Some aspects of analogy in mathematical reasoning

Abstract

An important research problem is the incorporation of “declarative” knowledge into an automated theorem prover that can be utilized in the search for a proof. An interesting proposal in this direction is Alan Bundy's approach of using explicit proof plans that encapsulate the general form of a proof and is instantiated into a particular proof for the case at hand. We give some examples that show how a “declarative” highlevel description of a proof can be used to find proofs of apparently “similiar” theorems by analogy. This “analogical” information is used to select the appropriate axioms from the database so that the theorem can be proved. This information is also used to adjust some options of a resolution theorem prover. In order to get a powerful tool it is necessary to develop an epistemologically appropriate language to describe proofs, for which a large set of examples should be used as a testbed. We present some ideas in this direction.