with each other? We apply Cartesian logic to reject not only divine intervention, preordained synchronization, and the eventual mass retreat to monism, but also an assumption Descartes himself somehow neglected to reject, that causal interaction within these planes is an easier problem than between. We use Chu spaces and residuation to derive all causal interaction, both between and within the two planes, from a uniform and algebraically rich theory of between-plane interaction alone. Lifting the two-valued Boolean logic of binary relations to the complex-valued fuzzy logic of quantum mechanics transforms residuation into a natural generalization of the inner product operation of a Hilbert space and demonstrates that this account of causal interaction is of essentially the same form as the Heisenberg-Schrödinger quantum-mechanical solution to analogous problems of causal interaction in physics.

- V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444454. IEEE Computer Society, June 1995.
We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete self-dual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a two-dimensional space we call the Stone gamut. The Stone gamut is coordinatized horizontally by coherence, ranging from -1 for sets to 1 for complete atomic Boolean algebras (CABA's), and vertically by complexity of language. Complexity 0 contains only sets, CABA's, and the inconsistent empty set. Complexity 1 admits noninteracting set-CABA pairs. The entire Stone duality menagerie of partial distributive lattices enters at complexity 2. Groups, rings, fields, graphs, and categories have all entered by level 16, and every category of relational structures and their homomorphisms eventually appears. The key is the identification of continuous functions and homomorphisms, which puts Stone-Pontrjagin duality on a uniform basis by merging algebra and topology into a simple common framework.

- V.R. Pratt. Linear logic complements classical logic. In Preliminary proceedings, Linear Logic '96, Tokyo, 1996.
Classical logic enforces the separation of individuals and predicates, linear logic draws them together via interaction; these are not right-or-wrong alternatives but dual or complementary logics. Linear logic is an incomplete realization of this duality. While its completion is not essential for the development and maintenance of logic, it is crucial for its application. We outline the ``four-square'' program for completing the connection, whose corners are set, function, number, and arithmetic, and define ordinal Set, a bicomplete

*equational*topos, meaning its canonical isomorphisms are identities, including associativity of product.- V.R. Pratt. Chu spaces from the representational viewpoint. In Parikh Festschrift. 1997.
We give an elementary introduction to Chu spaces. The perspective taken views their elements as represented by words of a fixed length over some alphabet. This perspective dualizes the alternative view of Chu spaces as generalized topological spaces, and has the advantage of substituting the intuitions of formal language theory for those of topology.

- V.R. Pratt. Towards full completeness for the linear logic of chu spaces. In Proc. Math. Foundations of Programming Semantics (MFPS'97, Pittsburgh), ENTCS (Electronic Notes of Theoretical Computer Science), 1997.
We prove full completeness for a fragment of the linear logic of the self-dual monoidal category of Chu spaces over 2, namely that the proofs between semisimple (conjunctive normal form) formulas of multiplicative linear logic without constants having two occurrences of each variable are in bijection with the dinatural transformations between the corresponding functors. The proof assigns to variables domains having at most four elements, demonstrating a uniform finite model property for this fragment. We define a notion of proof function analogous to the notion of truth function, determining a transformation between functors, and show that the transformation denoted by a proof net is dinatural if and only if the proof net is sound, namely acyclic and connected. Proof functions are of independent interest as a 2-valued model of MLL with MIX.

- V.R. Pratt. Types as processes, via Chu spaces. In EXPRESS'97 Proceedings, 1997.
We match up types and processes by putting values in correspondence with events, coproduct with (noninteracting) parallel composition, and tensor product with orthocurrence. We then bring types and processes into closer correspondence by broadening and unifying the semantics of both using Chu spaces and their transformational logic. Beyond this point the connection appears to break down; we pose the question of whether the failures of the corrrespondence are intrinsic or cultural.

- V.R. Pratt. Chu spaces as a semantic bridge between linear logic and mathematics. Theoretical Computer Science, 1998.
(Note: this supersedes "Broadening the Denotational Semantics of Linear Logic", doubling its length and adding much new material.) The motivating role of linear logic is as a ``logic behind logic.'' We propose a sibling role for it as a logic of transformational mathematics via the self-dual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition we introduce the dialectic lambda-calculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic.

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Given a model of the polymorphic typed lambda calculus based upon a Cartesian closed category ctK , there will be functors from ctK to ctK whose action on objects can be expressed by type expressions and whose action on morphisms can be expressed by ordinary expressions. We show that if T is such a functor then there is a weak initial T-algebra and if, in addition, ctK possesses equalizers of all subsets of its morphism sets, then there is an initial T-algebra. These results are used to establish the impossibility of certain models, including those in which types denote sets and morphs SS' denotes the set of all functions from S to S'.

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What I intend to show in this short paper is how one can translate in relational terms the concepts of deducibility and exactness which are the result of a sequence of works on homology theory and algebraic topology. As we shall see, we shall obtain as a final product the possibility to associate to an arbitrary binary relation R a difunctional relation difun R contained in R, in contrast with the difunctional closure of R which is larger that R. In [Rig49] we have built from a given Ferrers relation R the relation (I am using the notation S R for the result of the composition of the relations defined as setof pair xz pair xy in R wedge pair yz in S, whereas in [Schmidt and Ströhlein, 1993] are using the reverse notation R S. I also use the symbol dif for denoting the operation of Boolean difference. Give an arbitrary relation R, I denote by rect R the rectangular closure of R: rect R = dom R times cod R. By subst R I denote the substratction of R from its rectangular closure: subst R = rect R dif R. Obviously, convR = substR .) R dif R substRR and proved its difunctionality, but in fact, as already noticed by Schmidt and Ströhlein ( [SS93] p. 78 Prop. 4.4.14) R dif R substRR is difunctional even when R is arbitrary. I shall show that, in fact, R dif R substRR and difun R are identical. It is important to notice that the construction used here for the definition of difun R is made without using the Boolean difference operation.

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*resolution*), iterating which is vastly more efficient than the older cyclic procedures consisting of substitution stages alternating with truth-functional analysis stages. The theory of the resolution process is presented in the form of a system of first-order logic with just one inference principle (the resolution principle). The completeness of the system is proved; the simplest proof-procedure based on the system is then the direct implementation of the proof of completeness. However, this procedure is quite inefficient, and the paper concludes with a discussion of several principles (called search principles) which are applicable tho the design of efficient proof-procedures employing resolution as the basic logical process.- P. Roper. Intervals and tenses. J. null Philos. null Logic, 9, 1980.
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We briefly describe an algebraic approach to prove correctness of compilation and exemplify how proof obligations are discharged through algebraic transformations. The example is then used as a case study to explore the suitability of some formal systems to support the approach. The systems are compared and evaluated based on some stated criteria.

