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Tableau Methods for Modal and Temporal Logics
, 1995
"... This document is a complete draft of a chapter by Rajeev Gor'e on "Tableau Methods for Modal and Temporal Logics" which is part of the "Handbook of Tableau Methods", edited by M. D'Agostino, D. Gabbay, R. Hahnle and J. Posegga, to be published in 1998 by Kluwer, Dordrec ..."
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Cited by 126 (20 self)
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This document is a complete draft of a chapter by Rajeev Gor'e on "Tableau Methods for Modal and Temporal Logics" which is part of the "Handbook of Tableau Methods", edited by M. D'Agostino, D. Gabbay, R. Hahnle and J. Posegga, to be published in 1998 by Kluwer, Dordrecht. Any comments and corrections are highly welcome. Please email me at rpg@arp.anu.edu.au The latest version of this document can be obtained via my WWW home page: http://arp.anu.edu.au/ Tableau Methods for Modal and Temporal Logics Rajeev Gor'e Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Syntax and Notational Conventions . . . . . . . . . . . . 3 2.2 Axiomatics of Modal Logics . . . . . . . . . . . . . . . . 4 2.3 Kripke Semantics For Modal Logics . . . . . . . . . . . . 5 2.4 Known Correspondence and Completeness Results . . . . 6 2.5 Logical Consequence . . . . . . . . . . . . . . . . . . . . 8 2....
A tutorial on Stålmarck's proof procedure for propositional logic
 Formal Methods in System Design
, 1998
"... We explain Stalmarck's proof procedure for classical propositional logic. The method is implemented in a commercial tool that has been used successfully in real industrial verification projects. Here, we present the proof system underlying the method, and motivate the various design decisio ..."
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Cited by 64 (1 self)
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We explain Stalmarck's proof procedure for classical propositional logic. The method is implemented in a commercial tool that has been used successfully in real industrial verification projects. Here, we present the proof system underlying the method, and motivate the various design decisions that have resulted in a system that copes well with the large formulas encountered in industrialscale verification. 1
Are Tableaux an Improvement on TruthTables? CutFree proofs and Bivalence
, 1992
"... We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance bet ..."
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Cited by 12 (0 self)
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We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cutfree proofs and the Principle of Bivalence. Finally we discuss some ways in which this principle can be built into a tableaulike method without affecting its "analytic" nature. 1 Introduction The truthtable method, introduced by Wittgenstein in his Tractatus LogicoPhilosophicus, provides a decision procedure for propositional logic which is immediately implementable on a machine. However this timehonoured method is usually mentioned only to be immediately dismissed because of its incurable inefficiency. The wellknown tableau method (which is closely related to Gentzen's cutfree sequent calculus) is commonly regarded as a "shortcut" in testing the logical validity of complex propositions...
An Intensional Type Theory: Motivation and CutElimination
, 2001
"... By the theory TT is meant the higher order predicate logic with the following recursively defined types: (1) 1 is the type ofindividuals and [] is the type ofthe truth values; (2) [# 1 ,..., n ] is the type ofthe predicates with arguments ofthe types #1 ,...,# n . The theory ITT described in this pa ..."
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By the theory TT is meant the higher order predicate logic with the following recursively defined types: (1) 1 is the type ofindividuals and [] is the type ofthe truth values; (2) [# 1 ,..., n ] is the type ofthe predicates with arguments ofthe types #1 ,...,# n . The theory ITT described in this paper is an intensional version ofTT. The types ofITT are the same as the types ofTT, but the membership ofthe type 1 ofindividuals in ITT is an extension ofthe membership in TT. The extension consists ofallowing any higher order term, in which only variables oftype 1 have a free occurrence, to be a term oftype 1. This feature ofITT is motivated by a nominalist interpretation ofhigher order predication. In ITT both wellfounded and nonwellfounded recursive predicates can be defined as abstraction terms from which all the properties of the predicates can be derived without the use ofnonlogical axioms. The elementary syntax, semantics, and prooftheory for ITT are defined. A semantic consistency prooffor ITT is provided and the completeness proofofTakahashi and Prawitz for a version of TT without cut is adapted for ITT; a consequence is the redundancy of cut. 1.
A short survey of automated reasoning
"... Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so f ..."
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Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for
Limited Logical Belief Analysis
, 1996
"... . The process of rational inquiry can be defined as the evolution of a rational agent's belief set as a consequence of its internal inference procedures and its interaction with the environment. These beliefs can be modelled in a formal way using doxastic logics. The possible worlds model and i ..."
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. The process of rational inquiry can be defined as the evolution of a rational agent's belief set as a consequence of its internal inference procedures and its interaction with the environment. These beliefs can be modelled in a formal way using doxastic logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but they seem to commit us to model agents that are logically omniscient and perfect reasoners. These problems can be avoided with a syntactic view of possible worlds, defining them as arbitrary sets of sentences in a propositional doxastic logic. In this paper this syntactic view of possible worlds is taken, and a dynamic analysis of the agent's beliefs is suggested in order to model the process of rational inquiry in which the agent is permanently engaged. One component of this analysis, the logical one, is summarily described. This dimension of analysis is performed using a modified version of the analytic tableaux...
Logical and Psychological Analysis of Deductive Mastermind
"... Abstract. The paper proposes a way to analyze logical reasoning in a deductive version of the Mastermind game implemented within the Math Garden educational system. Our main goal is to derive predictions about the cognitive difficulty of gameplays, e.g., the number of steps needed for solving the l ..."
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Abstract. The paper proposes a way to analyze logical reasoning in a deductive version of the Mastermind game implemented within the Math Garden educational system. Our main goal is to derive predictions about the cognitive difficulty of gameplays, e.g., the number of steps needed for solving the logical tasks or the working memory load. Our model is based on the analytic tableaux method, known from proof theory. We associate the difficulty of the Deductive Mastermind gameitems with the size of the corresponding logical tree derived by the tableau method. We discuss possible empirical hypotheses based on this model, and preliminary results that prove the relevance of our theory.
Intuitionistic implication without disjunction
, 2010
"... We investigate fragments of intuitionistic propositional logic containing implication but not disjunction. These fragments are finite, but their size grows superexponentially with the number of generators. Exact models are used to characterize the fragments. ..."
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We investigate fragments of intuitionistic propositional logic containing implication but not disjunction. These fragments are finite, but their size grows superexponentially with the number of generators. Exact models are used to characterize the fragments.