Constructing Finite Least Kripke Models for Positive Logic Programs in Serial Regular Grammar Logics

A serial context-free grammar logic is a normal multimodal logicL characterized by the seriality axioms and a set of inclusionaxioms of the form _ts1..._sk. Such an inclusion axiom correspondsto the grammar rule t s₁... s_k. Thus the inclusion axioms ofL capture a context-free grammar . If for every modal index t, the set of words derivable fromt using is a regular language, then L is a serial regular grammar logic. In this paper, we present an algorithm that, given a positivemultimodal logic program P and a set of finite automata specifyinga serial regular grammar logic L, constructs a finite leastL-model of P. (A model M is less than or equal to model M' iffor every positive formula , if M then M' .) A least L-modelM of P has the property that for every positive formula , P iff M . The algorithm runs in exponential time and returnsa model with size 2^O(n³). We give examples of P and L, for bothof the case when L is fixed or P is fixed, such that every finiteleast L-model of P must have size 2⁽ⁿ⁾. We also prove that ifG is a context-free grammar and L is the serial grammar logiccorresponding to G then there exists a finite least L-modelof _s p iff the set of words derivable from s using G is a regularlanguage.