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Binary decision diagrams (BDDs, [Bry-1992]) have been recognized as an extremely efficient data structure for the representation of transition relations in the verification of finite-state reactive systems. With BDDs, it is possible to represent relations over domains with more than 2

^{100}elements [BCDM-1991], provided the represented relation is well-structured. Asynchronous parallel systems such as communication protocols often use implicit or explicit buffering of messages which are sent between the processes. In these notes, we analyze the complexity of various possibilities to model the transition relation of a bounded buffer with BDDs, and discuss alternative approaches to this problem.- Gunther Schmidt and Rudolf Berghammer, editors. Proc. null 17
^{th}Internat. null Workshop on Graph-Theoretic Concepts in Computer Science, volume 570 of LNCS, Fischbachau, June 1992. Springer.- Gunther Schmidt and Peter Kempf. Semantic domains with congruences. Technical Report 9201, Fakultät für Informatik, Universität der Bundeswehr München, 1992.
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- Gunther Schmidt and Thomas Ströhlein. On kernels of graphs and solutions of games a synopsis based on relations and fixpoints. SIAM J. null Algebraic Discrete Methods, 6:5465, 1985.
- Gunther Schmidt and Thomas Ströhlein. Relation algebras concept of points and representability. Discrete Math., 54:8392, 1985.
- Gunther Schmidt and Thomas Ströhlein. Diskrete Mathematik Relationen, Graphen und Programme II. Technical report, Inst. null für Informatik der Technischen Univ. null München, 1986. Internal Report.
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- Gunther Schmidt and Thomas Ströhlein. Relations and Graphs, Discrete Mathematics for Computer Scientists. EATCS-Monographs on Theoretical Computer Science. Springer, 1993.
Relational methods can be found at various places in computer science, notably in data base theory, relational semantics of concurrency, relational type theory, analysis of rewriting systems, and modern programming language design. In addition, they appear in algorithms analysis and in the bulk of discrete mathematics taught to computer scientists. This book devoted to the background of these methods. It is the first to explain how to use relational and graph- theoretic methods systematically in computer science. The powerful calculus of relation algebra is developed with respect to applications to a diverse range of problem areas. Results are first motivated by practical examples, often visualized by both Boolean 0-1-matrices and graphs, and then derived algebraically.

- Gunther Schmidt and Michael Winter. Is every tabular relation function dense? A note on relation algebras. Internal note, 1994. 2 p.
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- Gunther Schmidt, Rudolf Berghammer, and Hans Zierer. Describing semantic domains with sprouts. Technical Report TUM-I8611, Institut für Informatik, Technische Universität München, 1986.
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- Gunther Schmidt, Claudia Hattensperger, and Michael Winter. Heterogeneous relation algebra. In Brink et al. [Brink et al., 1997], chapter 3, pages 3953.
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^{th}International Colloquium on Automata, Languages and Programming, Graz, volume 71 of LNCS, pages 505519. Springer, 1979.- Gunther Schmidt. Investigating programs in terms of partial graphs (extended abstract). In Noltemeier [Noltemeier, 1980], pages 268269.
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?

- Gunther Schmidt. Programs as partial graphs II: Recursion. Theoretical Computer Science, 15(2):159179, 1981.
In part I of the paper, we have proposed a unified relational algebra approach using partial graphs for theoretical investigations on semantics, correctness and termination. This approach is extended here to systems of recursive programs, allowing not only sequencing and conditional branching as a control structure but also flow diagrams. An equivalence proof of operational and denotational semantics is obtained which is strictly based on axioms of relational algebra. A short new proof of an important completeness result is given in the generalized setting of systems of recursive flow diagram prog rams. Finally Hitchock Park's theorem on derivatives is formulated in the general case of nondeterministic recursive flow diagram programs.

- Gunther Schmidt. Relationen und Programme. In Broy [Broy, 1991], pages 98114.
- Renate A. Schmidt. Algebraic terminological representation. Technical Report MPI-I-91-216, Max-Planck-Institut für Informatik, Im Stadtwald, Saarbrücken, November 1991.
This thesis investigates terminological representation languages, as used in sc kl-one-type knowledge representation systems, from an algebraic point of view. Terminological representation languages are based on two primitive syntax types, called concepts and roles, which are usually interpreted model-theoretically as sets and relations, respectively. I propose an algebraic rather than a model-theoretic approach. I show that terminological representations can be naturally accomodated in equational algebras of sets interacting with relations, and I use equational logic as a vehicle for reasoning about concepts interacting with roles.

- Gunther Schmidt. Ordering isomorphism classes of semantic domains. Technical Report 9207, Fakultät für Informatik, Universität der Bundeswehr München, 1992.
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*Proceedings of the German Workshop on Artificial Intelligence (GWAI-92)*, Springer-Verlag, Berlin.In this paper I establish a link between sc kl-one-based knowledge representation concerned with

*terminological representation*and the work of P. Suppes (1976, 1979, 1981) and M. Böttner (1985, 1989) in computational linguistics. I show how this link can be utilised for the problem of finding adequate terminological representations for given information formulated in ordinary English.- Lothar Schmitz. An exercise in program synthesis: algorithms for computing the transitive closure of a relation. Science of Computer Programming, (1), 1982.
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Besides explicit axioms, an algebraic specification language contains model-theoretic constraints such as initiality. For proving proerties of specifications and refining them to programs, an axiomatization of these constraints is needed; unfortunaltely, no effective, sound and complete proof system can be constructed for initial models, and

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- Klaus Schulz. Event and Interval Structures: A Mathematical Comparison. Forschungsstelle für natürlich-sprachliche Systeme, Tübingen, Univ. null Tübingen, 1987. FNS-Bericht-87-18.
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^{st}International Workshop, IWWERT '90, volume 572 of LNCS, Tübingen, Germany, October 1990. Springer.- A. R. Schweitzer. A theory of geometrical relations. Amer. null J. null Math., 31:365410, 1909.
- Emil Sekerinski. A calculus for predicative programming. In Bird et al. [Bird et al., 1992], pages 302322.
A calculus for developing programs from specifications written as predicates that describe the relationship between the initial and final state is proposed. Such specifications are well known from the specification language Z. All elements of a simple sequential programming notation are defined in terms of predicates. Hence programs form a subset of specifications. In particular, sequential composition is defined by `demonic composition', nondeterministic choice by `demonic disjunction', and iteration by fixed points. Laws are derived which allow proving equivalence and refinement of specifications and programs by a series of steps. The weakest precondition calculus is also included. The approach is compared to the predicative programming approach of E. Hehner and to other refinement calculi.

