Decision Procedures For Modal Logics Research Articles

A Subset-Matching Size-Bounded Cache for Testing Satisfiability in Modal Logics

The implementation of efficient decision procedures for modal logics is a major research problem in automated deduction. Caching the result of intermediate consistency checks has experimentally proved to be a very important technique for attaining efficiency. Current state-of-the-art systems implement caching mechanisms based on hash tables. In this paper we present a data type – that we call “bit matrix” – for caching the (in)consistency of sets of formulas. Bit matrices have three distinguishing features: (i) they can be queried for subsets and supersets, (ii) they can be bounded in size, and (iii) if bounded, they can easily implement different policies to resolve which results have to be kept. We have implemented caching mechanisms based on bit matrices and hash tables in *SAT. In *SAT, the bit matrix cache is bounded, and keeps the latest obtained (in)consistency results. We experiment with the benchmarks proposed for the modal logic K at the “TABLEAUX Non Classical Systems Comparison (TANCS) 2000”. On the basis of the results, we conclude that *SAT performances are improved by (i) caching the results of intermediate consistency checks, (ii) using bit matrices instead of hash tables, and (iii) storing a small number of results in the bit matrices.

Building Decision Procedures for Modal Logics from Propositional Decision Procedures: The Case Study of Modal K(m)

The goal of this paper is to propose a new technique for developing decision procedures for propositional modal logics. The basic idea is that propositional modal decision procedures should be developed on top of propositional decision procedures. As a case study, we consider satisfiability in modal K ( m ), that is modal K with m modalities, and develop an algorithm, called K SAT , on top of an implementation of the Davis–Putnam–Longemann–Loveland procedure. K SAT is thoroughly tested and compared with various procedures and in particular with the state-of-the-art tableau-based system K RIS . The experimental results show that K SAT outperforms K RIS and the other systems of orders of magnitude, highlight an intrinsic weakness of tableau-based decision procedures, and provide partial evidence of a phase transition phenomenon for K ( m ).

SAT vs. Translation Based decision procedures for modal logics: a comparative evaluation

ABSTRACT This paper follows on previous papers which present and evaluate various decision procedures for modal logics. We consider new test sets and systems that have been recently proposed in the literature. This new experimental analysis confirm previous experimental results in showing that SAT based decision procedures, i.e., the procedures built on top of decision procedures for propositional satisfiability, are more efficient than tableau based decision procedures. They also confirm previous evidence of an easy-hard-easy pattern in the satisfiability curve for modal K. Finally, on these tests, SAT based decision procedures are also more efficient than the recently proposed decision procedures based on translation methods. These results contradict some of the claims presented in previous papers by other authors.

Tableaux Based Decision Procedures for Modal Logics of Confluence and Density

In this paper we present a decision procedures for modal logics with transitive models together with additionnal properties like confluence and density. These procedures are tableaux-based, and we show how to handle them in this well-known framework by generalizing usual tableaux (which are trees) to richer structures: directed acyclic graphs.

On control of systems modelled as deterministic Rabin automata

Recent results on the control of infinite behaviour of finite automata are extended to allow Rabin acceptance conditions as modelling assumptions as well as specifications. The key result is a fixpoint characterization of the automaton'scontrollability subset—the set of states from which it can be controlled to the satisfaction of its associated specification. The fixpoint characterization allows for straightforward computation of the subset and for effective synthesis of controllers. the results have potential applications to supervisory control synthesis, the synthesis of reactive systems, and decision procedures for modal logics.

Automata-theoretic techniques for modal logics of programs

We present a new technique for obtaining decision procedures for modal logics of programs. The technique centers around a new class of finite automata on infinite trees for which the emptiness problem can be solved in polynomial time. The decision procedures then consist of constructing an automaton A f for a given formula f such that A f accepts some tree if and only if f is satisfiable. We illustrate our technique by giving exponential decision procedures for several variants of deterministic propositional dynamic logic.