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Universal coalgebra: a theory of systems
, 2000
"... In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certa ..."
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In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (Non-Well-Founded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of non-wellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken
On the origins of bisimulation and coinduction.
- ACM Trans. Program. Lang. Syst.,
, 2009
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Are Types needed for Natural Language?
, 1996
"... Logic, due to the paradoxes, is absent from the type free -calculus. This makes such a calculus an unsuitable device for Natural Language Semantics. Moreover, the problems that arise from mixing the type free -calculus with logic lead to type theory and hence formalisations of Natural Language were ..."
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Logic, due to the paradoxes, is absent from the type free -calculus. This makes such a calculus an unsuitable device for Natural Language Semantics. Moreover, the problems that arise from mixing the type free -calculus with logic lead to type theory and hence formalisations of Natural Language were carried out in a strictly typed framework. It was shown however, that strict type theory cannot capture the self-referential nature of language ([Parsons 79], [Chierchia, Turner 88] and [Kamareddine, Klein 93]) and hence other approaches were needed. For example, the approach carried out by Parsons is based on creating a notion of floating types which can be instantiated to particular instances of types whereas the approaches of Chierchia, Turner and Kamareddine, Klein are based on a type free framework. In this paper, we will embed the typing system of [Parsons 79] into a version of the one proposed in [Kamareddine, Klein 93] giving an interpretation of Parsons' system in a type free theory...
On the Origins of Bisimulation and
"... Bisimulation and bisimilarity are coinductive notions, and as such are intimately related to fixed points, in particular greatest fixed points. Therefore also the appearance of coinduction and fixed points is discussed, though in this case only within Computer Science. The paper ends with some histo ..."
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Bisimulation and bisimilarity are coinductive notions, and as such are intimately related to fixed points, in particular greatest fixed points. Therefore also the appearance of coinduction and fixed points is discussed, though in this case only within Computer Science. The paper ends with some historical remarks on the main fixed-point theorems (such as Knaster-Tarski) that underpin the fixed-point theory presented.
Learning from Paradox
, 1995
"... The danger of paradoxes teaches us to check for consistency in (natural language) semantics. Paradoxes typically involve an element of self-reference (the set-theoretic paradoxes) or an element of self-reference combined with reference to truth (the semantic paradoxes). Self-reference need not be vi ..."
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The danger of paradoxes teaches us to check for consistency in (natural language) semantics. Paradoxes typically involve an element of self-reference (the set-theoretic paradoxes) or an element of self-reference combined with reference to truth (the semantic paradoxes). Self-reference need not be vicious and talking about truth need not be glib, but linguists who allow self-reference or a truth predicate in their representation languages should be aware of the dangers involved. 1 Russell's Paradox The logical and semantical paradoxes that were discovered at the beginning of this century arose in a context where formal languages were employed in a very loose sense. If Gottlob Frege is hailed in logic textbooks as the inventor of first order predicate logic, then it should be noted that in his Begriffsschrift (see [7]) proposal no distinction is made between language and meta-language, and indeed, no formal definition of the language is given. Here is a rational reconstruction of Frege'...
Reconciling Austinian and Russellian Accounts of the Liar Paradox
"... This paper was written at the Department of Mathematics, Manchester University, England under a doctoral grant provided by the Science and Engineering Research Council of the United Kingdom, and substantially revised at the Center for the Study of Language and Information, Stanford University, Calif ..."
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This paper was written at the Department of Mathematics, Manchester University, England under a doctoral grant provided by the Science and Engineering Research Council of the United Kingdom, and substantially revised at the Center for the Study of Language and Information, Stanford University, California under a postdoctoral fellowship provided by the Systems Development Foundation. I am deeply indebted to Peter Aczel, John Barwise and an anonymous referee for invaluable comments on earlier drafts of this paper. conventions" associated with the language. The statement A is true if s A is of type TA ; otherwise it is false.
Contents
, 2007
"... The origins of bisimulation and bisimilarity are examined, in the three fields where they have been independently discovered: Computer Science, Philosophical Logic (precisely, Modal Logic), Set Theory. Bisimulation and bisimilarity are coinductive notions, and as such are intimately related to fixed ..."
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The origins of bisimulation and bisimilarity are examined, in the three fields where they have been independently discovered: Computer Science, Philosophical Logic (precisely, Modal Logic), Set Theory. Bisimulation and bisimilarity are coinductive notions, and as such are intimately related to fixed points, in particular greatest fixed points. Therefore also the appearance of coinduction and fixed points are discussed, though in this case only within Computer Science. The paper ends with some historical remarks on the main fixed-point theorems (such
Are Intensions Necessary? Sense as the Construction of Reference*
"... More than half a century ago, Carnap (1947) defined the extension of an expression in a particular state of affairs as its reference in such a state, while construing the intension of an expression as a function that assigned, to each state ..."
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More than half a century ago, Carnap (1947) defined the extension of an expression in a particular state of affairs as its reference in such a state, while construing the intension of an expression as a function that assigned, to each state
Set Graphs VI: Logic Programming and Bisimulation?
"... Abstract. We analyze the declarative encoding of the set-theoretic graph property known as bisimulation. This notion is of central importance in non-well founded set theory, semantics of concurrency, model checking, and coinductive reasoning. From a modeling point of view, it is partic-ularly intere ..."
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Abstract. We analyze the declarative encoding of the set-theoretic graph property known as bisimulation. This notion is of central importance in non-well founded set theory, semantics of concurrency, model checking, and coinductive reasoning. From a modeling point of view, it is partic-ularly interesting since it allows two alternative high-level characteriza-tions. We analyze the encoding style of these modelings in various dialects of Logic Programming. Moreover, the notion also admits a polynomial-time maximum fix point procedure that we implemented in Prolog. Sim-ilar graph problems which are NP hard or not yet perfectly classified (e.g., graph isomorphism) can benefit from the encodings presented. 1
