Abstract. In this paper, a proof method based on a notion of transfinite semantic trees is presented and it is shown how to apply it to prove the completeness of ...
Proving
Refutational
Strategies:
Completeness
The Transfinite
Semantic
of Theorem-Proving Tree Method
JIEH HSIANG SUNY
at Stony Brook,
Stony Brook,
New
York
AND MICHAEL CRIN,
Nancy,
RUSINOWITCH France
Abstract. In this paper, a proof method based on a notion of transfinite semantic trees is presented and it is shown how to apply it to prove the completeness of refutational theorem proving methods for first order predicate calculus with equality. To demonstrate how this method is used, the completeness of two theorem-proving strategies, both refinements of resolution and paramodulation, are proved. Neither of the strategies need the functionally reflexive axioms nor paramodulating into variables. Therefore the Wos-Robinson conjecture follows as a corollary. Another strategy for Horn logic with equality is also presented. Categories and Subject Descriptors: F.4. 1: [Mathematical Logic and Formal Languages]: Mathematical Logic — mechanical theorem proving; proof theory; I. 2.3: [Artificial Intelligence]: Deduction and Theorem Proving —Deduction; resolution General Terms: Theory Additional Key Words and Phrases: Reftrtational Theorem Proving Strategies, Transfinite Ordinals, Transtlnite Semantic Trees, Resolution, Complete simplification orderings, completeness, first-order logic with equality, functional reflexive axioms, paramodulation, resolution. transfinite ordinals, transfinite semantic trees
The research for this paper was supported in part by National Science Foundation (NSF) grants CCR 88-05734, CCR 89-01322, and INT 87-15231, and by Greco Programmation of France. The research reported in this paper was initiated in 1985, when J. Hsiang was visiting CRIN, Nancy, France, as a visiting scientist of CNRS. Subsequently, M. Rusmowitch visited Stony Brook, partly sponsored by NSF. The generous supports of various agencies and laboratories are gratefully acknowledged. Authors’ present addresses: J. Hsiang, Department of Computer Science, SUNY at Stony Brook, Stony Brook, NY 11794-4400: M. Rusinowitch, CRIN, B. P. 239, 54506 Vandoeuvre-les-Nancy, Nancy, France. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its data appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. @ 1991 ACM 0004-5411 /91 /0700-0559
$01.50
Journal of the Asbocr.t,on for Comput,ng Machinery, Vol 38, No 3, July 1991. PP 559-5S7
560
J.
HSIANGAND M. RUSINOWITCH
1. Introduction The problem of proving the completeness of theorem-proving strategies has been prominent in automated theorem proving since its first conception in the late 50’s. Among the most difficult are the strategies involving equality. A notorious instance is the question of whether the inference system consisting of resolution and paramodulation is complete without the functionally reflexive axioms and without paramodulating into variables. In [2], an indirect proof (as a corollary of the completeness of the modification method) was given. A direct proof was given in [23]. However, Peterson’s proof requires the use of a simplification ordering that is also order isomorphic to u on ground atoms. This puts a serious restriction on the applicability of his method since most of the commonly used orderings do not have this property. Peterson conjectured that lifting this restriction requires a proof method that can successfully deal with transfinite semantic trees. In this paper, we present such a method. Our method differs from the usual semantic tree proof methods in several aspects. In addition to the apparent effect of being able to deal with transfinite objects, it is a refutational proof method rather than an inductive one. Furthermore, there is no need for Herbrand’s Theorem relating the inconsistence y of a set of clauses with a finite set of ground instantiation. At the core of our method is the notion of a complete simplification ordering, which we impose on the term structure of the language of first-order logic. Intuitively, this type of well-founded ordering takes into account the syntactic structure of terms when comparing them. The E-interpretations, that is, the Herbrand interpretations that preserve the equality axioms, can be built (transfinite) inductively with respect to these orderings. The collection of these E-interpretations forms a (transfinite) E-semantic tree. The proof method we present in this paper is a way to manipulate these E-semantic trees. Complete simplification orderings are much less restrictive than the orderings employed in [23]. In fact, almost every commonly used ordering in the term rewriting literature (see [5] for a survey) is either a complete simplification ordering or can be easily modified into one. Not only are these orderings essential for the completeness proofs, they can also be utilized to refine the inference rules. First, resolution and paramodulation need only be performed on the maximal literals. Second, when using an equality in a clause to paramodulate, only the larger of the two terms in the equality needs to be considered for paramodulation. This ordering strategy is complete without the functionally reflexive axioms and without paramodulating into variables. It is also compatible with deletion inference rules such as demodulation (simplification), subsumption, and tautology elimination. We do not know of any other ordering/indexing resolution-type strategies that enjoy these properties. As another example of how the proof method works, we give a complete strategy for first-order logic with equality that refines and extends the P1 -strategy [26]. The layout of the rest of the paper is as follows: In Section 2, we give the preliminary notions as well as the definition of complete simplification orderings. The orderings are used to build E-interpretations and transfinite E-semantic trees in Section 3. In Section 4, we introduce our proof method based on E-semantic trees. To demonstrate how the proof method is used, in Section 5, we present an ordered literals strategy for first-order logic with equality and prove that it is complete. Further refinements such as blocked inferences are
