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14
A Tableau Calculus for Multimodal Logics and Some (Un)Decidability Results
 IN PROC. OF TABLEAUX98
, 1998
"... In this paper we present a prefixed analytic tableau calculus for a class of normal multimodal logics and we present some results about decidability and undecidability of this class. The class is characterized by axioms of the form [t 1 ] : : : [t n ]' oe [s1 ] : : : [sm ]', called inclusion axio ..."
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Cited by 24 (8 self)
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In this paper we present a prefixed analytic tableau calculus for a class of normal multimodal logics and we present some results about decidability and undecidability of this class. The class is characterized by axioms of the form [t 1 ] : : : [t n ]' oe [s1 ] : : : [sm ]', called inclusion axioms, where the t i 's and s j 's are constants. This class of logics, called grammar logics, was introduced for the first time by Farinas del Cerro and Penttonen to simulate the behaviour of grammars in modal logics, and includes some wellknown modal systems. The prefixed tableau method is used to prove the undecidability of modal systems based on unrestricted, context sensitive, and context free grammars. Moreover, we show that the class of modal logics, based on rightregular grammars, are decidable by means of the filtration methods, by defining an extension of the FischerLadner closure.
Sequent Calculi for Nominal Tense Logics: A Step Towards Mechanization?
, 1999
"... . We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule ..."
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Cited by 15 (4 self)
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. We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule that is not eliminable. A nice computational property of the restriction is, for instance, that at any stage of the proof, only a finite number of potential cutformulae needs to be taken under consideration. Although restrictions on the proof search (preserving completeness) are given in the paper and most of them are theoretically appealing, the use of those calculi for mechanization is however doubtful. Indeed, we present sequent calculi for fragments of classical logic that are syntactic variants of the sequent calculi for the nominal tense logics. 1 Introduction Background. The nominal tense logics are extensions of Prior tense logics (see e.g. [Pri57, RU71]) by adding nomina...
A simple tableau system for the logic of elsewhere
 177– 192. LNAI 1071
, 1996
"... Abstract. We analyze different features related to the mechanization of von Wright’s logic of elsewhere E. First, we give a new proof of the NPcompleteness of the satisfiability problem (giving a new bound for the size of models of the satisfiable formulae) and we show that this problem becomes lin ..."
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Cited by 7 (3 self)
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Abstract. We analyze different features related to the mechanization of von Wright’s logic of elsewhere E. First, we give a new proof of the NPcompleteness of the satisfiability problem (giving a new bound for the size of models of the satisfiable formulae) and we show that this problem becomes lineartime when the number of propositional variables is bounded. Although E and the wellknown propositional modal S5 share numerous common features we show that E is strictly more expressive than S5 (in a sense to be specified). Second, we present a prefixed tableau system for E and we prove both its soundness and completeness. Two extensions of this system are also defined, one related to the logical consequence relation and the other related to the addition of modal operators (without increasing the expressive power). An example of tableau proof is also presented. Different continuations of this work are proposed, one of them being to implement the defined tableau system, another one being to extend this system to richer logics that can be found in the literature. 1
Labelled Proofs for Quantified Modal Logic
 Logics in Artificial Intelligence, pages 70–86, LNAI 1126
, 1996
"... Abstract. In this paper we describe a modal proof system arising from the combination of a tableaulike classical system, which incorporates a restricted (“analytic”) version of the cut rule, with a label formalism which allows for a specialised, logicdependent unification algorithm. The system pro ..."
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Cited by 7 (6 self)
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Abstract. In this paper we describe a modal proof system arising from the combination of a tableaulike classical system, which incorporates a restricted (“analytic”) version of the cut rule, with a label formalism which allows for a specialised, logicdependent unification algorithm. The system provides a uniform prooftheoretical treatment of firstorder (normal) modal logics with and without the Barcan Formula and/or its converse. 1
Prefixed Tableaux Systems for Modal Logics with Enriched Languages
, 1997
"... We present sound and complete prefixed tableaux systems for various modal logics with enriched languages including the "difference" modal operator] and the "only if" modal operator [—R]. These logics are of special interest in Artificial Intelligence since their expressive power is higher than the s ..."
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Cited by 5 (2 self)
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We present sound and complete prefixed tableaux systems for various modal logics with enriched languages including the "difference" modal operator] and the "only if" modal operator [—R]. These logics are of special interest in Artificial Intelligence since their expressive power is higher than the standard modal logics and for most of them the satisfiability problem remains decidable. We also include in the paper decision procedures based on these systems. In the conclusion, we relate our work with similar ones from the literature and we propose extensions to other logics.
A Prolog implementation of KEM
 Proceedings of the GULPPRODE’95 Joint Conference on Declarative Programming. Marina di Vietri, 11–14
, 1995
"... In this paper, we describe a Prolog implementation of a new theorem prover for (normal propositional) modal and multimodal logics. The theorem prover, which is called KEM , arises from the combination of a classical refutation system which incorporates a restricted ("analytic") version of the cut ..."
