Abstract not found

In automated deduction systems that are intended for human use, the presentation of a proof is no less important than its discovery. For most of today’s automated theorem proving systems, this requires a non-trivial translation procedure to extract human-oriented deductions from machine-oriented proofs. Previously known translation procedures, though complete, tend to produce unintuitive deductions. One of the major flaws in such procedures is that too often the rule of indirect proof is used where the introduction of a lemma would result in a shorter and more intuitive proof. We present an algorithm, symmetric simplification, for discovering useful lemmas in deductions of theorems in first- and higher-order logic. This algorithm, which has been implemented in the TPS system, has the feature that resulting deductions may no longer have the weak subformula property. It is currently limited, however, in that it only generates lemmas of the form C ∨ ¬C ′ , where C and C ′ have the same negation normal form. 1

Since logic programming systems directly implement search and unification and since these operations are essential for the implementation of most theorem provers, logic programming languages should make ideal implementation languages for theorem provers. We shall argue that this is indeed the case if the logic programming language is extended in several ways. We present an extended logic programming language where first-order terms are replaced with simply-typed A-terms, higher-order unification replaces firstorder unification, and implication and universal quantification are allowed in queries and the bodies of clauses. This language naturally specifies inference rules for various proof systems. The primitive search operations required to search for proofs generally have very simple implementations using the logical connectives of this extended logic programming language. Higher-order unification, which provides sophisticated pattern matching on formulas and proofs, can be used to determine when and at what instance an inference rule can be employed in the search for a proof. Tactics and tacticals, which provide a framework for high-level control over search, can also be directly implemented in this extended language. The theorem provers presented in this paper have been implemented in the higher-order logic programming language AProlog. 1