- J. P. Seldin and J. R. Hindley, editors. To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, New York/London, 1980.
- Peter Selinger. A note on bainbridge's powerset construction, 1998.
The category Rel of sets and relations has two natural traced monoidal structures: in (Rel,+,Tr), the tensor is given by disjoint union, and in (Rel, times ,Tr') by products of sets. Already in 1976, predating the definition of traced monoidal categories by 20 years, Bainbridge has shown how to model flowcharts and networks in these two respective settings. Bainbridge has also pointed out that one can move from one setting to the other via the powerset operation. However, Bainbridge's power operation is not functorial, and in this paper we show that there is no traced monoidal embedding of (Rel,+,Tr) into (Rel,x,Tr') whose object part is given by the powerset operation. On the other hand, we show that there is such an embedding whose object part is given by the power-multiset operation.

- G. Serény. Lower level connections between representations of relation algebras, 1985. Preprint, pp. null 3.
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*Still*Preempt System Design. In ? [?, 1992], pages 200211.It is a prime tenet of most programming language design that ``higher-level'' languages are a good thing indeed the higher the level, the better. The assumption is that the higher the level of the language the more abstract the abstractions the greater the leverage provided to the programmer. The language designer usually ensures that the higher-level constructs capture his intention by completely specifying the associated semantics. A decade ago, we challenged the ``higher-level is better'' assumption. The paper in which we did this has largely been ignored. Perhaps it should have been, but we don't think so. In fact we see this apparently benign assumption as aggressively interfering with good application design. Unfortunlately, the consequences of blind adherence to this tenet are spreading in both current language proposals and larger system designs. ... Language and systems designers continue to preempt details that should be controllable by the application programmer.

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^{th}National Conf. on Artificial Intelligence. Morgan Kaufmann, 1988.- H. E. Shrobe, editor. Proc. null of AAAI-87, the 6
^{th}National Conf. on Artificial Intelligence. Morgan Kaufmann, 1988.- Roman Sikorski. Boolean Algebras. Springer, Berlin, 1969. Third edition, Second edition published by Chelsea, Bronx, New York, 1966.
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Oz is an experimental higher-order concurrent constraint programming system under development at DFKI. It combines ideas from logic and concurrent programming in a simple yet expressive language. From logic programming Oz inherits logic variables and logic data structures, which provide for a programming style where partial information about the values of variables is imposed concurrently and incrementally. A novel feature of Oz is that it accommodates higher-order programming without sacrificing that denotation and equality of variables are captured by first-order logic. Another new feature of Oz is constraint communication, a new form of asynchronous communication exploiting logic variables. Constraint communication avoids the problems of stream communication, the conventional communication mechanism employed in concurrent logic programming. Constraint communication can be seen as providing a minimal form of state fully compatible with logic data structures. Based on constraint communication and higher-order programming, Oz readily supports a variety of object-oriented programming styles including multiple inheritance.

- Gert Smolka. A feature logic with subsorts. LILOG Report 33, IWBS, IBM Deutschland, Postfach 80 08 80, 7000 Stuttgart 80, Germany, May 1988.
This paper presents a set description logic with subsorts, feature selection (the inverse of unary function application), agreement, intersection, union and complement. We define a model theoretic open world semantics and show that sorted feature structures constitute a canonical model, that is, without loss of generality subsumption and consistency of set descriptions can be considered with respect to feature structures only. We show that deciding consistency of set descriptions is an NP-complete problem.

- Gert Smolka. Feature constraint logics for unification grammars. IWBS Report 93, IWBS, IBM Deutschland, Postfach 80 08 80, 7000 Stuttgart 80, Germany, November 1989. Published in Journal of Logic Programming 12, 5187, 1992.
This paper studies feature description languages that have been developed for use in unification grammars, logic programming and knowledge representation. The distinctive notational primitive of these languages are features that can be understood as unary partial functions on a domain of abstract objects. We show that feature description languages can be captured naturally as sublanguages of first-order predicate logic with equality and show the equivalence of a loose Tarski semantics with a fixed feature graph semantics for quantifier-free constraints. For quantifier-free constraints we give a constraint solving method and show the NP-completeness of satisfiability checking. For general feature constraints with quantifiers satisfiability is shown to be undecidable. Moreover, we investigate an extension of the logic with sort predicates and set-denoting expressions called feature terms.

- Gert Smolka. Logic Programming over Polymorphically Order-Sorted Types. PhD thesis, Univ. null Kaiserslautern, FB Informatik, Kaiserslautern, Germany, May 1989.
This thesis presents the foundations for relational logic programming over polymorphically order-sorted data types. This type discipline combines the notion of parametric polymorphism, which has been developed for higher-order functional programming, with the notion of order-sorted typing, which has been developed for equational first-order specification and programming. Polymorphically order-sorted types are obtained as canonical models of a class of specifications in a suitable logic accommodating sort functions. Algorithms for constraint solving, type checking and type inference are given and proven correct.

- Gert Smolka. Residuation and guarded rules for constraint logic programming. Research Report RR-91-13, German Research Center for Artificial Intelligence (DFKI), Stuhlsatzenhausweg 3, 6600 Saarbrücken 11, Germany, May 1991. Also available as PRL Research Report 12, Digital, 85 avenue Victor Hugo, 92563 Rueil-Malmaison Cedex, France.
A major difficulty with logic programming is combinatorial explosion: since goals are solved with possibly indeterminate (i.e., branching) reductions, the resulting search trees may grow wildly. Constraint logic programming systems try to avoid combinatorial explosion by building in strong determinate (i.e., non-branching) reduction in the form of constraint simplification. In this paper we present two concepts, residuation and guarded rules, for further strengthening determinate reduction. Both concepts apply to constraint logic programming in general and yield an operational semantics that coincides with the declarative semantics. Residuation is a control strategy giving priority to determinate reductions. Guarded rules are logical consequences of programs adding otherwise unavailable determinate reductions.

- Ugo Solitro. A typed calculus based on a fragment of linear logic. Theoretical Computer Science, 68:333342, 1989.
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- Roland Soltysiak. Die Projektion affiner Strukturen über Fastkörpern mit Hilfe relationentheoretischer Methoden. PhD thesis, Univ. null Duisburg, Germany, 1980.
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- Mike Spivey. A functional theory of exceptions. Science of Computer Programming, 14:2542, 1990.
Exceptions are a feature often provided by programming languages to deal with computations which may fail. This paper argues that lazy functional programming not only makes a built-in exception mechanism unnecessary, but provides a powerful tool for developing and transforming programs that use exceptions. The basic idea is the simple one of augmenting each type with a distinguished error value; this idea is made practical for writing programs and reasoning about them through the use of higher-order functions. An advantage is that simple equational arguments can be used to reason about the programs. Throughout the paper, the problem of simplifying algebraic expressions using rewriting rules is used as a source of motivation and examples.

