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Cutfree Sequent and Tableau Systems for Propositional Diodorean Modal Logics
"... We present sound, (weakly) complete and cutfree tableau systems for the propositional normal modal logics S4:3, S4:3:1 and S4:14. When the modality 2 is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of po ..."
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Cited by 20 (3 self)
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We present sound, (weakly) complete and cutfree tableau systems for the propositional normal modal logics S4:3, S4:3:1 and S4:14. When the modality 2 is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points. Although cutfree, the last two systems do not possess the subformula property. But for any given finite set of formulae X the "superformulae" involved are always bounded by a finite set of formulae X L depending only on X and the logic L. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive. Each tableau system has a cutfree sequent analogue proving that Gentzen's cutelimination theorem holds for these latter systems. The techniques are due to Hi...
A New Method for Bounding the Complexity of Modal Logics
, 1997
"... . We present a new prooftheoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility r ..."
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Cited by 12 (2 self)
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. We present a new prooftheoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility relation of the corresponding Kripke structures, yields decision procedures with bounded space requirements. As examples we give O(n log n) space procedures for the modal logics K and T. 1 Introduction We present a new prooftheoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as cutfree labelled sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility relation of the corresponding Kripke structures, yields decision procedures with space requirements that are easily bounded. As examples we give O(n log n) space decision procedures f...
Semianalytic Tableaux For Propositional Normal Modal Logics with Applications to Nonmonotonicity
, 1991
"... The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden in ..."
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Cited by 5 (4 self)
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The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden in modal logic where \Gamma is required to be empty, resulting in a weaker notion of deducibility. Recently, Marek, Schwarz and Truszczi'nski have given algorithms to compute the stable expansions of a finite theory \Gamma in various such nonmonotonic formalisms. Their algorithms assume the existence of procedures for deciding (strong) deducibility in these monotonic modal logics and consequently such decision procedures are important for automating nonmonotonic deduction. We first give a sound, (weakly) complete and cutfree, semianalytic tableau calculus for monotonic S4R, thus extending the cut elimination results of Schwarz for monotonic K45 and K45D. We then give sound and complete semi...
Games for Modal and Temporal Logics
, 2002
"... Every logic comes with several decision problems. One of them is the model checking problem: does a given structure satisfy a given formula? Another is the satisfiability problem: for a given formula, is there a structure fulfilling it? For modal and temporal logics; tableaux, automata and games ar ..."
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Cited by 1 (0 self)
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Every logic comes with several decision problems. One of them is the model checking problem: does a given structure satisfy a given formula? Another is the satisfiability problem: for a given formula, is there a structure fulfilling it? For modal and temporal logics; tableaux, automata and games are commonly accepted as helpful techniques that solve these problems. The fact that these logics possess the tree model property makes tableau structures suitable for these tasks. On the other hand, starting with Buchi's work, intimate connections between these logics and automata have been found. A formula can describe an automaton's behaviour, and automata are constructed to accept exactly the word or tree models of a formula.
Semianalytic Tableaux For Propositional Modal Logics of Nonmonotonicity
, 1993
"... The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden ..."
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Cited by 1 (0 self)
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The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden in modal logic where \Gamma is required to be empty, resulting in a weaker notion of deducibility. Recently, Marek, Schwarz and Truszczi'nski have given algorithms to compute the stable expansions of a finite theory \Gamma in various such nonmonotonic formalisms. Their algorithms assume the existence of procedures for deciding (strong) deducibility in these monotonic modal logics and consequently such decision procedures are important for automating nonmonotonic deduction. We first give a sound, (weakly) complete and cutfree, semianalytic tableau calculus for monotonic S4R, thus extending the cut elimination results of Schwarz for monotonic K45 and K45D. We then give sound and co...
Frege systems for extensible modal logics
, 2006
"... By a wellknown result of Cook and Reckhow [4, 12], all Frege systems for the Classical Propositional Calculus (CPC) are polynomially equivalent. Mints and Kojevnikov [11] have recently shown pequivalence of Frege systems for the Intuitionistic Propositional Calculus (IPC) in the standard language, ..."
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By a wellknown result of Cook and Reckhow [4, 12], all Frege systems for the Classical Propositional Calculus (CPC) are polynomially equivalent. Mints and Kojevnikov [11] have recently shown pequivalence of Frege systems for the Intuitionistic Propositional Calculus (IPC) in the standard language, building on a description of admissible rules of IPC by Iemhoff [8]. We prove a similar result for an infinite family of normal modal logics, including K4, GL, S4, and S4Grz.
Rules with parameters in modal logic I Emil Jeˇrábek ∗
, 2013
"... We study admissibility of inference rules and unification with parameters in transitive modal logics (extensions of K4), in particular we generalize various results on parameterfree admissibility and unification to the setting with parameters. Specifically, we give a characterization of projective f ..."
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We study admissibility of inference rules and unification with parameters in transitive modal logics (extensions of K4), in particular we generalize various results on parameterfree admissibility and unification to the setting with parameters. Specifically, we give a characterization of projective formulas generalizing Ghilardi’s characterization in the parameterfree case, leading to new proofs of Rybakov’s results that admissibility with parameters is decidable and unification is finitary for logics satisfying suitable frame extension properties (called clusterextensible logics in this paper). We construct explicit bases of admissible rules with parameters for clusterextensible logics, and give their semantic description. We show that in the case of finitely many parameters, these logics have independent bases of admissible rules, and determine which logics have finite bases. As a sideline, we show that clusterextensible logics have various nice properties: in particular, they are finitely axiomatizable, and have an exponentialsize model property. We also give a rather general characterization of logics with directed (filtering) unification. In the sequel, we will use the same machinery to investigate the computational complexity of admissibility and unification with parameters in clusterextensible logics, and we will adapt the results to logics with unique top cluster (e.g., S4.2) and superintuitionistic logics. 1