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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 139 (16 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
The Complexity of Poor Man’s Logic
, 1999
"... Motivated by description logics, we investigate what happens to the complexity of modal satisfiability problems if we only allow formulas built from literals, ∧, ✸, and ✷. Previously, the only known result was that the complexity of the satisfiability problem for K dropped from PSPACEcomplete to co ..."
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Cited by 16 (0 self)
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Motivated by description logics, we investigate what happens to the complexity of modal satisfiability problems if we only allow formulas built from literals, ∧, ✸, and ✷. Previously, the only known result was that the complexity of the satisfiability problem for K dropped from PSPACEcomplete to coNPcomplete (SchmidtSchauss and Smolka [8] and Donini et al. [3]). In this paper we show that not all modal logics behave like K. In particular, we show that the complexity of the satisfiability problem with respect to frames in which each world has at least one successor drops from PSPACEcomplete to P, but that in contrast the satisfiability problem with respect to the class of frames in which each world has at most two successors remains PSPACEcomplete. As a corollary of the latter result, we also solve the open problem from Donini et al.’s complexity classification of description logics [2]. In the last section, we classify the complexity of the satisfiability problem for K for all other restrictions on the set of operators. 1
Simulating without Negation
, 1997
"... Although negationfree languages are widely used in logic and computer science, relatively little is known about their expressive power. To address this issue we consider kinds of nonsymmetric bisimulations called directed simulations, and use these to analyse the expressive power and model theory ..."
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Cited by 12 (4 self)
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Although negationfree languages are widely used in logic and computer science, relatively little is known about their expressive power. To address this issue we consider kinds of nonsymmetric bisimulations called directed simulations, and use these to analyse the expressive power and model theory of negationfree modal and temporal languages. We first use them to obtain preservation, safety and definability results for a simple negationfree modal language. We then obtain analogous results for stronger negationfree languages. Finally, we extend our methods to deal with languages with nonBoolean negation. Keywords: Expressive power, modal logic, negationfree languages. 1
Natural Deduction for NonClassical Logics
, 1996
"... We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke m ..."
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Cited by 11 (3 self)
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We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework.
A New Coalgebraic Semantics for Positive Modal Logic
 Coalgebraic Methods in Computer Science (CMCS’03), volume 82.1 of ENTCS
, 2002
"... Positive Modal Logic is the restriction of the modal local consequence relation defined by the class of all Kripke models to the propositional negationfree modal language. The class of positive modal algebras is the one canonically associated with PML according to the theory of Abstract Algebraic L ..."
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Cited by 8 (2 self)
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Positive Modal Logic is the restriction of the modal local consequence relation defined by the class of all Kripke models to the propositional negationfree modal language. The class of positive modal algebras is the one canonically associated with PML according to the theory of Abstract Algebraic Logic. In [4], a Priestleystyle duality is established between the category of positive modal algebras and the category of K spaces.In this paper, we establish a categorical equivalence between the category K spaces and the category Coalg(V) of coalgebras of a suitable endofunctor V on the category of Priestley spaces. 2000 Mathematics Subject Classification: 06D22 Keywords and Phrases: Positive Modal Logic, Positive Modal Algebra, Priestley space, coalgebra, Vietoris space, equivalence of categories.
Automated theorem proving by resolution in nonclassical logics
 Annals of Mathematics and Artificial Intelligence
, 2007
"... This paper is an overview of a variety of results, all centered around a common theme, namely embedding of nonclassical logics into first order logic and resolution theorem proving. We present several classes of nonclassical logics, many of which are of great practical relevance in knowledge repre ..."
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Cited by 8 (4 self)
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This paper is an overview of a variety of results, all centered around a common theme, namely embedding of nonclassical logics into first order logic and resolution theorem proving. We present several classes of nonclassical logics, many of which are of great practical relevance in knowledge representation, which can be translated into tractable and relatively simple fragments of classical logic. In this context, we show that refinements of resolution can often be used successfully for automated theorem proving, and in many interesting cases yield optimal decision procedures. 1
Representation Theorems and the Semantics of NonClassical Logics , and Applications to Automated Theorem Proving
, 2002
"... We give a uniform presentation of representation and decidability results related to the Kripkestyle semantics of several nonclassical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, d ..."
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Cited by 4 (2 self)
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We give a uniform presentation of representation and decidability results related to the Kripkestyle semantics of several nonclassical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, distributive lattices and semilattices) extends in a natural way to several classes of operators and allows to establish a relationship between algebraic and Kripkestyle models. We illustrate the ideas on several examples. We conclude by showing how the Kripkestyle models thus obtained can be used (if rstorder axiomatizable) for automated theorem proving by resolution for some nonclassical logics.
Implementing Modal and Relevance Logics in a Logical Framework
, 1996
"... We present a framework for machine implementation of both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, ..."
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Cited by 2 (2 self)
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We present a framework for machine implementation of both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework. 1 INTRODUCTION The origins of natural deduction (ND) are both philosophical and practical. In philosophy, it arises from an analysis of deductive inference in an attempt to provide a theory of meaning for the logical connectives [24, 33]. Practically, it provides a language for building proofs, which can be seen as providing the deduction theorem directly, rather than as a derived result. Our interest is on this practical side, and a development of our work on ap...
Learning in a changing world via algebraic modal logic. http://www.comlab.ox.ac.uk/files/2815/mehrnoosh prakash.pdf and http://www.cs.mcgill.ca/ prakash/Pubs/mehrnoosh prakash.pdf
 The Completeness of Propositional Dynamic Logic. LNCS 64:403–415
, 1978
"... We develop an algebraic modal logic that combines epistemic modalities with dynamic modalities with a view to modelling information acquisition (learning) by automated agents in a changing world. Unlike most treatments of dynamic epistemic logic, we have transitions that “change the state ” of the u ..."
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Cited by 1 (1 self)
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We develop an algebraic modal logic that combines epistemic modalities with dynamic modalities with a view to modelling information acquisition (learning) by automated agents in a changing world. Unlike most treatments of dynamic epistemic logic, we have transitions that “change the state ” of the underlying system and not just the state of knowledge of the agents. The key novel feature that emerges is the need to have a way of “inverting transitions” and distinguishing between transitions that “really happen ” and transitions that are possible. Our approach is algebraic, rather than being based on a Kripkestyle semantics. The semantics are given in terms of quantales. We study a class of quantales with the appropriate inverse operations and prove soundness and completeness theorems. We illustrate the ideas with a simple game as well as a toy robotnavigation problem. The examples illustrate how an agent discovers information by taking actions. 1
Perp and Star in the Light of Modal Logic
, 2004
"... This paper is an exploration in the light of modal logic of Dunn’s ideas about two treatments of negation in nonclassical logics: perp and star. We take negation as an impossibility modal operator and choose the base positive logic to be distributive lattice logic (DLL). It turns out that, if we ad ..."
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This paper is an exploration in the light of modal logic of Dunn’s ideas about two treatments of negation in nonclassical logics: perp and star. We take negation as an impossibility modal operator and choose the base positive logic to be distributive lattice logic (DLL). It turns out that, if we add one De Morgan law and contraposition to DLL (call this system K−), then we can prove a natural completeness and hence treat perp in this modal setting. Moreover, star can be dealt with in the extensions of K−. Based on these results, a complete table of star and perp semantics for Dunn’s kite of negations is given. In the last section, we discuss perp and star in relevance logic and their related logics. The Routley star is interpreted at the end of this paper. Keywords: perp, Routley star, modal logic, relevance logic, MeyerRoutley semantics 1