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Monodic temporal resolution
 ACM Transactions on Computational Logic
, 2003
"... Until recently, FirstOrder Temporal Logic (FOTL) has been only partially understood. While it is well known that the full logic has no finite axiomatisation, a more detailed analysis of fragments of the logic was not previously available. However, a breakthrough by Hodkinson et al., identifying a f ..."
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Cited by 27 (15 self)
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Until recently, FirstOrder Temporal Logic (FOTL) has been only partially understood. While it is well known that the full logic has no finite axiomatisation, a more detailed analysis of fragments of the logic was not previously available. However, a breakthrough by Hodkinson et al., identifying a finitely axiomatisable fragment, termed the monodic fragment, has led to improved understanding of FOTL. Yet, in order to utilise these theoretical advances, it is important to have appropriate proof techniques for this monodic fragment. In this paper, we modify and extend the clausal temporal resolution technique, originally developed for propositional temporal logics, to enable its use in such monodic fragments. We develop a specific normal form for monodic formulae in FOTL, and provide a complete resolution calculus for formulae in this form. Not only is this clausal resolution technique useful as a practical proof technique for certain monodic classes, but the use of this approach provides us with increased understanding of the monodic fragment. In particular, we here show how several features of monodic FOTL can be established as corollaries of the completeness result for the clausal temporal resolution method. These include definitions of new decidable monodic classes, simplification of existing monodic classes by reductions, and completeness of clausal temporal resolution in the case of
On the Products of Linear Modal Logics
 JOURNAL OF LOGIC AND COMPUTATION
, 2001
"... We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz: ..."
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Cited by 24 (9 self)
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We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz:3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman [7]. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square K4:3 K4:3 of the minimal liner logic using nonstructural Gabbaytype inference rules.
Temporalising Tableaux
 STUDIA LOGICA
, 2004
"... As a remedy for the bad computational behaviour of firstorder temporal logic (FOTL), it has recently been proposed to restrict the application of temporal operators to formulas with at most one free variable thereby obtaining socalled monodic fragments of FOTL. In this paper, we are concerned with ..."
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Cited by 17 (5 self)
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As a remedy for the bad computational behaviour of firstorder temporal logic (FOTL), it has recently been proposed to restrict the application of temporal operators to formulas with at most one free variable thereby obtaining socalled monodic fragments of FOTL. In this paper, we are concerned with constructing tableau algorithms for monodic fragments based on decidable fragments of firstorder logic like the twovariable fragment or the guarded fragment. We present a general framework that shows how existing decision procedures for firstorder fragments can be used for constructing a tableau algorithm for the corresponding monodic fragment of FOTL.
Towards the Implementation of FirstOrder Temporal Resolution: the Expanding Domain Case
"... Firstorder temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic firstorder temporal logics has identified important enumerable and even decidable frag ..."
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Cited by 11 (7 self)
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Firstorder temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic firstorder temporal logics has identified important enumerable and even decidable fragments. In this paper, we develop a clausal resolution method for the monodic fragment of firstorder temporal logic over expanding domains. We first define a normal form for monodic formulae and show how arbitrary monodic formulae can be translated into the normal form, while preserving satisfiability. We then introduce novel resolution calculi that can be applied to formulae in this normal form and state correctness and completeness results for the method. We illustrate the method on a comprehensive example. The method is based on classical firstorder resolution and can, thus, be efficiently implemented.
A Complete Quantified Epistemic Logic for Reasoning about Message Passing Systems
 PROCEEDINGS OF THE 8TH INTERNATIONAL WORKSHOP ON COMPUTATIONAL LOGIC IN MULTIAGENT SYSTEMS (CLIMA VIII
, 2008
"... We introduce quantified interpreted systems, a semantics to reason about knowledge in multiagent systems in a firstorder setting. Quantified interpreted systems may be used to interpret a variety of firstorder modal epistemic languages with global and local terms, quantifiers, and individual and ..."
