A First Order Nonmonotonic Extension of Constructive Logic

Abstract

Certain extensions of Nelson's constructive logic N with strong negation have recently become important in artificial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's first-order logic as a basis for defining nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for first-order here-and-there logics, and their minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QN c 5 , as a basis for defining a first-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QN c 5 , including Skolem forms and Herbrand theorems and Interpolation, and show that the first-oder version of equilibrium logic can be used as a foundation for answer set inference.

A Revised Concept of Safety for General Answer Set Programs

“…In terms of satisfiability and validity this logic is equivalent to the logic previously introduced in [17]. To simplify notation we drop the labels for static domains and equality and refer to this logic simply as quantified here-and-there, QHT.…”

“…Two variations of this semantics, the open [8] and generalised open answer set [9] semantics, consider non-ground programs and open domains, thereby relaxing the PNA. For the present version of QEL the correspondence to answer sets can be summarised as follows (see [17,18,3]). If ϕ is a universal sentence in L = C, P (see §3 below), a total QHT model U, T, T of ϕ is an equilibrium model of ϕ iff T, T is a propositional equilibrium model of the grounding of ϕ with respect to the universe U .…”

“…It follows that if Π is a logic program (of any form), a total QHT model U, T, T of Π is an equilibrium model of Π iff it is a generalised open answer set of Π in the sense of [9]. If we assume all models are UNA-models, we obtain the version of QEL found in [17]. There, the following relation of QEL to (ordinary) answer sets for logic programs with variables was established.…”

“…A universal theory is a set of universal sentences. For QHT, normal forms such as prenex and Skolem forms were studied in [17]. In particular it is shown there that in quantified here-and-there logic every sentence is logically equivalent to a sentence in prenex form.…”

“…A stable model semantics for first-order structures and languages was defined in the framework of equilibrium logic in [16,17,18]. As a logical basis the non-classical logic of quantified here-and-there, QHT, is used (see also [14]).…”