Providing declarative semantics for HH extended constraint logic programs

Abstract

This paper is focused on a double extension of traditional Logic Programming which enhances it following two different approaches. On one hand, extending Horn logic to hereditary Harrop formulas (HH ), in order to improve the expressive power; on the other, incorporating constraints, in order to increase the efficiency. For this combination, called HH(C), an operational semantics exists, but no declarative semantic for it has been defined so far.One of the main features of (Constraint) Logic Programming is that the algorithmic behavior of (constraint) logic programs and its mathematical interpretations agree with each other, in the sense that the declarative meaning of a program can be interpreted operationally as a goal-oriented search for solutions. Both operational (algorithmic) and declarative (mathematical) semantics for programs are useful and widely developed in the frame of Logic Programming as well as in its extension, Constraint Logic Programming.For these reasons, HH(C) was in need of a more mathematical interpretation of programs. In this paper some fixed point semantics for HH(C) are presented. Taking as a starting point the techniques used by Miller to interpret the fragment of HH that incorporates intuitionistic implication in goals, we have formulated two novel extensions capable of dealing with the whole HH logic, including universal quantifiers, as well as with the presence of constraints. Those semantics are proved to be sound and complete w.r.t. the operational semantics of HH(C).

Formalizing a Constraint Deductive Database Language Based on Hereditary Harrop Formulas with Negation

“…The proof is an adaptation of those presented in [7] to the definition of the forcing relation defined now for HH ¬ (C). Notice that, since we are assuming that str(G) = 1, then the case G ≡ ¬A has not to be considered.…”

“…The original formalisms in which HH(C) is founded [10,7] are not enough expressive to represent set difference, so it is incomplete with respect to RA. We will extend the scheme including negation to obtain a complete Constraint Deductive Database (CDDB ) language w.r.t.…”

“…Several kinds of semantics have been defined for HH(C) without negation, including proof theoretic, operational [10] and fixed point semantics [7], as well as for its higher-order version [11]. The simplest way for explaining the meaning of programs and goals in the present framework is by using a proof theoretic semantics.…”

“…We have extended the semantics presented in [7] in order to interpret full HH ¬ (C). The semantics there defined is based on a forcing relation among programs, sets of constraints and goals that states whether an interpretation makes true a goal G in the context Δ, Γ of a program and a set of constraints.…”

“…The fixed point semantics defined in [7] for HH(C) was proved to be sound and complete with respect to the calculus UC. Our interest now is to prove soundness and completeness of the new fixed point semantics for HH ¬ (C), with respect to the extended calculus UC ¬ .…”