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Cited by 7 (6 self)
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We introduce quantified interpreted systems, a semantics to reason about knowledge in multiagent systems in a firstorder setting. Quantified interpreted systems may be used to interpret a variety of firstorder modal epistemic languages with global and local terms, quantifiers, and individual and distributed knowledge operators for the agents in the system. We define firstorder modal axiomatisations for different settings, and show that they are sound and complete with respect to the corresponding semantical classes. The expressibility potential of the formalism is explored by analysing two MAS scenarios: an infinite version of the muddy children problem, a typical epistemic puzzle, and a version of the battlefield game. Furthermore, we apply the theoretical results here presented to the analysis of message passing systems [17,41], and compare the results obtained to their propositional counterparts. By doing so we find that key known metatheorems of the propositional case can be expressed as validities on the corresponding class of quantified interpreted systems.
Mechanising FirstOrder Temporal Resolution
, 2003
"... Firstorder temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic firstorder temporal logics has identified important enumerable and even decidable frag ..."
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Cited by 7 (5 self)
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Firstorder temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic firstorder temporal logics has identified important enumerable and even decidable fragments. Although a complete and correct resolutionstyle calculus has already been suggested for this specific fragment, this calculus involves constructions too complex to be of a practical value. In this paper, we develop a machineoriented clausal resolution method which features radically simplified proof search. We first define a normal form for monodic formulae and then introduce a novel resolution calculus that can be applied to formulae in this normal form. The calculus is based on classical firstorder resolution and can, thus, be efficiently implemented. We prove correctness and completeness results for the calculus and illustrate it on a comprehensive example. An implementation of the method is briefly discussed.
Monodic Epistemic Predicate Logic
 In Proceedings of JELIA, Lecture Notes in Arti Intelligence
, 2000
"... We consider the monodic formulas of common knowledge predicate logic, which allow applications of epistemic operators to formulas with at most one free variable. We provide finite axiomatizations of the monodic fragment of the most important common knowledge predicate logics (the full logics are kno ..."
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Cited by 5 (0 self)
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We consider the monodic formulas of common knowledge predicate logic, which allow applications of epistemic operators to formulas with at most one free variable. We provide finite axiomatizations of the monodic fragment of the most important common knowledge predicate logics (the full logics are known to be not recursively enumerable) and single out a number of their decidable fragments. On the other hand, it is proved that the addition of the equality symbol to the monodic fragment makes it not recursively enumerable. 1 Introduction Ever since it became common knowledge that intelligent behaviour of an agent is based not only on her knowledge about the world but also on knowledge about both her own and other agents' knowledge, logical formalisms designed for reasoning about knowledge have attracted attention in artificial intelligence, computer science, economic theory, and philosophy (cf. e.g. the books [5, 16, 13] and the seminal works [8, 1]). In all these areas, one of the most s...
Undecidability of firstorder intuitionistic and modal logics with two variables
 Bulletin of Symbolic Logic
, 2005
"... Abstract. We prove that the twovariable fragment of firstorder intuitionistic logic is undecidable, even without constants and equality. We also show that the twovariable fragment of a quantified modal logic L with expanding firstorder domains is undecidable whenever there is a Kripke frame for L ..."
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Cited by 5 (3 self)
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Abstract. We prove that the twovariable fragment of firstorder intuitionistic logic is undecidable, even without constants and equality. We also show that the twovariable fragment of a quantified modal logic L with expanding firstorder domains is undecidable whenever there is a Kripke frame for L with a point having infinitely many successors (such are, in particular, the firstorder extensions of practically all standard modal logics like K, K4, GL, S4, S5, K4.1, S4.2, GL.3, etc.). For many quantified modal logics, including those in the standard nomenclature above, even the monadic twovariable fragments turn out to be undecidable. §1. Introduction. Ever since the undecidability of firstorder classical logic became known [5], there has been a continuing interest in establishing the ‘borderline ’ between its decidable and undecidable fragments; see [2] for a detailed exposition. One approach to this classification problem is to consider fragments with finitely many individual variables. The