Gazing: An approach to the problem of definition and lemma use

Abstract

A major outstanding problem in automated theorem proving research is determining the appropriate use of definitions and previously proved theorems within a proof. Presenting the theorem prover with only the formulae that are necessary for the proof might be viewed as ‘cheating’ and requires that the user prove the theorem ahead of time. In real world applications of theorem proving, this is almost certain to be infeasible. On the other hand providing the prover with all formulae that might be relevant rapidly swamps the prover with unnecessary information. A technique for the selective retrieval of formulae based on features of those formulae and the conjecture at hand is required to solve this problem.

In this paper I describe an abstraction-based technique which addresses this problem. Implicit hypotheses such as definitions, axioms and previously proved theorems are stored in a database which may be accessed by a heuristic rule of inference calledgazing. Before accessing this database the gazing rule plans the use of these formulae in a hierarchy of abstraction spaces. When the planning phase is complete, the system can use the indicated formulae with some confidence that they are relevant to the proof.

Since the technique is abstraction-based, no guarantee that the plant will be eventually applicable or successful can be made. However, because the plans are built by considering increasing amounts of detail, the number of ways in which the application of a plan can fail is limited. Plan failures may be ‘patched’ in a uniform way.