Circumscription Implies Predicate Completion (Sometimes)

Predicate completion is an approach to closed world reasoning which assumes that the given sufficient conditions on a predicate are also necessary. Circumscription is a formal device characterizing minimal reasoning i.e. reasoning in minimal models, and is realized by an axiom schema. The basic result of this paper is that for first order theories which are Horn in a predicate P, the circumscription of P logically implies P's completion axiom. Predicate completion [Clark 1978, Kowalski 19781 is a device for "closing off" a first order representation. This concept stems from the observation that frequently a world description provides sufficient, but not necessary, conditions on one or more of its predicates and hence is an incomplete description of that world. In reasoning about such worlds, one often appeals to a convention of common sense reasoning which sanctions the assumption the so-called closed world assumption [Reiter 19781 that the information given about a certain predicate is all and only the relevant information about that predicate. Clark interprets this assumption formally as the assumption that the sufficient conditions on the predicate, which are explicitly given by the world description, are also necessary. The idea is best illustrated by an example, so consider the following simple blocks world description: A and B are distinct blocks. A is on the table. B is on A. (1) These statements translate naturally into the following first order theory with equality, assuming the availability of general knowledge to the effect that blocks cannot be tables: BLOCK (A) BLOCK (B) 0~ (A,TABLE) ON (B,A) (2) A#B A# TABLE B # TABLE Notice that we cannot, from (2), prove that nothing is on B, i.e., (2) /+ (x> -ON(x,B), yet there is a common sense convention about the description (1) which should admit this conclusion. This convention holds that, roughly speaking, (1) is a description of all and only the relevant information about this world. To see how Clark understands this convention, consider the formulae (x) .x = A V x = B ZJ BLOCK (x) (3) (xy).x = A 6 y = TABLE V x * B d y = A 3 ON(x,y) which are equivalent, respectively, to the facts about the predicate BLOCK, and the predicate ON in (2). These can be read as "if halves", or sufficient conditions, of the predicates BLOCK and ON. Clark identifies the closed world assumption with the assumption that these sufficient conditions are also necessary. This assumption can be made explicit by augmenting the representation (2) by the "only if halves" or necessary conditions, of BLOCK and ON: (x). BLOCK(x) 1 x=A V x=B (xy). ON (x,y) 3 x=A & y=TABLE V x=B & y=A Clark refers to these "only if" formulae as the completions of the predicates BLOCK and ON respectively. It now follows that the first order representation (1) under the closed world assumption is (x). BLOCK(x) : x=A V x=B (xy). ON(x,y) = x=A & y=TABLE V x=B & y=A AfB A # TABLE B # TABLE From this theory we can prove that nothing is on B (x)-ON(x,B) a fact which was not derivable from the original theory (2). Circumscription [McCarthy 19801 is a different approach to the problem of "closing off" a first order representation. McCarthy's intuitions about the closed world assumption are essentially semantic. For him, those statements derivable from a first order theory T under the closed world assumption about a predicate P are just the statements true in all models of T which are minimal with respect to P. Roughly speaking, these are models in which P's extension is minimal. McCarthy forces the consideration of only such models by augmenting T with the following axiom schema, called the circumscription of P in T: T(t) & b> .4(x) = P(x)] = (xl. P(x) = @(xl From: AAAI-82 Proceedings. Copyright ©1982, AAAI (www.aaai.org). All rights reserved. Here, if P is an n-ary predicate, then $ is an n-ary predicate parameter. T(4) is the conjunction of the formulae of T with each occurrence of P replaced by 9. Reasoning about the theory T under the closed world assumption about P is formally identified with first order deductions from the theory T together with this axiom schema. This enlarged theory, denoted by CLOSUREp(T), is called the closure of T with respect to P. Typically, the way this schema is used is to "guess" a suitable instance of 4, one which permits the derivation of something useful. a fact which is not derivable from the original theory T. Notice that in order to make this work, a judicious choice of the predicate parameter 4, namely (5), was required. Notice also that this choice of $ is precisely the antecedent of the "if half" (3) of ON and that, by (6), the "only if half" the completion of ON is derivable from the closure of T with respect to ON. For this example, circumscription is at least as powerful as predicate completion. To see how this all works in practice, consider the blocks world theory (2), which we shall denote by T. To close T with repect to ON, augment T with the circumscription schema X,Y> = ON(x,y)l ON(x,y) = $(x,Y) (4) Here 4 is a 2-place predicate parameter. Intuitively, this schema says that if 4 is a predicate satisfying the same axioms in T as does ON, and if 4's extension is a subset of ON's, then ON's extension is a subset of 4's, i.e., ON has the minimal extension of all predicates satisfying the same axioms as ON. In fact, this example is an instance of a large class of first order theories for which circumscription implies predicate completion. Let T be a first order theory in clausal form (so that existential quantifiers have been eliminated in favour of Skolem functions, all variables are universally quantified, and each formula of T is a disjunct of literals). If P is a predicate symbol occurring in some clause of T, then T is said to be Horn in P iff every clause of T contains at most one positive literal in the predicate P. Notice that the definition allows any number of positive literals in the clauses of T so long as their predicates are distinct from P. Any such theory T may be partitioned into two disjoint sets To see how one might reason with the theory CLOSUREON( consider the following choice of the parameter + in the schema (4): Tp: those clauses of T containing exactly one positive literal in P, and T-Tp: those clauses of T containing no positive $(x,y) q x=A & y=TABLE V x=B 6 y=A (5) (but possibly negative) literals in P.