- John Staples and Peter J. Robinson. Unification of quantified terms. In Fasel and Keller [Fasel and Keller, 1987], pages 426450.
Unification algorithms for quantified terms are needed for the implementation of extended functional and logic programming languages, and also for the implementation of other symbolic computation systems such as theorem provers and proof editors. This paper describes and proves correct such a unification algorithm. Although discussed here in a theoretically convenient way, the algorithm is suitable for enhancement of conventional unification algorithms for free variable terms, such as are found in, for example, Prolog interpreters. The algorithm has been demonstrated by modifying a conventional Prolog interpreter so as to interpret formulas which include previously declared quantifiers.

- John Staples and Peter J. Robinson. Efficient unification of quantified terms. J. null Logic Programming, 5(2):133150, June 1988.
Conventional logic-programming languages rely fundamentally on symbolic computation with quantifier-free terms. Much theoretical logic uses the richer vocabulary of quantified terms, however. In this paper we sketch some first steps in a program of research for developing data structures and algorithms to support efficient computation directly on quantified terms. We describe a simple concept of quantified term, and efficient unification algorithms for both structure-sharing and non-structure-sharing representations of those terms. The efficiency of the approach results from the techniques used to represent terms, which enable naive substitution to implement correct substitution for quantified terms. The non-structure-sharing unification algorithm described here has been prototyped by modification of a conventional logic-programming interpreter.

- Eugene W. Stark. Compositional relational semantics for indeterminate dataflow networks. Technical report, State University of New York Stony Brook CS, 1991. A version of this paper appeared as: E. W. Stark, Compositional Relational Semantics for Indeterminate Dataflow Networks Category Theory and Computer Science, Manchester, England pp. 52-74 Volume 389 of Lecture Notes in Computer Science Springer-Verlag, 1989.
def ubar #1#1 def obar #1#1 def AUTO

**Auto**def EVDOM**EvDom**def C**C**def T**T**Given suitable categories T , C and functor F: T rightarrow C , if X, Y are objects of T , then we define an (X, Y)-*relation in*C to be a triple (R, ubar r, obar r), where R is an object of C and ubar r: R rightarrow FX and obar r: R rightarrow FY are morphisms of C . We define an algebra of relations in C , including operations of ``relabeling,'' ``sequential composition,'' ``parallel composition,'' and ``feedback,'' which correspond intuitively to ways in which processes can be composed into networks. Each of these operations is defined in terms of composition and limits in C , and we observe that any operations defined in this way are preserved under the mapping from relations in C to relations in C ' induced by a continuous functor G: C rightarrow C '. To apply the theory, we define a category AUTO of concurrent automata, and we give an operational semantics of dataflow-like networks of processes with indeterminate behaviors, in which a network is modeled as a relation in AUTO . We then define a category EVDOM of ``event domains,'' a (non-full) subcategory of the category of Scott domains and continuous maps, and we obtain a coreflection between AUTO and EVDOM . It follows, by the limit-preserving properties of coreflectors, that the denotational semantics in which dataflow networks are represented by relations in EVDOM , is ``compositional'' in the sense that the mapping from operational to denotational semantics preserves the operations on relations. Our results are in contrast to examples of Brock and Ackerman, which imply that no compositional semantics is possible in terms of set-theoretic relations.- Richard Statman. Logical relations and the typed &lgr;-calculus. Information and Control, 65:8597, 1985.
- Gh. Stefanescu. Reaction and control I. mixing additive and multiplicative network algebras. Logic Journal of the IGPL, 6(2):349368, 1998.
This paper is included in a series aiming to contribute to the algebraic theory of distributed computation. The key problem in understanding Multi-Agent Systems is to find a theory which integrates the

*reactive*part and the*control*part of such systems. To this end we use the*calculus of flownomials*. It is a polynomial-like calculus for representing flowgraphs and their behaviours. An `additive' interpretation of the calculus was intensively developed to study control flowcharts and finite automata. For instance, regular algebra and iteration theories are included in a unified presentation. On the other hand, a `multiplicative' interpretation of the calculus of flownomials was developed to study dataflow networks. %Such networks consist of a collection of concurrent asynchronous processes %which communicate by sending data over FIFO channels. The claim of this series of papers is that the mixture of the additive and multiplicative network algebras will contribute to the understanding of distributed computation. The r^ole of this first paper is to present a few motivating examples.- Gh. Stefanescu. Network Algebra. Springer, London, April 2000.
Network Algebra considers the algebraic study of networks and their behaviour. It contains general results on the algebraic theory of networks, recent results on the algebraic theory of models for parallel programs, as well as results on the algebraic theory of classical control structures. The results are presented in a unified framework of the calculus of flownomials, leading to a sound understanding of the algebraic fundamentals of the network theory. Network Algebra will be of interest to anyone interested in network theory or its applications and provides them with the results needed to put their work on a firm basis. Graduate students will also find the material within this book useful for their studies.

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Commutativity is very useful in showing the Church-Rosser property for the union of term rewriting systems. This paper studies the critical pair technique for proving commutativity of term rewriting systems. Extending the concept of critical pairs between two term rewriting systems, a sufficient condition for commutativity is proposed. Using this condition, a new sufficient condition is offered for the Church-Rosser property of left-linear term rewriting systems.

- Ralf Treinen. A new method for undecidability proofs of first order theories. Internal Report A09/90, Universität des Saarlandes, Fachbereich 14: Informatik, 6600 Saarbrücken 11, May 1990.
We claim that the reduction of Post's Correspondence Problem to the decision problem of a theory provides a useful tool for proving undecidability of first order theories given by an interpretation. The goal of this paper is to propose a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo AC and the theory of a partial lexicographic path ordering.

- Ralf Treinen. First order data types and first order logic. Interner Bericht A01/91, Universität des Saarlandes, Fachbereich 14: Informatik, 6600 Saarbrücken 11, January 1991.
This paper concerns the relation between parameterized first order data types and first order logic. Augmenting first order logic by data type definitions yields in general a strictly stronger logic than first order logic. We consider some properties of the new logic for fixed data type definitions. While our new logic always fulfills the downward Skolem-Löwenheim property, compactness is fulfilled if and only if for the given data type definition the new logic has the same expressive power than first order logic. We show that this last property is undecidable.

