Analytic Tableaux for KLM Preferential and Cumulative Logics

Lecture Notes in Computer Science, 2006

In this paper we present a tableau calculus for the rational logic R of default reasoning, introduced by Kraus, Lehmann and Magidor. Our calculus is obtained by introducing suitable modalities to interpret conditional assertions, and makes use of labels to represent possible worlds. We also provide a decision procedure for R, and study its complexity.

ACM Transactions on Computational Logic, 2009

We present a tableau calculus for some fundamental systems of propositional conditional logics. We consider the conditional logics that can be characterized by preferential semantics (i.e. possible world structures equipped with a family of preference relations). For these logics, we provide a uniform completeness proof of the axiomatization w.r.t. the semantics, and a uniform labelled tableau procedure.

2007

We present FreeP 1.0, a theorem prover for the KLM preferential logic P of nonmonotonic reasoning. FreeP 1.0 is a SICStus Prolog implementation of a free-variables, labelled tableau calculus for P, obtained by introducing suitable modalities to interpret conditional assertions. The performances of FreeP 1.0 are promising. FreeP 1.0 can be downloaded at http://www.di.unito.it/~pozzato/FreeP1.0 .

arXiv: Logic, 2016

In this paper we present analytic tableau proof systems for various justification logics. We show that the tableau systems are sound and complete with respect to Mkrtychev models. In order to prove the completeness of the tableaux, we give a syntactic proof of cut elimination. We also show the subformula property for our tableaux, and prove the decidability of justification logics for finite constant specifications.

Studia Logica, 2011

Grammar logics were introduced by Fariñas del Cerro and Penttonen in [7] and have been widely studied. In this paper we consider regular grammar logics with converse (REG c logics) and present sound and complete tableau calculi for the general satisfiability problem of REG c logics and the problem of checking consistency of an ABox w.r.t. a TBox in a REG c logic. Using our calculi we develop ExpTime (optimal) tableau decision procedures for the mentioned problems, to which various optimization techniques can be applied. We also prove a new result that the data complexity of the instance checking problem in REG c logics is coNP-complete. This work is an extension of the conference paper [25]. It is supported by grants N N206 3982 33 and N N206 399334 from the Polish Ministry of Science and Higher Education. 3 i.e., the satisfiability problem, where a specification of the logic is also given as an input of the problem can use that translation together with the tableau decision procedure for CPDL given by De Giacomo and Massacci [2] for deciding REG c logics. This method uses the translation approach and, additionally, has the disadvantage that the formal decision procedure given in [2] for CPDL has non-optimal NExpTime complexity. Although De Giacomo and Massacci [2] described also a transformation of their NExpTime algorithm into an ExpTime decision procedure for CPDL, the description is informal and unclear. Namely, the transformation is based on Pratt's global caching method formulated for PDL [26], but no global caching method has been formalized and proved sound for labeled tableaux that allow modifying labels of ancestor nodes in order to deal with converse. 7 In this work we develop a sound and complete tableau calculus for deciding the general satisfiability problem of REG c logics. Our calculus is an extension of the tableau calculus for regular grammar logics given by Goré and Nguyen in [11]. To deal with converse, we use an analytic cut rule. Similarly to [21,12], our cut rule is a kind of "guessing the future" for nodes in traditional (unlabeled) tableaux. Besides, there is a substantial difference comparing to [11]. Namely, Goré and Nguyen introduced only universal automaton-modal operators for regular grammar logics, while using cuts to deal with converse we have to use also existential automaton-modal operators. As a consequence, our calculus for REG c deals also with "eventualities" (like operators α * of PDL). For that we adopt the tableau method given by Pratt for PDL [26], but with a more direct formulation. Our tableaux in REG c logics are "and-or" graphs constructed using traditional tableau rules and global caching. The idea of global caching appeared in Pratt's work [26] on PDL and has been formalized and proved sound by Goré and Nguyen for traditional tableaux in a number of other modal and description logics [11,12,13,14]. Similarly as for PDL [26] but in contrast with [11,12,13,14], checking satisfiability in REG c logics deals not only with the local consistency but also with a global consistency property of the constructed "and-or" graph. Using our tableau calculus, we give an ExpTime (optimal) tableau decision procedure for the general satisfiability problem of REG c logics. We also briefly discuss optimizations for the procedure. REG c logics can also be used as description logics. Two basic components of description logic theories are ABoxes and TBoxes. An ABox (assertion box) consists of facts, and a TBox (terminological box) consists of formulas expressing relationships between concepts. In [9], by encoding the ABox by "nominals" and "internalizing" the TBox, De Giacomo showed that the complexity of checking consistency of an ABox w.r.t. a TBox in CPDL is ExpTime-complete. Using the translation of REG c logics into CPDL given by Demri and de Nivelle [4], one can show that the problem of checking consistency of an ABox w.r.t. a TBox in a REG c logic is ExpTime-complete. Extending our method to deal with ABox assertions, we give the first ExpTime tableau decision procedure not based on transformation for checking consistency of an ABox w.r.t. a TBox in a REG c logic.