A New Method of Proving the Existence of Answer Sets for Disjunctive Logic Programs: a Metric Fixed Point Theorem for Multi-valued Maps

After the success of stable model semantics [GL88] and its generalization for programs with classical negations [GL90], the same ideas have been used in [GL91] to include disjunction into the resulting formalization of commonsense reasoning. The resulting semantics of disjunctive logic programs is relatively new, and therefore, few results are known. An additional problem with this semantics is as follows: since it incorporates a larger number of logical connectives, it is inevitably more complicated, and therefore, usual syntactic proofs are far less intuitive than the ones for stable models. That these proofs become really complex, one can see from the fact that even for the simplest case of stratified programs, it is not so easy to prove that there always exists an answer set becomes really complicated. So, what is needed is a new methodology that would enable us to prove the existence of answer sets without going into syntactical details. As a basis for this new methodology, we decided to use fixed point theorems. Traditionally, logic programming uses fixed point theorems a lot (see, e.g., [L87]), but these are fixed point theorems for monotonic mappings of ordered sets (to be more precise, Tarski’s theorem on lattices). In disjunctive logic programming, an answer set can also be formulated as a fixed point, but this is a fixed point of a multi-valued mapping, and for such mappings neither Tarski’s theorem, not other known result is applicable. To overcome this difficulty, we decided to have a look at another case when Tarski’s theorem was not directly applicable. Such a case was analyzed by M. Fitting in [F93] and [F93a]. Fitting showed that in some cases, we can use metric fixed point theorems to prove the existence of stable models. Fitting’s theorem uses historically the first general metric fixed point theorem: so-called Banach’s contraction principle [AK90], [K55]. This theorem is also not directly applicable to disjunctive logic programs, because it is about singlevalued mappings, and we are dealing with multi-valued ones. However, Fitting theorem turned out to be a prefect starting point for us. In this paper, we describe a generalization of the contraction principle to multi-valued mappings, and show that the resulting generalization can be used to produce a simple proof that every stratified disjunctive logic program has an answer set. This result is easily generalizable to locally stratified disjunctive logic programs [P88]. 1. BASIC DEFINITIONS General remark. Since the main purpose of this paper is to promote a new methodology, we have tried to make it as understandable as possible. For that reasons, we are including all the definitions; readers who are already familiar with these definitions, can skip the corresponding subsection. 1.1. What is a disjunctive logic program and what is an answer set [GL91] Remark. The definition of an answer set follows the tradition of a stable model semantics in that it is done in two steps: • first, if we have variables in the original logic program, we substitute all possible terms instead of them; as a result, we get a new program that contains only ground instances of all the rules; • after that, we apply a special procedure to this new logic program to check whether a guess is an answer set (or, in case of a stable model semantics, a stable model). Because of that, when defining answer set, we can safely assume that our program already has no atoms and contains only ground instances of the rules. Thus, we arrive at the following definition. Definition 1 [GL91]. Assume that a set A is given. Its elements will be called atoms. By a literal we mean either an atom, or an expression of the type ¬p, where p is an atom. This symbol ¬ will be called classical negation. The set of all literals is denoted by Lit. An extended rule (or rule for short) is an expression of the form L1 ∨ L2 ∨ ... ∨ Lk ←− Lk+1 ∧ Lk+2 ∧ ... ∧ Lk+m ∧ not Lk+m+1 ∧ ... ∧ not Lk+m+n, where n,m, k are non-negative integers. We will say that literals L1, ..., Lk are in the head, and literals Lk+1, ..., Lk+m+n are in the body of this rule. An extended disjunctive logic program Π is a set of extended rules. Remarks. 1. It is usually assumed that the set Lit coincides with all the literals that can be formulated in the language of the original logic program (with variables). 2. A rule with k = 1 and m = n = 0 is called a fact. Definition 2 (of an answer set) [GL91] 1. Let Π be an extended disjunctive program which doesn’t contain not. An answer set of Π is a minimal set S ⊆ Lit such that (i) for each rule L1∨L2∨...∨Lk ←− Lk+1∧Lk+2∧...∧Lk+m from Π, if Lk+1, .., Lk+m ∈ S, then, for some i ∈ {1, .., k}, Li ∈ S (sets that satisfy (i) are called closed under the rules of Π). (ii) if S contains a pair of complimentary literals (a and ¬a), then S = Lit. 2. Now, let Π be an arbitrary extended disjunctive logic program. For any set S ⊆ Lit, let Π denote the extended disjunctive program obtained from Π by deleting (i) each rule that has a formula not L in its body with L ∈ S, and (ii) all formulas of the form not L in the bodies of the remaining rules. Clearly Π doesn’t contain not, so for this program, we can use part 1 of this Definition to define the set of all its answer sets. We will denote this set by α ( Π ) . If S ∈ α ( Π ) , then we say that S is an answer set for Π. 1.2. What is a stratified disjunctive logic program Definition 3. Let Π be an extended logic program. We will say that Π is stratified if for some integer α > 1, Lit = ⋃