Logic programming: laxness and saturation

TL;DR

This paper unifies lax and saturated semantics in logic programming, offering a comprehensive framework that captures derivations and proof-search, and extends to models with existential variables and local state.

Contribution

It unifies two semantic approaches in logic programming, refining lax semantics for finitary models and relating variables to local states.

Findings

Unified lax and saturated semantics as complementary

Refined lax semantics for finitary models with existential variables

Established semantic link between variables and local states

Abstract

A propositional logic program $P$ may be identified with a $P_fP_f$-coalgebra on the set of atomic propositions in the program. The corresponding $C(P_fP_f)$-coalgebra, where $C(P_fP_f)$ is the cofree comonad on $P_fP_f$, describes derivations by resolution. That correspondence has been developed to model first-order programs in two ways, with lax semantics and saturated semantics, based on locally ordered categories and right Kan extensions respectively. We unify the two approaches, exhibiting them as complementary rather than competing, reflecting the theorem-proving and proof-search aspects of logic programming. While maintaining that unity, we further refine lax semantics to give finitary models of logic programs with existential variables, and to develop a precise semantic relationship between variables in logic programming and worlds in local state.