Modal operator

About: Modal operator is a research topic. Over the lifetime, 1151 publications have been published within this topic receiving 22865 citations. The topic is also known as: modal connective.

On the Deterministic Horn Fragment of Test-free PDL

Abstract: We study the deterministic Horn fragment of test-free proposi- tional dynamic logic (PDL(0)). This fragment adopts the restriction that, in bodies of program clauses and goals, special universal modal operators which are a kind of combination of ! and " are used instead of ! . The fragment contains deterministic positive logic programs and deterministic negative clauses, whose negations form serial positive formulae. A least Kripke model for a deterministic positive logic program in PDL (0) may not exist, because PDL(0) is a non-serial modal logic. In this work, we present an algorithm that, given a deterministic positive logic program P in PDL (0) , constructs a least pseudo-model of P. A pseudo-model is similar to a Kripke model except that it contains two sets of accessibility relations, one for dealing with existential modal operators and the other for dealing with universal modal operators. A least pseudo-model M of P has the property that, for every serial positive formula ! , P |= ! i! M |= ! . Fur- thermore, checking whether M |= ! is solvable in polynomial time in the sizes of M and ! . Our algorithm runs in exponential time and returns a pseudo-model with size 2 O(n 2 ) . We give a deterministic positive logic pro- gram in PDL(0) such that every pseudo-model characterizing it must have size 2 ! (n) .

A Fixpoint Semantics and an SLD-Resolution Calculus for Modal Logic Programs

Abstract: We propose a modal logic programming language called MProlog, which is as expressive as the general modal Horn fragment. We give a fixpoint semantics and an SLD-resolution calculus for MProlog in all of the basic serial modal logics KD, T, KDB, B, KD4, S4, KD5, KD45, and S5. For an MProlog program P and for L being one of the mentioned logics, we define an operator T_{L,P}, which has the least fixpoint I_{L,P}. This fixpoint is a set of formulae, which may contain labeled forms of the modal operator d, and is called the least L-model generator of P. The standard model of I_{L,P} is shown to be a least L-model of P. The SLD-resolution calculus for MProlog is designed with a similar style as for classical logic programming. It is sound and complete. We also extend the calculus for MProlog in the almost serial modal logics KB, K5, K45, and KB5.

Bilattices and modal operators

Abstract: A bilattice is a set equipped with two partial orders and a negation operation that inverts one of them while leaving the other unchanged; it has been suggested that the truth values used by inference systems should be chosen from such a structure instead of the two-point set {t, f}. Given such a choice, we redefine a modal operator to be a function on the bilattice selected, and show that this definition generalizes both Kripke's possible worlds approach and Moore's autoepistemic logic. Extensions to causal and temporal reasoning are also discussed.