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An introduction to substructural logics
, 2000
"... Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1 ..."
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Cited by 182 (17 self)
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Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1
Coalgebraic Logic
 Annals of Pure and Applied Logic
, 1999
"... We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The ..."
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Cited by 108 (0 self)
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We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a frangment of infinitary modal logic which contains the characterizing formulas and is closed under infinitary conjunction and an operation called 4. This fragment generalizes to a wide range of coalgebraic logics. We then apply the characterization result to get representation theorems for final coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of the final coalgebra. Keywords: infinitary modal logic, characterization theorem, functor on sets, coalgebra, greatest fixed point. 1 Intr...
Models of Sharing Graphs: A Categorical Semantics of let and letrec
, 1997
"... To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sha ..."
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Cited by 75 (9 self)
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To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is firstorder acyclic sharing graphs represented by letsyntax, and others are extensions with higherorder constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced
On the Origins of Bisimulation and Coinduction
"... The origins of bisimulation and bisimilarity are examined, in the three fields where they have been ..."
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Cited by 61 (0 self)
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The origins of bisimulation and bisimilarity are examined, in the three fields where they have been
Dynamic Epistemic Logic
 Logic, Language, and Information 2, Stanford University, CSLI Publication
, 1997
"... This paper is the result of combining two traditions in formal logic: epistemic logic and dynamic semantics. Dynamic semantics is a branch of formal semantics that is concerned with ..."
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Cited by 43 (1 self)
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This paper is the result of combining two traditions in formal logic: epistemic logic and dynamic semantics. Dynamic semantics is a branch of formal semantics that is concerned with
Dynamic Bits And Pieces
, 1997
"... Arrow Logic remains PSPACEcomplete. Further arrow axioms can easily lead to 36 undecidability. (In private correspondence, Marx has also announced EXPTIME complexity for the original Guarded Fragment, via a reduction to CRS over 'locally cube' models.) Marx and Venema 1996 is a systemat ..."
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Cited by 29 (3 self)
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Arrow Logic remains PSPACEcomplete. Further arrow axioms can easily lead to 36 undecidability. (In private correspondence, Marx has also announced EXPTIME complexity for the original Guarded Fragment, via a reduction to CRS over 'locally cube' models.) Marx and Venema 1996 is a systematic stateoftheart presentation of manydimensional modal logic, including bridges with algebraic logic, as well as many key techniques for dynamic logic, broadly conceived. Ter Meulen 1995 proposes a concise framework for temporal representation in natural language that may be viewed as an alternative dynamification of temporal logic, using an extra, intermediate level of representation. Successive formulas algorithmically generate successive 'dynamic aspect trees', for which there is a notion of 'succesful embedding' into standard temporal models. Valid inference can then be defined as verification of the conclusion by any succesful embedding for the DAT of the premise sequence. This alternative d...
Turing Machines, Transition Systems, and Interaction
 Information and Computation
, 2004
"... We present Persistent Turing Machines (PTMs), a new way of interpreting Turingmachine computation, one that is both interactive and persistent. A PTM repeatedly receives an input token from the environment, computes for a while, and then outputs the result. Moreover, it can \remember" its p ..."
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Cited by 29 (4 self)
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We present Persistent Turing Machines (PTMs), a new way of interpreting Turingmachine computation, one that is both interactive and persistent. A PTM repeatedly receives an input token from the environment, computes for a while, and then outputs the result. Moreover, it can \remember" its previous state (worktape contents) upon commencing a new computation. We show that the class of PTMs is isomorphic to a very general class of eective transition systems, thereby allowing one to view PTMs as transition systems \in disguise." The persistent stream language (PSL) of a PTM is a coinductively dened set of interaction streams : innite sequences of pairs of the form (w i ; w o ), recording, for each interaction with the environment, the input token received by the PTM and the corresponding output token. We dene an innite hierarchy of successively ner equivalences for PTMs over nite interactionstream prexes and show that the limit of this hierarchy does not coincide with PSLequivalence. The presence of this \gap" can be attributed to the fact that the transition systems corresponding to PTM computations naturally exhibit unbounded nondeterminism. We also consider amnesic PTMs, where each new computation begins with a blank work tape, and a corresponding notion of equivalence based on amnesic stream languages (ASLs). We show that the class of ASLs is strictly contained in the class of PSLs. Amnesic stream languages are representative of the classical view of Turingmachine computation. One may consequently conclude that, in a streambased setting, the extension of the Turingmachine model with persistence is a nontrivial one, and provides a formal foundation for reasoning about programming concepts such as objects with static elds. We additional...
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
Proof Methods for Corecursive Programs
 Fundamenta Informaticae Special Issue on Program Transformation
, 1999
"... This article is a tutorial on four methods for proving properties of corecursive programs: fixpoint induction, the approximation lemma, coinduction, and fusion. ..."
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Cited by 27 (8 self)
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This article is a tutorial on four methods for proving properties of corecursive programs: fixpoint induction, the approximation lemma, coinduction, and fusion.
Distributive laws for the coinductive solution of recursive equations
 Information and Computation
"... This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributi ..."
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Cited by 23 (1 self)
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This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributive laws. 1