Proving
Refutational
Completeness
of Theorem-Proving
Strategies
561
also described. Comparisons with ordering/indexing strategies are given. As another example, we introduce the positive strategy in Section 6. The positive strategy is a mixture of the P 1-strategy and the ordered literals strategy. Its completeness is also proved. As a consequence, we have also proved the completeness of a positive unit strategy for Horn logic with equality. In addition to presenting a new proof method, the new results in this paper are the completeness of the strategies in Sections 5 and 6. We emphasize that our strategies need neither the functionally reflexive axioms nor paramodulating into variables. The completeness of resolution and paramodulation without the functionally reflexive axioms is a straightforward corollary of the completeness theorem in Section 5. The proof method in this paper has also been used to give a refutationally complete Knuth – Bendix type procedure [9]. It can also be extended to a more general framework for proving the completeness of special inference rules for other axioms and operators. One such extension, to incorporate the cancellation axioms, is given in [1 O], where complete sets of inference rules for various forms of the cancellation axioms are given. 2. Preliminaries In this section, we review some basic concepts and notations. Let F be a finite set of functions with arities, and let X be a countably infinite set of variables. The algebra of terms composed from F and X is denoted by T(F, X). We use T(F) for the set of ground terms (the Herbrand universe). Let P be a finite set of predicate symbols including the equality . . . . t.): pGP and predicate = . The set of atoms A(P, F, X) is {p(t,, tie T( F, X)}. We denote the set of ground atoms (the Herbrand base) by A( P, F). An equality atom is an atom whose predicate symbol is = . Throughout this paper, we assume that = is commutative in the sense that we do not distinguish between the atoms s = t and t = s. A literal is either an atom or the negation of an atom, and a clause is a disjunction of literals. In general, we use the term object to indicate a term, an atom, a literal, or a clause, and the term ground object to indicate a ground term, atom, literal, or a ground clause. Let V(t) denote the set of variables appearing in an object t.A substitution is a mapping o from X to T( F, X) such that a(x) # x for only finitely many variables. We use Dom( a) to denote the set { x : O(x) # x}. We further assume whenever possible that for every x e Dom( o), V( O( x)) n Dom( o) = ~. The substitution o is applied to an object t if all variables x in t are replaced by a ( x). The result is denoted as ta. A substitution a is a unifier of two objects s and t if so = tc. A unifier o of s and t is the most general unifier ( mgu) if for every unifier 0 of s and t, there exists a substitution p such that 19= OP. The mgu is unique up to renaming of variables [26]. To express subterms and substitutions more effectively, we sometimes use positions. Envision a term represented as a tree; a position in a term indicates a node in the tree. Positions are usually represented as sequences of integers. Letting u be a position, we use t/u for the subterm of t at u. More precisely, t/A = t where h is the empty position, and g(tl, . . . . t~)/i. u = ti/u. We also use s = S[ u - t]to denote that s is a term whose subterm at position u is t. For convenience, we sometimes express it by s = S[ t] if the particular position is not important. A subterm of t is proper if it is distinct from t.
562
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HSIANGAND M. RUSINOWITCH
2.1. AXIOMS FOR EQUALITY. The use of the equality assumes the axioms of a congruence relation: Vx(x
=x).
=yAy=Z)3x=
Z].
Given any predicate Vxy((x= Given any functionf, We call resolution able effort the use of example. 2.2.
yAp(.
,.,
vxy(x=y
implicitly
=x).
vxy(x=y3y Vxyz((x
predicate
x,.
symbolp,
..))3
of(.
...
P(.
...
x,.
..)=
J))..)). f(.
...
y,...)).
this set of axioms K following [3]. Using these axioms in the framework usually produces many redundant resolvents. Considerhas been spent on designing special inference rules for eliminating these axioms. The rule of paramodzdation [32] is the most notable
ORDERINGSON THE FIRST ORDER OBJECTS
2.2.1. Simple Facts about Ordinal Numbers. Before introducing the orderings, we present some simple properties about ordinal numbers that are needed later. Our default reference for all set-theoretic concepts, constructions, and theorems is [17]. A partiai order on a set A is a transitive, irreflexive binary relation on A x A. A partial order is linear (or total) if any two distinct elements in A are related to each other. A partial order is well-founded if any subset of A has at least one minimal element with respect to the ordering. A well-ordered set is a pair (A, < ) where A is a nonempty set and < is a well-founded total order on A. Two well-ordered sets (A, ~, or —P = Q, P is not the equality predicate, and (sl, . . . . Sfl) >t (t].....t,,,) compared lexicographically. Intuitive y, in > ~ all equality atoms are smaller than any nonequality atom. As we shall see later. such predicate-first orderings are quite effective in restricting the number of inferences. It is easy to show that >P is indeed a complete simplification ordering. In general, it is not order-isomorphic to o. Suppose there are only two predicate symbols = and P, one constant a. and two unary functions g and h. Assume further that the recursive path ordering [4], with a < g < h is used in ordering the terms T( F, X). In other words, the Herbrand universe is ordered as (recall ‘Multlset ordering [6] can be detined as. (1) A U {a} >.f 0, [2) A U {a} anct (3) A U {a] >., BU {b} If a>, band ,4 U {a} >~,B.
.> ~ B
U {a} If .4 >., B,
Proving
Refutational
Completeness
of Theorem-Proving
once again that atoms s = t and t = s are considered a cf
w
