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We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective i... more We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via " declarative resource reading " , intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-) abduction.
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Outstanding Contributions to Logic, 2014
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数理解析研究所講究録, Nov 1, 2006
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Electronic Proceedings in Theoretical Computer Science, 2013
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Journal of Applied Logic, 2010
We provide complete axiomatizations for several fragments of XPath: sets of equivalences from whi... more We provide complete axiomatizations for several fragments of XPath: sets of equivalences from which every other valid equivalence is derivable. Specically, we axiomatize downward single axis fragments of Core XPath (that is, Core XPath(↓) and Core XPath(↓+)) as well as the full Core XPath. We make use of techniques from modal logic.
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Bulletin of the Section of Logic, 2007
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This paper provides a complete algebraic axiomatization of node and path equivalences in Core XPa... more This paper provides a complete algebraic axiomatization of node and path equivalences in Core XPath 1.0. Our completeness proof builds on a completeness result of Blackburn et al. (3) for a modal logic of nite trees.
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We are going to show that the standard notion of Kripke completeness is the strongest one among m... more We are going to show that the standard notion of Kripke completeness is the strongest one among many provably distinct algebraically motivated completeness properties, some of which seem to be of intrinsic interest. More specically, we are going to investigate notions of completeness with respect to algebras which are either atomic, complete, completely additive or admit residuals (the last notion of completeness coincides with conservativity of minimal tense extensions); we will be also interested in combinations of these properties. 1 Motivation It is known that Kripke frames correspond to complete, atomic and completely additive Boolean algebras with operators (baos). This fact became the basis of duality theory for Kripke frames, developed in the 1970's by Thomason (13), Goldblatt (4) and others. In this paper, we are going to investigate notions of completeness and con- sequence weaker than those associated with standard Kripke frames from an algebraic perspective. Our star...
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This paper generalizes the 1977 paper of V.B. Shehtman, which constructed the first Kripke incomp... more This paper generalizes the 1977 paper of V.B. Shehtman, which constructed the first Kripke incomplete intermediate logic, by presenting a continuum of such logics. This version fixes an error in my simplified proof of incompleteness of Shehtman's original logic.
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Supervisor:小野 寛晰 情報科学研究科 博士 An algebraic approach to incompleteness in modal logic Tadeusz Litak
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Journal of Logic and Computation, 2015
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Lecture Notes in Computer Science, 2011
ABSTRACT We define Boolean algebras over nominal sets with a function-symbol N mirroring the N &a... more ABSTRACT We define Boolean algebras over nominal sets with a function-symbol N mirroring the N 'fresh name' quantifier. We also define dual notions of nominal topology and Stone space, prove a representation theorem over fields of nominal sets, and extend this to a Stone duality.
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Lecture Notes in Computer Science, 2013
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Lecture Notes in Computer Science, 2006
... Algebraic operators formalizing substitutions in first-order logic have been studied since Ha... more ... Algebraic operators formalizing substitutions in first-order logic have been studied since Halmos started working on polyadic algebras [4]. In particular, they play a prominent role in formalisms developed by Pinter in the 1970's, cf., eg, [5]. Nevertheless, algebras studied in the ...
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We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective i... more We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via " declarative resource reading " , intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-) abduction.
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Outstanding Contributions to Logic, 2014
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数理解析研究所講究録, Nov 1, 2006
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Electronic Proceedings in Theoretical Computer Science, 2013
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Journal of Applied Logic, 2010
We provide complete axiomatizations for several fragments of XPath: sets of equivalences from whi... more We provide complete axiomatizations for several fragments of XPath: sets of equivalences from which every other valid equivalence is derivable. Specically, we axiomatize downward single axis fragments of Core XPath (that is, Core XPath(↓) and Core XPath(↓+)) as well as the full Core XPath. We make use of techniques from modal logic.
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Bulletin of the Section of Logic, 2007
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This paper provides a complete algebraic axiomatization of node and path equivalences in Core XPa... more This paper provides a complete algebraic axiomatization of node and path equivalences in Core XPath 1.0. Our completeness proof builds on a completeness result of Blackburn et al. (3) for a modal logic of nite trees.
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We are going to show that the standard notion of Kripke completeness is the strongest one among m... more We are going to show that the standard notion of Kripke completeness is the strongest one among many provably distinct algebraically motivated completeness properties, some of which seem to be of intrinsic interest. More specically, we are going to investigate notions of completeness with respect to algebras which are either atomic, complete, completely additive or admit residuals (the last notion of completeness coincides with conservativity of minimal tense extensions); we will be also interested in combinations of these properties. 1 Motivation It is known that Kripke frames correspond to complete, atomic and completely additive Boolean algebras with operators (baos). This fact became the basis of duality theory for Kripke frames, developed in the 1970's by Thomason (13), Goldblatt (4) and others. In this paper, we are going to investigate notions of completeness and con- sequence weaker than those associated with standard Kripke frames from an algebraic perspective. Our star...
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This paper generalizes the 1977 paper of V.B. Shehtman, which constructed the first Kripke incomp... more This paper generalizes the 1977 paper of V.B. Shehtman, which constructed the first Kripke incomplete intermediate logic, by presenting a continuum of such logics. This version fixes an error in my simplified proof of incompleteness of Shehtman's original logic.
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Supervisor:小野 寛晰 情報科学研究科 博士 An algebraic approach to incompleteness in modal logic Tadeusz Litak
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Journal of Logic and Computation, 2015
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Lecture Notes in Computer Science, 2011
ABSTRACT We define Boolean algebras over nominal sets with a function-symbol N mirroring the N &a... more ABSTRACT We define Boolean algebras over nominal sets with a function-symbol N mirroring the N 'fresh name' quantifier. We also define dual notions of nominal topology and Stone space, prove a representation theorem over fields of nominal sets, and extend this to a Stone duality.
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Lecture Notes in Computer Science, 2013
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Lecture Notes in Computer Science, 2006
... Algebraic operators formalizing substitutions in first-order logic have been studied since Ha... more ... Algebraic operators formalizing substitutions in first-order logic have been studied since Halmos started working on polyadic algebras [4]. In particular, they play a prominent role in formalisms developed by Pinter in the 1970's, cf., eg, [5]. Nevertheless, algebras studied in the ...
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We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective i... more We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via "declarative resource reading", intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-)abduction.
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Motivated by the recent interest in models of guarded (co-)recursion, we study their equational p... more Motivated by the recent interest in models of guarded (co-)recursion, we study their equational properties. We formulate axioms for guarded fixpoint operators generalizing the axioms of iteration theories of Bloom and \'Esik. Models of these axioms include both standard (e.g., cpo-based) models of iteration theories and models of guarded recursion such as complete metric spaces or the topos of trees studied by Birkedal et al. We show that the standard result on the satisfaction of all Conway axioms by a unique dagger operation generalizes to the guarded setting. We also introduce the notion of guarded trace operator on a category, and we prove that guarded trace and guarded fixpoint operators are in one-to-one correspondence. Our results are intended as first steps leading, hopefully, towards future description of classifying theories for guarded recursion.
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C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical p... more C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.
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We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a... more We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and completeness results for several natural classes of such logics. Moreover, we show that an entirely general completeness result is not possible. We study the expressive power of our language, both in comparison with coalgebraic hybrid logics and with existing first-order proposals for special classes of Set-coalgebras (apart from relational structures, also neighbourhood frames and topological spaces). Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes that allow completeness---and in some cases beyond that. Finally, we discuss a basic sequent system, for which we establish a syntactic cut-elimination result.
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Studia Logica
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