- Ralf Treinen. First order data types and first order logic. In Ito and Meyer [Ito and Meyer, 1991], pages 594614.
This paper concerns the relation between parameterized first order data types and first order logic. Augmenting first order logic by data type definitions yields in general a strictly stronger logic than first order logic. Some modeltheoretic properties of the new logic are investigated. While the new logic always fulfills the downward Skolem-Löwenheim property, compactness is fulfilled if and only if for the given data type definition the new logic has the same expressive power than first order logic. This last property is shown to be undecidable.

- Ralf Treinen. Modulare Datentypdefinitionen und Ihre Beziehungen zur Logik erster Stufe. PhD thesis, Universität des Saarlandes, December 1991. In german.
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Feature Constraint Systems have been proposed as a logical data structure for constraint (logic) programming. They provide a record-like view to trees by identifying subtrees by keyword rather than by position. Their atomic constraints are finer grained than in the constructor-based approach. The recently proposed sl CFT [ST:RecordsLogProg92] in fact generalizes the rational tree system of Prolog II. We propose a new feature constraint system sl EF which extends sl CFT by considering features as first class values. As a consequence, sl EF contains constraints like x[v]w where v is a variable ranging over features, while sl CFT restricts v to be a fixed feature symbol. We show that the satisfiability of conjunctions of atomic EF -constraints is NP-complete. Satisfiability of quantifier-free EF -constraints is shown to be decidable, while the exists ^* forall ^* exists ^* fragment of the first order theory is undecidable.

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The theory of computable functions on abstract data types is outlined. Methods for extending the theory to establish the scope and limits of computation on streams over abstract data types are described. Applications of these methods to the theory of synchronous concurrent algorithms are discussed

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This paper studies termination properties of rewrite systems that are typeable using intersection types. It introduces a notion of partial assignment on Currified Term Rewrite Systems, that consists of assigning intersection types to function symbols, and specifying the way in which types can be assigned to nodes and edges between nodes in the tree representation of term. Using a more liberal approach to recursion, a general scheme for recursive definitions is presented, that generalizes primitive recursion, but has full Turing-machine computational power. It will be proved that, for all systems that satisfy this scheme, every typeable term is strongly normailizable.

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This paper introduces a notion of partial type assignment on left linear applicative term rewriting systems that is based on the extension defined by Mycroft of Curry's type assignment system. The left linear applicative TRS we consider are extensions to those suggested by most functional programming languages in that they do not discriminate against the varieties of function symbols that can be used in patterns. As such there is no distinction between function symbols (such as helv append and helv plus) and constructor symbols (such as helv cons and helv succ). Terms and rewrite rules will be written as trees, and type assignment will consist of assigning types to function symbols, nodes and edges between nodes. The only constraints on this system are imposed by the relation between the type assigned to a node and those assigned to its incoming and out-going ewdges. We will show that every typeable term has a principal type, and formulate a needed and sufficient condition typeable rewrite rules should satisfy in order to gain preservance of types under rewriting. As an example we will show that the optimisation function performed after bracket abstraction is typeable. Finally we will present a type check algorithm that checks if rewrite rules are correctly typed, and finds the principal pair for typeable terms.

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*Origin Tracking*is a technique which, in the framework of first-order term rewriting systems, establishes relations between each subterm t of a normal form and a set of subterms, the*origins of t*, in the initial term. Origin tracking is based on the notion of residuals. It has been used successfully for the generation of error handlers and debuggers from algebraic specifications of programming languages. Recent experiments with the use of higher-order algebraic specifications for the definition of programming languages, reveaked a need to extend origin tracking to higher-order term rewriting systems. This extension is discussed, covering a definition and some alternatives, as well as an assessment with respect to existing specifications.- Jean van Heijenoort. From Frege to Gödel: A Source Book in Mathematical Logic, 18791931. Harvard Univ. null Press, Cambridge, MA, 1967.
- Ivo van Horebeck and Johan Lewi. Algebraic Specifications in Software Engineering, An Introduction. Springer, 1989.
- Jan van Leeuwen, editor. Handbook of Theoretical Computer Science, volume B. Elsevier Science Publishers B. V., 1990.
- Vincent van Oostrom and Femke van Raamsdonk. Comparing combinatory reduction systems and higher-order rewrite systems. In Heering et al. [Heering et al., 1993], pages 276304.
In this paper two formats of higher-order rewriting are compared: Combinatory Reduction Systems introduced by Klop and Higher-order Rewrite Systems defined by Nipkow. Although it always has been obvious that both formats are closely related to each other, up to now the exact relationship between them has not been clear. This was an unsatisfying situation since it meant that proofs for much related frameworks were given twice. We present two translations, one from Combinatory Reduction Systems into Higher-order Rewrite Systems and one vice versa, based on a detailed comparison of both formats. Since the translations are very `neat' in the sense that the rewrite relation is preserved and (almost) reflected, we can conclude that as far as theory is concerned, Combinatory Reduction Systems and Higher-order Rewrite Systems are equivalent, the only difference being that Combinatory Reduction Systems employ a more `lazy' evaluation strategy. Moreover, due to this result it is the case that some syntactic properties derived for one class also hold for the other.

- Paulo A. S. Veloso and Armando M. Haeberer. A finitary relational algebra for classical first-order logic. Bull. null Polish Acad. null Sci. null Math., Sect. null on Logic, 20(2):5262, 1991.
- Paulo A. S. Veloso and Armando M. Haeberer. A new algebra of first-order logic. In LMPS '91 [LMPS '91, 1991], pages ????
- Paulo A. S. Veloso, Armando M. Haeberer, and Gabriel A. Baum. Formal program construction within an extended calculus of binary relations. Res. Rept. MCC 19, Pontifícia Universidade Católica do Rio de Janeiro, 1992. Submitted to an special issue on Automatic Programming of the J. null Symbolic Comput.
- Paulo A.S. Veloso, Armando Martín Haeberer, and Marcelo F. Frias. Fork algebras as algebras of logic. Bull. null Symbolic Logic, pages 265266, June 1995.
- Paulo A.S. Veloso. The history of an error in the theory of representations of relation algebras. Journal of Symbolic Logic, 42, 1977.
- Paulo A. S. Veloso. Outline of a mathematical theory of general problems. Philosophia Naturalis, 21:354365, 1984.
- F. Veltman. Defaults in update semantics. J. null Philos. null Logic. to appear.
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- Yde Venema. Many-Dimensional Modal Logic. PhD thesis, Faculteit Wiskunde en Informatica, Amsterdam Univ., 1991.
- Yde Venema. Completeness through flatness. In D.M. Gabbay and Hans Jürgen Ohlbach, editors, Temporal Logic, 1
^{st}Internat. null Conf., ICTL'94, volume 827 of LNCS, pages 149164, Berlin, 1994. Springer.- Yde Venema. A crash course in arrow logic. In M. Marx and L. Polos, editors, Arrow Logic and Multi-Modal Logic, Studies in Logic, Language and Information. CSLI Publications, Stanford, 1995.
- Richard Verhoeven and Roland Backhouse. Towards tool support for program verification
*and*construction. In Wing et al. [Wing et al., 1999], pages 11281146.MathSpad is a document preparation system designed and developed by the authors and oriented towards the calculational construction of programs . PVS (Prototype Verification System) is a theorem checker developed at SRI that has been extensively used for verifying software, in particular in safety-critical applications. This paper describes how these two systems have been combined into one. We discuss the potential benefits of the combination seen from the viewpoint of someone wanting to use formal methods for the construction of computer programs, and we discuss the architecture of the combined system for the benefit of anyone wanting to investigate combining the MathSpad system with other programming tools.

