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66
Parametric Corecursion
 Theoretical Computer Science
"... This paper gives a treatment of substitution for "parametric" objects in final coalgebras, and also presents principles of definition by corecursion for such objects. The substitution results are coalgebraic versions of wellknown consequences of initiality, and the work on corecursion is a gener ..."
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Cited by 14 (3 self)
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This paper gives a treatment of substitution for "parametric" objects in final coalgebras, and also presents principles of definition by corecursion for such objects. The substitution results are coalgebraic versions of wellknown consequences of initiality, and the work on corecursion is a general formulation which allows one to specify elements of final coalgebras using systems of equations. One source of our results is the theory of hypersets, and at the end of this paper we sketch a development of that theory which calls upon the general work of this paper to a very large extent and particular facts of elementary set theory to a much smaller extent. 1 Introduction This paper has two overall goals. The first is a general theory of substitution and corecursion having to do with final coalgebras. To give an example of the kind of phenomena we have in mind, consider any set S and form the functor F on sets defined by Fa = S \Theta a \Theta a. F is defined on functions in the u...
Modal Logic: A Semantic Perspective
 ETHICS
, 1988
"... This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimul ..."
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Cited by 13 (1 self)
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This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimulations) which underly it. We introduce the syntax and semantics of basic modal logic, discuss its expressivity at the level of models, examine its computational properties, and then consider what it can say at the level of frames. We then move beyond the basic modal language, examine the kinds of expressivity offered by a number of richer modal logics, and try to pin down what it is that makes them all ‘modal’. We conclude by discussing an example which brings many of the ideas we discuss into play: games.
Coinductive Models of Finite Computing Agents
 Electronic Notes in Theoretical Computer Science
, 1999
"... This paper explores the role of coinductive methods in modeling nite interactive computing agents. The computational extension of computing agents from algorithms to interaction parallels the mathematical extension of set theory and algebra from inductive to coinductive models. Maximal xed points ..."
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Cited by 13 (6 self)
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This paper explores the role of coinductive methods in modeling nite interactive computing agents. The computational extension of computing agents from algorithms to interaction parallels the mathematical extension of set theory and algebra from inductive to coinductive models. Maximal xed points are shown to play a role in models of observation that parallels minimal xed points in inductive mathematics. The impact of interactive (coinductive) models on Church's thesis and the connection between incompleteness and greater expressiveness are examined. A nal section shows that actual software systems are interactive rather than algorithmic. Coinductive models could become as important as inductive models for software technology as computer applications become increasingly interactive.
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 12 (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
Interpolation, Preservation, and Pebble Games
 Journal of Symbolic Logic
, 1996
"... Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focuse ..."
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Cited by 12 (5 self)
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Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focused on L!1! and its fragments (see e.g. Keisler [19]), since countable formulas seemed best behaved. The past decade has seen a renewed interest in L1! and its finite variable fragments L (k) 1! (for 2 k ! !) and the modal fragment L \Pi 1! (see e.g. Ebbinghaus and Flum [17] on the former and Barwise and Moss [9] on the latter), due to various connections with topics in computer science. These logics form a hierarchy of increasingly powerful logics L \Pi 1! ae L (2) 1! ae L (3) 1! ae : : : ae L (k) 1! ae : : : ae L1! ; with each of these inclusions being proper. Moreover, there is a useful and elegant algebraic characterization of equivalence in L in each of these logics L, from b...
Proof Methods for Structured Corecursive Programs
, 1999
"... Corecursive programs produce values of greatest fixpoint types, in contrast to recursive programs, which consume values of least fixpoint types. There are a number of widely used methods for proving properties of corecursive programs, including fixpoint induction, the take lemma, and coinduction. Ho ..."
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Cited by 12 (4 self)
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Corecursive programs produce values of greatest fixpoint types, in contrast to recursive programs, which consume values of least fixpoint types. There are a number of widely used methods for proving properties of corecursive programs, including fixpoint induction, the take lemma, and coinduction. However, these methods are all rather lowlevel, in the sense that they do not exploit the common structure that is often present in corecursive definitions. We argue for a more structured approach to proving properties of corecursive programs. In particular, we show that by writing corecursive programs using an operator called unfold that encapsulates a common pattern of corecursive de nition, we can then use highlevel algebraic properties of this operator to conduct proofs in a purely calculational style that avoids the use of either induction or coinduction.
Sum and Product in Dynamic Epistemic Logic
, 2007
"... The SumandProduct riddle was first published in the reference H. Freudenthal (1969, Nieuw Archief voor Wiskunde 3, 152) [6]. We provide an overview on the history of the dissemination of this riddle through the academic and puzzlemath community. This includes some references to precursors of the ..."
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Cited by 10 (5 self)
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The SumandProduct riddle was first published in the reference H. Freudenthal (1969, Nieuw Archief voor Wiskunde 3, 152) [6]. We provide an overview on the history of the dissemination of this riddle through the academic and puzzlemath community. This includes some references to precursors of the riddle, that were previously (as far as we know) unknown. We then model the SumandProduct riddle in a modal logic called public announcement logic. This logic contains operators for knowledge, but also operators for the informational consequences of public announcements. The logic is interpreted on multiagent Kripke models. The information in the riddle can be represented in the traditional way by number pairs, so that Sum knows their sum and Product their product, but also as an interpreted system, so that Sum and Product at least know their local state. We show that the different representations are isomorphic. We also provide characteristic formulas of the initial epistemic state of the riddle. We analyse one of the announcements towards the solution of the riddle as a socalled unsuccessful update: a formula that becomes false because it is announced. The riddle is then implemented and its solution verified in the epistemic model checker DEMO. This can be done, we think, surprisingly elegantly. The results are compared with other work in epistemic model checking and the complexity is experimentally investigated for several representations and parameter settings.
Typical ambiguity: trying to have your cake and eat it too
 the proceedings of the conference Russell 2001
"... Would ye both eat your cake and have your cake? ..."
STS: A Structural Theory of Sets
 Logic Journal of the IGPL
, 1999
"... and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of ..."
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Cited by 7 (0 self)
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and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of
The AntiFoundation Axiom In Constructive Set Theories
 Stanford University Press
, 2003
"... . The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial inte ..."
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Cited by 6 (5 self)
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. The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called nonwellfounded sets, or hypersets (cf. [17], [5]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [7]). Instead of the Foundation Axiom these set theories adopt the socalled AntiFoundation Axiom, AFA, which gives rise to a rich universe of sets. AFA provides an elegant tool for modeling all sorts of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and pr...