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On the Foundations of Final Semantics: NonStandard Sets, Metric Spaces, Partial Orders
 PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are ..."
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Cited by 48 (10 self)
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Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are also used here for giving a new comprehensive presentation of the (still) nonstandard theory of nonwellfounded sets (as nonstandard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages  concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single `coalgebraic' definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.
SetTheoretical and Other Elementary Models of the lambdacalculus
 Theoretical Computer Science
, 1993
"... Part 1 of this paper is the previously unpublished 1972 memorandum [43], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary settheoretical model of the ficalculus. F ..."
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Cited by 40 (0 self)
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Part 1 of this paper is the previously unpublished 1972 memorandum [43], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary settheoretical model of the ficalculus. Functions are modeled in a similar way to that normally employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the fijcalculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those nontrivial models of the fijcalculus which are continuous complete lattices. The second part begins with a brief discussion of models of the calculus in set theories with antifoundation axioms. Next the model of the fi calculus of Part 1 and also the closely relatedbut different!models of Scott [53, 54] and of Engeler [21, 22] are reviewed....
On the Foundations of Final Coalgebra Semantics: nonwellfounded sets, partial orders, metric spaces
, 1998
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A Fast Bisimulation Algorithm
 PROC. OF INT. CONFERENCE ON COMPUTER AIDED VERIFICATION (CAV’01), VOLUME 2102 OF LNCS
, 2000
"... In this paper we propose an efficient algorithmic solution to the problem of determining a Bisimulation Relation on a finite structure. ..."
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Cited by 29 (15 self)
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In this paper we propose an efficient algorithmic solution to the problem of determining a Bisimulation Relation on a finite structure.
A Cook’s tour of the finitary nonwellfounded sets
 Invited Lecture at BCTCS
, 1988
"... It is a great pleasure to contribute this paper to a birthday volume for Dov. Dov and I arrived at imperial College at around the same time, and soon he, Tom Maibaum and I were embarked on a joint project, the Handbook of Logic in Computer Science. We obtained a generous advance from Oxford Universi ..."
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Cited by 21 (0 self)
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It is a great pleasure to contribute this paper to a birthday volume for Dov. Dov and I arrived at imperial College at around the same time, and soon he, Tom Maibaum and I were embarked on a joint project, the Handbook of Logic in Computer Science. We obtained a generous advance from Oxford University Press, and a grant from the Alvey Programme, which allowed us to develop the Handbook in a rather unique, interactive way. We held regular meetings at Cosener’s House in Abingdon (a facility run by what was then the U.K. Science and Engineering Research Council), at which contributors would present their ideas and draft material for their chapters for discussion and criticism. Ideas for new chapters and the balance of the volumes were also discussed. Those were a remarkable series of meetings — a veritable education in themselves. I must confess that during this long process, I did occasionally wonder if it would ever terminate.... But the record shows that five handsome volumes were produced [6]. Moreover, I believe that the Handbook has proved to be a really valuable resource for students and researchers. It has been used as the basis for a number of summer schools. Many of the chapters have become standard references for their topics. In a field with rapidly changing fashions, most of the material has stood the test of time — thus
From Bisimulation to Simulation  Coarsest Partition Problems
 J. Automated Reasoning
, 2002
"... The notions of bisimulation and simulation are used for graph reduction and are widely employed in many areas: Modal Logic, Concurrency Theory, Set Theory, Formal Verification, etc. In particular, in the context of Formal Verification they are used to tackle the socalled stateexplosion problem. ..."
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Cited by 18 (1 self)
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The notions of bisimulation and simulation are used for graph reduction and are widely employed in many areas: Modal Logic, Concurrency Theory, Set Theory, Formal Verification, etc. In particular, in the context of Formal Verification they are used to tackle the socalled stateexplosion problem.
Final Semantics for a Higher Order Concurrent Language
 CAAP'96 Conference Proceedings, H.Kirchner ed., Springer LNCS
, 1995
"... We show that adequate semantics can be provided for imperative higher order concurrent languages simply using syntactical final coalgebras. In particular we investigate and compare various behavioural equivalences on higher order processes defined by finality using hypersets and c.m.s.'s. Correspond ..."
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Cited by 15 (11 self)
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We show that adequate semantics can be provided for imperative higher order concurrent languages simply using syntactical final coalgebras. In particular we investigate and compare various behavioural equivalences on higher order processes defined by finality using hypersets and c.m.s.'s. Correspondingly, we derive various coinduction and mixed inductioncoinduction proof principles for establishing these equivalences.
From Settheoretic Coinduction to Coalgebraic Coinduction: some results, some problems
, 1999
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On the Foundations of Corecursion
 Logic Journal of the IGPL
, 1997
"... We consider foundational questions related to the definition of functions by corecursion. This method is especially suited to functions into the greatest fixed point of some monotone operator, and it is most applicable in the context of nonwellfounded sets. We review the work on the Special Final C ..."
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Cited by 13 (1 self)
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We consider foundational questions related to the definition of functions by corecursion. This method is especially suited to functions into the greatest fixed point of some monotone operator, and it is most applicable in the context of nonwellfounded sets. We review the work on the Special Final Coalgebra Theorem of Aczel [1] and the Corecursion Theorem of Barwise and Moss [4]. We offer a condition weaker than Aczel's condition of uniformity on maps, and then we prove a result relating the operators satisfying the new condition to the smooth operators of [4]. Keywords: corecursion, coalgebra, operator on sets 1 Introduction By a stream of natural numbers we mean a pair hn; si where n 2 N and s is again a stream of natural numbers. Let f : N ! N . Consider the following function which purports to define a function from N into the streams: iter f (n) = hn; iter f f(n)i (1.1) For each n, iter f (n) is a stream, so iter f itself is a function from numbers to streams. This is an examp...