Satisfiability Problem For Logics Research Articles

Reasoning about Data Repetitions with Counter Systems

We study linear-time temporal logics interpreted over data words with multiple attributes. We restrict the atomic formulas to equalities of attribute values in successive positions and to repetitions of attribute values in the future or past. We demonstrate correspondences between satisfiability problems for logics and reachability-like decision problems for counter systems. We show that allowing/disallowing atomic formulas expressing repetitions of values in the past corresponds to the reachability/coverability problem in Petri nets. This gives us 2EXPSPACE upper bounds for several satisfiability problems. We prove matching lower bounds by reduction from a reachability problem for a newly introduced class of counter systems. This new class is a succinct version of vector addition systems with states in which counters are accessed via pointers, a potentially useful feature in other contexts. We strengthen further the correspondences between data logics and counter systems by characterizing the complexity of fragments, extensions and variants of the logic. For instance, we precisely characterize the relationship between the number of attributes allowed in the logic and the number of counters needed in the counter system.

Satisfiability in composition-nominative logics

Abstract Composition-nominative logics are algebra-based logics of partial predicates constructed in a semantic-syntactic style on the methodological basis, which is common with programming. They can be considered as generalizations of traditional logics on classes of partial predicates that do not have fixed arity. In this paper we present and investigate algorithms for solving the satisfiability problem in various classes of composition-nominative logics. We consider the satisfiability problem for logics of the propositional, renominative, and quantifier levels and prove the reduction of the problem to the satisfiability problem for classical logics. The method developed in the paper enables us to leverage existent state-of-the-art satisfiability checking procedures for solving the satisfiability problem in composition-nominative logics, which could be crucial for handling industrial instances coming from domains such as program analysis and verification. The reduction proposed in the paper requires extension of logic language and logic models with an infinite number of unessential variables and with a predicate of equality to a constant.

Characterizing the NP-PSPACE Gap in the Satisfiability Problem for Modal Logic

There has been a great deal of work on characterizing the complexity of the satisfiability and validity problem for modal logics. In particular, Ladner showed that the satisfiability problem for all logics between K and S4 is PSPACE-hard, while for S5 it is NP-complete. We show that it is negative introspection, the axiom ¬Kp ⇒ K¬Kp, that causes the gap: if we add this axiom to any modal logic between K and S4, then the satisfiability problem becomes NP-complete. Indeed, the satisfiability problem is NP-complete for any modal logic that includes the negative introspection axiom.

Complexity Results for First-Order Two-Variable Logic with Counting

Let $C^2_p$ denote the class of first-order sentences with two variables and with additional quantifiers exists exactly (at most, at least) $i$ for $i\leq p$, and let $C^2$ be the union of $C^2_p$ taken over all integers $p$. We prove that the satisfiability problem for $C^2_1$ sentences is NEXPTIME-complete. This strengthens the results by [E. Gradel, Ph. Kolaitis, and M. Vardi, Bull. Symbolic Logic, 3 (1997), pp. 53--69], who showed that the satisfiability problem for the first-order two-variable logic $L^2$ is NEXPTIME-complete and by [E. Gradel, M. Otto, and E. Rosen, 12th Annual IEEE Symposium on Logic in Computer Science, 1997, pp. 306--317], who proved the decidability of $C^2$. Our result easily implies that the satisfiability problem for $C^2$ is in nondeterministic, doubly exponential time. It is interesting that $C^2_1$ is in NEXPTIME in spite of the fact that there are sentences whose minimal (and only) models are of doubly exponential size. It is worth noticing that by a recent result of [E. Gradel, M. Otto, and E. Rosen, Proceedings of 14th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 1200, Springer-Verlag, Berlin, 1997], extensions of two-variable logic $L^2$ by a weak access to cardinalities through the Hartig (or equicardinality) quantifier is undecidable. The same is true for extensions of $L^2$ by very weak forms of recursion. The satisfiability problem for logics with a bounded number of variables has applications in artificial intelligence, notably in modal logics (see, e.g., [W. van der Hoek and M. De Rijke, J. Logic Comput., 5 (1995), pp. 325--345]), where counting comes in the context of graded modalities and in description logics, where counting can be used to express so-called number restrictions (see, e.g., [A. Borgida, Artificial Intelligence, 82 (1996), pp. 353--367]).