- R. M. Verma. Strings, trees, and patterns. Inform. null Process. null Lett., 41:157161, March 1992.
- M. Vilain and H. Kautz. Constraint propagation algorithms for temporal reasoning. In Shrobe [Shrobe, 1988a], pages 377382.
- A. Visser and K. Vermeulen. Dynamic bracketing and discourse representation. Technical report, Dept. null of Philosophy, Utrecht Univ. null , 1995.
- Ed Voermans and Jaap van der Woude. A relational perspective on types with laws. unpublished?, July 1993.
With relational transformational programming in mind, an extension of a ``lawless'' relational theory of datatypes is proposed in order to study and manipulate quotient types within a Tarski-like calculus of relations. The extended notion of type, pertype (from partial equivalence relation), is shown to admit a complete lattice structure by constructing the order via a Galois connection. A pertyping of relations is developed and inductive pertypes generated by equations are discussed. Pertypes do occur in model theory for &lgr;-calculus but we are unaware of manipulations with inductive ``lawful'' types based on a simple relational calculus.

- Ed Voermans. Inductive Datatypes with Laws and Subtyping A Relational Model. PhD thesis, Eindhoven University of Technology, January 1999.
Inductive datatypes, datatypes where elements of the type occur as ``subcomponents'' of other elements of the type, are an essential feature of all modern programming languages. Commonly used examples of such types are for example binary trees where, a tree can have other binary trees as subtrees, or cons-lists, where the tail of a cons-list is another cons-list. A standard mathematical method for reasoning about such datatypes and programs operating with these types was developed by Malcolm. He constructed an elegant generic theory of free inductive datatypes using category theory based on the concepts of functors and initial algebras. By generic we mean parameterised by the shape of the datatype. A limitation of this theory is that it only deals with free datatypes, types without rules specifying equality of elements or restrictions on the construction of elements. In practice there are many common datatypes that are not free. For example, join-lists have associativity laws on the join operation, and height-balanced trees can not be constructed using arbitrary subtrees. Fokkinga extended Malcolm's theory to datatypes with laws, but was not able to handle restrictions on the construction of elements (subtyping). Other, set-theoretical, theories about inductive datatypes can handle both laws and subtyping but have as disadvantage that they treat laws and subtyping as dual concepts. This complicates reasoning about datatypes that combine both laws and subtyping. An example of a type combining both concepts is the AVL-tree, where different trees can be used to represent the same set of values (law), but where it is not allowed to join two arbitrary AVL-trees to construct a new valid AVL tree (restriction). The goal of this thesis is to develop a theory about inductive datatypes that can handle laws and subtyping in a uniform way. The theory should predict when (recursively defined) operations are well-defined and when they are uniquely defined. The theory should also provide a sound basis for the construction and verification of generic programs. The theory of inductive datatypes presented in this thesis was inspired by the category-theoretical approach but uses a point-free relational calculus to model both datatypes and programs. One of the main advantages of using the relational calculus is that it opens up the possibility of working with lattices where extreme solutions to equations are uniquely defined. Category theory always gives solutions ``up to isomorphism'' that are often less suitable for direct manipulation. The extreme solutions of lattice equations provide unique, canonical representations of the concepts that are being modelled. Datatypes and programs are usually specified as solutions to equations Another advantage of the lattice structures that are available when working with relations is the abundant possibility for using Galois connections. Identifying Galois connections and using their calculational properties is a recurring theme throughout the thesis. We prefer a calculational style for constructing and presenting proofs and Galois connections are a great tool for this purpose. We identify a special class of relations that can be used as representatives for datatypes. These datatypes are the elements of a complete lattice where the ordering represents (the combination of) subtyping and quotient formation. Combining these aspects in a single ordering allows us to find solutions for specifications involving both restrictions (subtyping) and laws (quotients). Combining these features is often difficult in other formalisms for datatypes. This ordering is a vital tool for achieving our goal of a uniform treatment of laws and subtyping. Our datatype construction methods are inspired by categorical datatype theories and we will construct a category where objects and arrows are relations. Categorical notions like functors, natural transformations and F-algebras lead to relational constructions that are useful for the construction of datatypes and programs. A variant of F-algebras is used for the introduction of inductive datatypes and structural recursion. An important aspect of datatype construction is simulation, implementing one datatype using another datatype. The notion of simulation can easily be formulated in our theory. Inductive types that simulate each other form equivalence classes. We prove the remarkable result that every equivalence class contains one special representative.The special representatives form a complete lattice, using our special ordering of datatypes. The elements of the lattice represent all inductively defined datatypes for a given induction structure. Using this lattice, we can describe inductive datatypes with both laws and restrictions as an extreme fixpoint. We will give an equivalent characterization of the extreme fixpoint using a Galois connection. This Galois connection, which defines a closure operation, turns out to be very convenient for proving properties of inductive datatypes. Laws and restrictions can be specified with equations, which can be combined to a single specification of the datatype. Not only are datatypes described as solutions of equations, but recursively defined operations on these inductive datatypes are also specified as solutions of equations. We will show that a large class of ``recursion structure'' equations for operations on inductive datatypes have at most one solution, so they are suitable as a specification. Another subject investigated in this thesis is conditions under which parameterisation of inductive datatypes with laws and restrictions is possible. Here we demonstrate that, if the law and restriction equations satisfy certain naturality (``polymorphy'') criteria, parameterisation is possible.

