A case study in automated theorem proving: Finding sages in combinatory logic

Abstract

We present in this article an application of automated theorem proving to a study of a theorem in combinatory logic. The theorem states: the strong fixed point property is satisfied in a system that contains the B and W combinators. The theorem can be stated in terms of Smullyan's forests of birds: a sage exists in a forest that contains a bluebird and a warbler. Proofs of the theorem construct sages from B and W. Prior to the study, one sage, discovered by Statman, was known to exist. During the study, with much assistance from two automated theorem-proving programs, four new sages were discovered. The study was conducted from a syntactic point of view because the authors know very little about combinatory logic. The uses of the automated theorem-proving programs are described in detail.