As a remedy for the bad computational behaviour of first-order temporal logic (FOTL), it has recently been proposed to restrict the application of temporal operators to formulas with at most one free variable thereby obtaining so-called monodic fragments of FOTL. In this paper, we are concerned with constructing tableau algorithms for monodic fragments based on decidable fragments of first-order logic like the two-variable fragment or the guarded fragment. We present a general framework that shows how existing decision procedures for first-order (...) fragments can be used for constructing a tableau algorithm for the corresponding monodic fragment of FOTL. Some example instantiations of the framework are presented. (shrink)
In 1984, Danecki proved that satisfiability in IPDL, i.e., Propositional Dynamic Logic (PDL) extended with an intersection operator on programs, is decidable in deterministic double exponential time. Since then, the exact complexity of IPDL has remained an open problem: the best known lower bound was the ExpTime one stemming from plain PDL until, in 2004, the first author established ExpSpace-hardness. In this paper, we finally close the gap and prove that IPDL is hard for 2-ExpTime, thus 2-ExpTime-complete. We then sharpen (...) our lower bound, showing that it even applies to IPDL without the test operator interpreted on tree structures. (shrink)
Propositional dynamic logic is one of the most successful variants of modal logic. To make it even more useful for applications, many extensions of PDL have been considered in the literature. A very natural and useful such extension is with negation of programs. Unfortunately, as long-known, reasoning with the resulting logic is undecidable. In this paper, we consider the extension of PDL with negation of atomic programs, only. We argue that this logic is still useful, e.g. in the context of (...) description logics, and prove that satisfiability is decidable and EXPTIME-complete using an approach based on Büchi tree automata. (shrink)
We study satisfiability and infinite-state model checking in ICPDL, which extends Propositional Dynamic Logic (PDL) with intersection and converse operators on programs. The two main results of this paper are that (i) satisfiability is in 2EXPTIME, thus 2EXPTIME-complete by an existing lower bound, and (ii) infinite-state model checking of basic process algebras and pushdown systems is also 2EXPTIME-complete. Both upper bounds are obtained by polynomial time computable reductions to ω-regular tree satisfiability in ICPDL, a reasoning problem that we introduce specifically (...) for this purpose. This problem is then reduced to the emptiness problem for alternating two-way automata on infinite trees. Our approach to (i) also provides a shorter and more elegant proof of Danecki's difficult result that satisfiability in IPDL is in 2EXPTIME. We prove the lower bound(s) for infinite-state model checking using an encoding of alternating Turing machines. (shrink)
The aim of this paper is to construct a tableau decision algorithm for the modal description logic K ALC with constant domains. More precisely, we present a tableau procedure that is capable of deciding, given an ALC-formula with extra modal operators (which are applied only to concepts and TBox axioms, but not to roles), whether is satisfiable in a model with constant domains and arbitrary accessibility relations. Tableau-based algorithms have been shown to be practical even for logics of rather high (...) complexity. This gives us grounds to believe that, although the satisfiability problem for K ALC is known to be NEXPTIME-complete, by providing a tableau decision algorithm we demonstrate that highly expressive description logics with modal operators have a chance to be implementable. The paper gives a solution to an open problem of Baader and Laux [5]. (shrink)
In this paper, we investigate the relationship between two decidable interval-based temporal description logics that have been proposed in the literature, T L-ALCF and ALCF. Although many aspects of these two logics are quite similar, the two logics suggest two rather different paradigms for representing temporal conceptual knowledge. In this paper, we exhibit a reduction from T L-ALCF concepts to ALCF concepts that serves two purposes: first, it nicely illustrates the relationship between the two knowledge representation paradigms; and second, it (...) provides a tight PSPACE upper bound for T L-ALCF concept satisfiabiliy, whose complexity was previously unknown. (shrink)
