The absence and the presence of fixed point combinators

Abstract

In this article we give some classes of fragments of weak combinatory logic and show that they fail to admit various fixed point properties. We also present the results of a new technique, the kernel strategy, for using an automated theorem-proving program to search for fixed point combinators within a given fragment. A key aspect of the work is that experimentation with the kernel strategy led indirectly to proofs of theorems concerning the absence of the fixed point properties.