Automated Theorem-Proving for Theories with Simplifiers Commutativity, and Associativity

Abstract

To prove really difficult theorems, resolution principle programs need to make better inferences and to make them faster. An approach is presented for taking advantage of the structure of some special theories. These are theories with simplifiers, commutativity, and associativity, which are valuable concepts to build in, since they so frequently occur in important theories, for example, number theory (plus and times) and set theory (union and intersection). The object of the approach is to build in such concepts in a (refutation) complete, valid, efficient (in time) manner by means of a “natural” notation and/or new inference rules. Some of the many simplifiers that can be built in are axioms for (left and right) identities, inverses, and multiplication by zero. As for results, commutativity is built in by a straightforward modification to the unification (matching) algorithm. The results for simplifiers and associativity are more complicated. These theoretical results can be combined with one another and/or extended to either C -linear refutation completeness or theories with partial ordering, total ordering, or sets. How these results can serve as the basis of practical computer programs is discussed.

Completion and Its Applications

“…The effect of completion. in this case, is to narrow sand t until a substitution (j is generated which equates the two terms in the theory presented by R. Thus, narrowing, as defined by Slagle (1974) and used for this purpose by Fay (1979) and Lankford and Ballantyne (1979), is a restricted form of completion (as was understood in Lankford (1975) and is noted in Dershowitz (1985a) and Rety et al (1985)). Since the equality symbol is presumed not to appear in R, any critical pair between a rule I ~ r in R and a generated rule eq(u, v) ~ F(wJ,"" w n ), must involve an overlap of I on a (not necessarily proper) nonvariable subterm u!p of u or v!p of v. The resultant critical pair can be oriented into a new rule, eq (u[r!,]…”

Multi-paradigm Declarative Languages

“…On the operational side, the resolution principle of logic programming must be extended to deal with replacements of subterms. Narrowing, originally introduced in automated theorem proving [81], is a constructive method to deal with such replacements. For this purpose, defining equations are interpreted as rewrite rules that are only applied from left to right (as in functional programming).…”

“…Often, non-constructor-based rules specify properties of functions rather than providing a constructive definition (compare rule assoc above that specifies the associativity of "++"), or they can be transformed into constructor-based rules by moving non-constructor terms in left-hand side arguments into the condition (e.g., rule last ). Although there exist narrowing strategies for non-constructor-based rewrite rules (see [38,74,81] for more details), they often put requirements on the rewrite system that are too strong or difficult to check in universal programming languages, like termination or confluence. An important insight from recent works on functional logic programming is the restriction to constructor-based programs since this supports the development of efficient and practically useful evaluation strategies (see below).…”

From Boolean Equalities to Constraints

“…Functions are defined by (conditional) equations and are evaluated lazily. Function calls with free variables are evaluated by a possibly non-deterministic instantiation of demanded arguments which corresponds to narrowing [28,25]. Curry narrows with possibly non-most-general unifiers to ensure the optimality of computations [4].…”