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On transition systems and nonwellfounded sets
"... (Labelled) transition systems are relatively common in theoretical computer science, chie y as vehicles for operational semantics. The rst part of this paper constructs a hierarchy of canonical transition systems and associated maps, aiming to give a strongly extensional theory of transition systems ..."
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spaces. The resulting hierarchy hasvery rich combinatorial (and topological) structure, and a lot of the rst part of the paper is devoted to its study. We also discuss xed points in this framework. This kind of study of transition systems is very closely connected to nonwellfounded set theory
Classification of nonwellfounded sets and an application
"... In set theory, the foundation axiom (or regularity)(F) says that the relation ∈ is wellfounded, that is, there is no infinite descending ∈sequence · · · ∈ x2 ∈ x1 ∈ x0. If we identify ∈ in a set with ← in a graph, the set is identified with a graph. The set 1 is 1 = {φ}, that is, φ ∈ 1. It corr ..."
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refusing (F). In 1988, Aczel([A]) introdeced nonwellfounded ZFC − +(AF) set theory, and studied various kinds of antifoundation axioms and hence the associated nonwellfounded set theories, which include Aczel set theory, Scott set theory, Finsler set theory and Boffa set theory.
Infinitarily Definable NonWellFounded Sets
"... This paper is my second approach to set theory conceived as a maximal consistent theory of set comprehension. The principle innovation in this version is to simplify the syntax by removing ..."
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This paper is my second approach to set theory conceived as a maximal consistent theory of set comprehension. The principle innovation in this version is to simplify the syntax by removing
On Modal µCalculus and NonWellFounded Set Theory
"... A finitary characterization for nonwellfounded sets with finite transitive closure is established in terms of modal µcalculus. This result generalizes the standard approach in the literature where a finitary characterization is only provided for wellfounded sets with finite transitive closure ..."
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A finitary characterization for nonwellfounded sets with finite transitive closure is established in terms of modal µcalculus. This result generalizes the standard approach in the literature where a finitary characterization is only provided for wellfounded sets with finite transitive
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 28 (1 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
Models of nonwellfounded sets via an indexed final coalgebra theorem
 J. Symbolic Logic
"... Abstract The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on finitely complete and cocomplete categories. As an instance of this result, we build the final coalge ..."
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coalgebra for the powerclass functor, in the context of a Heyting pretopos with a class of small maps. This is then proved to provide models for various nonwellfounded set theories, depending on the chosen axiomatisation for the class of small maps.
Modal Logic and nonwellfounded Set Theory: translation, bisimulation, interpolation.
, 1998
"... ..."
Models of nonwellfounded sets via an indexed final coalgebra theorem
, 2006
"... Abstract The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on categories with finite limits and colimits. This is then put to use in the context of a Heyting pretop ..."
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pretopos with a class of small maps, in order to build the final coalgebra for the Ps functor. This is then proved to provide a model for various set theories with the AntiFoundation Axiom, depending on the chosen axiomatisation for the class of small maps. 1
A Nonwellfounded Sets Semantics for Observation Congruence over Full CCS
, 1994
"... . In the present paper we study the semantics of a minor variant of Milner's CCS process calculus. We use a compact semantic domain for processes which has been described by Aczel in [P. Aczel. Nonwellfounded Sets. Stanford University, 1988.] On the basis of the operational semantics of CC ..."
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. In the present paper we study the semantics of a minor variant of Milner's CCS process calculus. We use a compact semantic domain for processes which has been described by Aczel in [P. Aczel. Nonwellfounded Sets. Stanford University, 1988.] On the basis of the operational semantics
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