On the Complexity of Fragments of Modal Logics

Electronic Proceedings in Theoretical Computer Science

This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the µ-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce a terminating tableau system for the logics we study, based on Kozen's tableau for the µ-calculus, and the one of Fitting and Massacci for modal logic.

Logic Journal of the IGPL, 2018

While finite-variable fragments of the propositional modal logic S5-complete with respect to reflexive, symmetric, and transitive frames-are polynomialtime decidable, the restriction to finite-variable formulas for logics of reflexive and transitive frames yields fragments that remain "intractable." The role of the symmetry condition in this context has not been investigated. We show that symmetry either by itself or in combination with reflexivity produces logics that behave just like logics of reflexive and transitive frames, i.e., their finite-variable fragments remain intractable, namely PSPACE-hard. This raises the question of where exactly the borderline lies between modal logics whose finite-variable fragments are tractable and the rest.

Lecture Notes in Computer Science, 1999

We propose so called clausal tableau systems for the common modal logics K4, KD4 and S4. Basing on these systems, we give more efficient decision procedures than those hitherto known for the considered logics. In particular space requirements for our logics are reduced from the previously established bound O(n 2. log n) to O(n. log n).

Theory and Practice of Logic Programming

A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames has already been established. By “traditional” classes of frames, we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper, we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular, we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the u...

Lecture Notes in Computer Science, 2019

A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames have already been established. By "traditional" classes of frames, we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper, we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular, we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the undecidability of the language with graded converse modalities.

Mathematical Foundations of Computer Science, 2010

Many tractable algorithms for solving the Constraint Satisfaction Problem (Csp) have been developed using the notion of the treewidth of some graph derived from the input Csp instance. In particular, the incidence graph of the Csp instance is one such graph. We introduce the notion of an incidence graph for modal logic formulae in a certain normal form. We investigate

2018

SAT solvers have become efficient for solving NP-complete problems (and beyond). Usually those problems are solved by direct translation to SAT or by solving iteratively SAT problems in a procedure like CEGAR. Recently, a new recur-sive CEGAR loop working with two abstraction levels, called RECAR, was proposed and instantiated for modal logic K. We aim to complete this work for modal logics based on axioms (B), (D), (T), (4) and (5). Experimental results show that the approach is competitive against state-of-the-art solvers for modal logics K, KT and S4.

We present two families of exponential lower bounds on the size of modal formulae and use them to establish the following succinctness results. We show that the logic of contingency (ConML) is exponentially more succinct than basic modal logic (ML). We strengthen the known proofs that the so-called public announcement logic (PAL) in a signature containing at least two different diamonds and one propositional symbol is exponentially more succinct than ML by showing that this is already true for signatures that contain only one diamond and one propositional symbol. As a corollary of these results, we obtain an alternative proof of the fact that modal circuits are exponentially more succinct than ML-formulae.

arXiv (Cornell University), 2015

We investigate the computational complexity of the satisfiability problem of modal inclusion logic. We distinguish two variants of the problem: one for the strict and another one for the lax semantics. Both problems turn out to be EXPTIME-complete on general structures. Finally, we show how for a specific class of structures NEXPTIMEcompleteness for these problems under strict semantics can be achieved.

Fundamenta Informaticae, 2013

We introduce a fragment of multi-sorted stratified syllogistic, called 4LQS R , admitting variables of four sorts and a restricted form of quantification, and prove that it has a solvable satisfiability problem by showing that it enjoys a small model property. Then, we consider the sublanguage (4LQS R) k of 4LQS R , where the length of quantifier prefixes (over variables of sort 1) is bounded by k ≥ 0, and prove that its satisfiability problem is NP-complete. Finally we show that modal logics S5 and K45 can be expressed in (4LQS R) 1. 1 We recall that, for any set s, pow(s) denotes the powerset of s, i.e., the collection of all subsets of s.