- N. D. Volkov. The transition from a relation algebra to a halmos algebra. In Algebra and Discrete Mathematics: Theoretical Foundations of Software. Latv. null Gos. null Univ., Riga, 1986. (Russian).
- B. von Karger and R. Berghammer. A relational model for temporal logic. Logic Journal of the IGPL, 6(2):157173, 1998.
We use Tarski's relational calculus to construct a model of linear temporal logic. Both discrete and dense time are covered and we obtain denotational domains for a large variety of reactive systems.

- David von Oheimb and Thomas F. Gritzner. RALL: Machine-supported proofs for relation algebra. In William McCune, editor, Conference on Automated Deduction CADE-14, LNCS 1249, pages 380394. Springer-Verlag, Berlin, 1997.
We present a theorem proving system for abstract relation algebra called RALL (= Relation-Algebraic Language and Logic), based on the generic theorem prover Isabelle. On the one hand, the system is an advanced case study for Isabelle/HOL, and on the other hand, a quite mature proof assistant for research on the relational calculus. RALL is able to deal with the full language of heterogeneous relation algebra including higher-order operators and domain constructions, and checks the type-correctness of all formulas involved. It offers both an interactive proof facility, with special support for substitutions and estimations, and an experimental automatic prover. The automatic proof method exploits an isomorphism between relation-algebraic and predicate-logical formulas, relying on the classical universal-algebraic concepts of atom structures and complex algebras.

- Andrei Voronkov, editor. Logic Programming First Russian Conference on Logic Programming, Irkutsk, Russia, September 1990; Second Russian Conference on Logic Programming, St. Petersburg, Russia, September 1991; Proceedings, volume 592 of LNAI. Springer, 1992.
- Andrei Voronkov, editor. Logic Programming and Automated Reasoning 4th International Conference LPAR '93, St. Petersburg, Russia, July 1993, volume 698 of LNAI. Springer-Verlag, 1993.
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This paper presents an algebraic and categorical approachto the mathematical modelling of imparative programming languages. In particular we model languages with block structure, records and variants, user definable recursive types, and pointers, etc., and with ``control constructs'' such as primitive recursion (generalized to recursive types), while-do, if-then-else, and assignment. In our earlier papers on this subject ([4,5,6]) we showed how data types and operations can be defined in an algebraic framework. In this paper we present a more mathematically sophisticated version of that framework, and we show how it can be used to provide a new approach to languages that have block structure together with objects, such as pointers, which are dynamically declared and may persist outside the block in which they are declared. The main new mathematical concept, and the key to the development, is the concept of an EDHT-category which is an extension of the DHT-symmetric categories introduced by Hoehnke [13] as a categorical framework for partial algebras.

- I. Walukiewicz. Completeness of Kozens axiomatization of the propositional &mgr;-calculus. In Annual Sympos. null on Logic in Computer Science. IEEE Computer Society Press, 1995.
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Incremental modification is a fundamental mechanism not only in software systems, but also in physical and mathematical systems. inheritance owes its importance in large measure to its flexibility as a discrete incremental modification mechanism. Four increasingly permissive properties of incremental modification realizable by inheritance are examined: behaviour compatibility, signature compatibility, name compatibility, and cancellation. inheritance for entities with finite sets of attributes is defined and characterized as incremental modification with deferred binding of self-reference. Types defined as predicates for type checking are contrasted with classes defined as templates for object generation. Mathematical, operational, and conceptual models of inheritance are then examined in detail, leading to a discussion of algebraic models of behavioral compatibility, horizontal and vertical signature modification, algorithmically defined name modification, additive and subtractive exceptions, abstract inheritance networks, and parametric polymorphism. Liketypes are defined as a symmetrical general form of incremental modification that provide a framework for modelling similarity. The combination of safe behaviorally compatible changes and less safe radical incremental changes in a single programming language is considered.

- Alfred North Whitehead and Bertrand Russell. Principia Mathematica, Volume I. Cambridge Univ. null Press, Cambridge, England, 1910.
- Alfred North Whitehead and Bertrand Russell. Principia Mathematica, Volume II. Cambridge Univ. null Press, Cambridge, England, 1912.
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- Michael Winter. A relation algebraic approach to interaction categories. Information Sciences, 119(34):301314, December 1999.
Samson Abramsky introduced in the concept of Interaction Categories. His motivating example of a synchronous Interaction Category is the category

**SProc**of synchronisation trees between concurrent system specifications. In this paper we show that this category is a unitary division allegory in the sense of Peter Freyd. Furthermore, we want to introduce the notion of time-extended allegories, i.e., a theory for relations extended in time. Some properties of this kind of allegories and strongly guarded functors are proven and the connections to Interaction Categories are discussed.- Martin Wirsing. Algebraic specifications. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science B, pages 675788. Elsevier, 1990.
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Inspired by the work of S. Kaplan about positive/negative conditional rewriting, we investigate initial semantics for algebraic specifications with Gentzen-formulas. Since the standard initial approach is limited to conditional equations (i.e. positive Horn-formulas), the notion of semi-initial and quasi-initial algebras is introduced and it is shown that all specifications with (positive) Gentzen-formulas admit quasi-initial models. The whole approach is generalized to the parametric case where quasi-initiality generalizes to quasi-freeness. Since quasi-free objects need not be isomorphic, the persistence requirement is added to obtain a

*unique*semantics for many interesting practical examples. Unique persistent quasi-free semantics can be described as a free construction when the parameter category is restricted to injective homomorphisms. An example which does not admit a correct initial semantics but a correct unique persistent quasi-initial semantics demonstrates that the concepts introduced in this paper might be of some importance w.r.t. practical application.- U. Wooyenaka. On postulate sets for relation algebras. Notices Amer. null Math. null Soc., 6:534535, 1959.
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Hindley/Milner-style polymorphism is a simple, natural, and flexible type discipline for functional languages, but incorporating imperative extensions is difficult. We present a new technique for typing references in the presence of polymorphism by inferring a concise summary of each expression's allocation behavioura

*type effect*. A simple technique for proving soundness with respect to a reduction semantics demonstrates that the type system prevents type errors. By establishing that the system corresponds to an alternate system better suited to implementation, we obtain an algorithm to perform type and effect inference.- J. Würtz. Unifying cycles. Research Report RR-92-22, German Research Center for Artificial Intelligence (DFKI), Stuhlsatzenhausweg 3, 6600 Saarbrücken 11, Germany, March 1992.
Two-literal clauses of the form L leftarrowR occur quite frequently in logic programs, deductive databases, anddisguised as an equationin term rewriting systems. These clauses define a cycle if the atoms L and R are weakly unifiable, i.e., if L unifies with a new variant of R. The obvious problem with cycles is to control the number of iterations through the cycle. In this paper we consider the cycle unification problem of unifying two literals G and F modulo a cycle. We review the state of the art of cycle unification and give new results for a special type of cycles called unifying cycles, i.e., cycles L leftarrowR for which there exists a substitution &sgr; such that &sgr; L = &sgr; R . Altogether, these results show how the deductive process can be efficiently controlled for special classes of cycles without losing completeness.

