Projection and scope-determined circumscription

Abstract

We develop a semantic framework that extends first-order logic by literal projection and a novel second semantically defined operator, raising, which is only slightly different from literal projection and can be used to define a generalization of parallel circumscription with varied predicates in a straightforward and compact way. We call this variant of circumscription scopedetermined, since like literal projection and raising its effects are controlled by a so-called scope, that is, a set of literals, as parameter. We work out formally a toolkit of propositions about projection, raising and circumscription and their interaction. It reveals some refinements of and new views on previously known properties. In particular, we apply it to show that wellfoundedness with respect to circumscription can be expressed in terms of projection, and that a characterization of the consequences of circumscribed propositional formulas in terms of literal projection can be generalized to first-order logic and expressed compactly in terms of new variants of the strongest necessary and weakest sufficient condition.

Abduction in Logic Programming as Second-Order Quantifier Elimination

“…It allows to express variants of parallel predicate circumscription where the effects on each atom are controlled by a scope argument [37]. Atoms that occur just in a positive literal in the scope are minimized, atoms that occur just in a negative literal are maximized, atoms that occur in both polarities are fixed and atoms that do not at all occur in the scope are varying.…”

“…The globally weakest sufficient condition (GWSC) [37] is the application pattern of second-order quantification by which explanations with respect to classical logic are characterized as in (i) in the introduction. We specify it formally in terms of literal projection, such that also polarity can be constrained:…”

“…Aside of the consideration of polarity, GWSCs differ from WSCs in the sense of [24] in that for a given formula and scope only GWSCs are unique up to equivalence [37].…”

“…In this paper, we focus on the skeptical view for the other semantics. We consider finite normal ground programs, but the material should generalize to programs with disjunctive heads, negation as failure in the head, and first-order quantification, as indicated in [35,37].…”

“…Predicate quantification can be applied to express predicate circumscription [4]. We express circumscription by a dedicated second-order operator with a syntax analogous to projection [37]. We will develop the "constructive" characterizations of abductive explanations for the three considered logic programming semantics in parallel.…”