Derivation Procedures for Extended Stable Models

We present derivation proof procedures for extended stable model semantics. Given program II and goal G, G belongs to the well founded model of El iff there is a WFM-derivation for G in II. Likewise, given program II and goal G, G belongs to some extended stable model of II iff there is a XSM-derivation for G in II. Cor-rectness (completeness and soundness) of these procedures is discussed. Example derivations are exhibited, as well as a simple Prolog implementation that directly mirrors the procedures. 1 Introduction Well Founded Semantics (WFS) [Van Gelder et a/., 1990] adequately captures various forms of hypothetical reasoning [Pereira et a/., 1991c, Pereira et a/., 1991d, Pereira ct a/., 1991b, Pereira et al., 1991a] if we interpret the well-founded model (WFM) of a program II as a (possibly incomplete) core view of the world, the extended stable models (XSMs) specifying alternative complementary consistent views of the world, all of each containing the core WFM. The paper is organized as follows: in section 2 we review well founded semantics. In section 3 we define WFM-derivations, discuss their correctness, and give examples. Next, in section 4, we define XSM-derivations and discuss their correctness. Finally, in section 5, a Prolog implementation is produced, directly reflecting the derivation procedures mentioned. More details can be found in an extended version of this paper [Pereira et al., 1990]. By a logic program II we mean a finite set of universally closed rules of the form: where H is an atom and the Li's are literals. When n = 0, we also write where t stands for an atom satisfied in all models. Negative literals are expressed as 4 where A is an atom. We denote by Lit(II) the set of literals in ground(II), the Herbrand instantiation of program II. We denote the well founded model of a program II by WFM (II), and an extended stable model (which may be the well founded one) by XSM(II). 2 The Extended Stable Model Semantics In this section we characterize the Well Founded and Extended Stable Models of a program, based on [Przy-musinska and Przymusinski, 1990]. Alternative definitions of the Well Founded Semantics can be found in [Van Gelder ct a/., 1990] or in [Przymusinski, 1989]. Because the semantics is 3-valued, we begin by defining 3-valued interpretations. Definition 2.1 A 3-valued Herbrand interpretation I of a first-order language L is any pair (T; F), where …