- J. Würtz. Unifying cycles. In B. Neumann, editor, Proceedings of the European Conference on Artificial Intelligence, pages 6064. Wiley, August 1992.
Two-literal clauses of the form L leftarrowR occur quite frequently in logic programs, deductive databases, and disguised as an equation in term rewriting systems. These clauses define a cycle if the atoms L and R are weakly unifiable, i.e., if L unifies with a new variant of R. The obvious problem with cycles is to control the number of iterations through the cycle. In this paper we consider the problem of unifying two literals G and F modulo a cycle. We review the state of the art of cycle unification and give new results for a special type of cycles called unifying cycles, i.e., cycles L leftarrowR for which there exists a substitution &sgr; such that &sgr; L = &sgr; R . Altogether, these results show how the deductive process can be efficiently controlled for special classes of cycles without losing completeness.

- S. Ben Yahia, H. Ounalli, and Ali Jaoua. An extension of classical functional dependency: dynamic fuzzy functional dependency. Information Sciences, 119(34):219234, December 1999.
Relational data model has constituted an incontestable success in database history. In this context, a lot of attention has been paid to functional dependencies due to their paramount importance in the design of relational database. For about fifteen years, several attempts to formalize (soft) real world constraints imposed on the data has been made, leading to the emergence of the concept of fuzzy functional dependency. In this paper, an overview of the different proposals of the fuzzy functional dependency is presented. A new extension of classical functional dependency based on the Lukasiewicz implication is presented and called dynamic fuzzy functional dependency. The associated axiomatic system is introduced and proved to be sound.

- D. Yetter. Quantales and (noncommutative) linear logic. Journal of Symbolic Logic, 55:4164, 1990.
- Hirofumi Yokouchi. Church-rosser theorem for a rewriting system on categorical combinators. Theoretical Computer Science, 65:271290, 1989.
This paper develops the Church-Rosser theorem for the rewriting system CCL&bgr; on type-free categorical combinators introduced by Curien. The system CCL&bgr; is not confluent. However we show that there are various sets D of categorical combinator terms such that each D satisfies the following two conditions: (1) D is closed under reduction by CCL&bgr;; (2) CCL&bgr; is confluent on D. Moreover we examine the relation among these sets.

- Maria Zamfir. Initial algebra semantics and concurrency. In Main et al. [Main et al., 1987], pages 528549.
The purpose of this paper is to show that

*initial algebra semantics*has an immediate and useful application in the area of communicating computing systems. The major technical feature is a category of continuous many-sorted algebras called*parallel-nondeterministic*algebras. In this setting parallel and nondeterministic behaviour of communicating computing systems can be rigorously formulated as sequences of rewritings on abstract objects called*parallel-nondeterministic terms*or*diamonds*. It is shown that diamonds are free in teh category of continuous*parallel-nondeterministic*algebras. (To demonstrate this fact, some results concerning categories of continuous algebras, which can be found in the ork of the ADJ group, are presented in a self-contained form.) Nondeterminism and parallelism are modeled explicitely by introducing a*choice*operator and a*parallel*operator, respectively. In a companion paper [10]*flow nets*are introduced to describe parallel and nondeterministic behaviours of computing systems that communicate with each other, just as conventional flowcharts are used to describe sequential computations. In a continuous parallel-nondeterministic algebra a flow net is represented by its unfoldment the solution of a finite system of recursive equations.- Jay Zeman. Peirce on the algebra of logic: Some comments on Houser. Trans. null of the Charles S. null Peirce Society, 25:5156, 1989.
- Kaizhong Zhang, Dennis Shasha, and Jason Tsong-Li Wang. Fast serial and parallel algorithms for approximate tree matching with VLDC's. In Apostolico et al. [Apostolico et al., 1992], pages 151161. (extended abstract).
Ordered, labeled trees are trees in which each node has a label and the left-to-right order of its children (if it has any) is fixed. Suppose we define the distance between two ordered trees to be the weighted number (the user chooses the weighting) of edit operations (insert, delet, and relabel) to transform one tree to the other. This paper presents algorithms to perform approximate matching for such trees with variable-length don't cares (VLDC's). As far as we know, these are the first algorithms ever to be presented.

- Guo-Qiang Zhang. Some monoidal closed categories of stable domains and event structures. Mathematical Structures in Computer Science, 1992.
This paper introduces the following new constructions on stable domains and event structures: the tensor product, the linear function space, and the exponential. It results in a monoidal closed category of dI-domains as well as one of stable event structures which can be used to interpret intuitionistic linear logic. Finally, the usefulness of the category of stable event structures for modeling concurrency and its relation to other models is discussed.

- M. Zhu, N. K. Loh, and P. Siy. Towards the minimum set of primitive relations in temporal logic. Inform. null Process. null Lett., 26:121126, 1987/88.
- Hans Zierer, Gunther Schmidt, and Rudolf Berghammer. An interactive graphical manipulation system for higher objects based on relational algebra. In Tinhofer and Schmidt [Tinhofer and Schmidt, 1986], pages 6881.
- Hans Zierer. Relationale Semantik. Master's thesis, Techn. Univ. München, 1983.
- Hans Zierer. Programmierung mit Funktionsobjekten: Konstruktive Erzeugung semantischer Bereiche und Anwendung auf die partielle Auswertung. Dissertation, Technische Univ. null München, Fakultät für Informatik, 1988. Report TUM-I8803.
- Hans Zierer. Relation-algebraic domain constructions. Theoretical Computer Science, 87:163188, 1991.
Aiming at a constructive approach to domain theory, the definition of domains with deflations is presented. This class of domains is closed with respect to the common domain constructions. Another concern of this paper is to provide a formal calculus for a uniform algebraic treatment of order theoretic and functional aspects of domain theory. The abstract relation algebra turns out to be an appropriate technical means for the characterization and construction of domains. As partial functions present no problem in relation algebra, domains need not contain an additional bottom -element and functions between domains are generally not total. Using symmetric quotients the relation algebraic approach is extended to cope with higher order functions.

- J.I. Zucker. Transformations of normal and inverted function tables. Formal Aspects of Computing, 1996. to appear (Also as CRL Report No. null 291, August 1994, McMaster University, Communications Research Laboratory and Telecommunications Research Inst. null of Ontario.).