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Cited by 3 (3 self)
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In this paper, we describe a Prolog implementation of a new theorem prover for (normal propositional) modal and multimodal logics. The theorem prover, which is called KEM , arises from the combination of a classical refutation system which incorporates a restricted ("analytic") version of the cut rule with a label formalism which allows for a specialised, logicdependent unification algorithm. An essential feature of KEM is that it yields a rather simple and efficient proof search procedure which offers many computational advantages over the usual tableaubased proof search methods. This is due partly to the use of linear 2premise # rules in place of the branching # rules of the standard tableau method, and partly to the crucial role played by the analytic cut (the only branching rule) in eliminating redundancy from the search space. It turns out that KEM method of proof search is not only computationally more efficient but also intuitively more natural than other (e.g. resolutionbased) methods leading to simple and easily implementable procedures (two KEM Theorem Proverlike systems have been implemented: an LPA interpreter on Macintosh, and a Quintus compiler on SunSparcstation) which make it well suited for efficient automated proof search in modal logics.
System description: leanK 2.0
 In Proceedings, 15th International Conference on Automated Deduction
, 1998
"... Abstract. leanK is a “lean”, i.e., extremely compact, Prolog implementation of a free variable tableau calculus for propositional modal logics. leanK 2.0 includes additional search space restrictions and fairness strategies, giving a decision procedure for the logics K, KT, and S4. Overview. leanK i ..."
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Cited by 3 (1 self)
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Abstract. leanK is a “lean”, i.e., extremely compact, Prolog implementation of a free variable tableau calculus for propositional modal logics. leanK 2.0 includes additional search space restrictions and fairness strategies, giving a decision procedure for the logics K, KT, and S4. Overview. leanK is a “lean ” Prolog implementation of the free variable tableau calculus for propositional modal logics reported in [1]. It performs depth first search and is based upon leanTAP [2]. Formulae are annotated with labels containing variables, which capture the universal and existential nature of the box and diamond modalities, respectively. leanK 2.0 includes additional search space restrictions and fairness strategies, giving a decision procedure for the logics K, KT, and S4. It has 87, 51, and 132 lines of code for K, KD, and S4, respectively. The main advantages of leanK are its modularity and its versatility. Due to its small size, leanK is easier to understand than a complex prover, and hence easier to adapt to special needs. Minimal changes in the rules give provers for all the 15 basic normal modal logics. By sacrificing modularity we can obtain specialised (faster) provers for particular logics like K45D, G and Grz. It is easy to obtain an explicit counterexample from a failed proof attempt. The leanK (2.0) SICStus Prolog 3 code is at:
Fibred Modal Tableaux (preliminary report)
 Tilburg University
, 1998
"... We describe a general and uniform tableau methodology for multimodal logics arising from Gabbay's methodology of fibring and Governatori's tableau system KEM. ..."
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Cited by 1 (1 self)
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We describe a general and uniform tableau methodology for multimodal logics arising from Gabbay's methodology of fibring and Governatori's tableau system KEM.
A Labelled Sequent System for Tense Logic K t
 In Australian Joint Conference of Articifial Intelligence
, 1998
"... . The method of labelled tableaux for proof search in modal logics is extended and modified to give a labelled sequent system for the tense logic K t . Soundness and completeness proofs are sketched, and results of an initial lean Prolog implementation in the programming style of leanTAP are presen ..."
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Cited by 1 (0 self)
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. The method of labelled tableaux for proof search in modal logics is extended and modified to give a labelled sequent system for the tense logic K t . Soundness and completeness proofs are sketched, and results of an initial lean Prolog implementation in the programming style of leanTAP are presented. The sequent system is modular in that small modifications capture any combination of the reflexive, transitive, euclidean, symmetric and serial extensions of K t . Keywords: automated deduction, labelled deductive system, lean deduction, sequent system, tense logic. Introduction Modal and temporal logics are widely used in computer science in areas as diverse as Artificial Intelligence [MST91], Models for Concurrency [Sti92], and Hardware Verification [NFKT87]. In these applications, automated proof search in these logics is of fundamental importance. We present a sound and complete proof search system, K seq t , for the minimal tense logic K t [RU71]. Our system K seq t is an extens...
Clausal Tableaux for Multimodal Logics of Belief
"... Abstract. We develop clausal tableau calculi for seven multimodal logics variously designed for reasoning about multidegree belief, reasoning about distributed systems of belief and for reasoning about epistemic states of agents in multiagent systems. Our tableau calculi are sound, complete, cutf ..."
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Abstract. We develop clausal tableau calculi for seven multimodal logics variously designed for reasoning about multidegree belief, reasoning about distributed systems of belief and for reasoning about epistemic states of agents in multiagent systems. Our tableau calculi are sound, complete, cutfree and have the analytic superformula property, thereby giving decision procedures for all of these logics. We also use our calculi to obtain complexity results for five of these logics. The complexity of one was known and that of the seventh remains open.