The kernel strategy and its use for the study of combinatory logic

Abstract

Barendregt defines combinatory logic as an equational system satisfying the combinatorsS andK with ((Sx)y)z=(xz)(yz) and (Kx)y=x; the set consisting ofS andK provides abasis for all of combinatory logic. Rather than studying all of the logic, logicians often focus onfragments of the logic, subsets whose basis is obtained by replacingS orK or both by other combinators. In this article, we present a powerful new strategy, called thekernel strategy, for studying fragments in the context of questions concerned with fixed point properties. Interest in such properties rests in part with their relation to normal forms and paradoxes. We show how the kernel strategy was used to answer a number of open questions, offering abundant evidence that the availability of the kernel strategy marks a singular advance for automated reasoning. In all of our experiments with this strategy applied by an automated reasoning program, the rate of success has been impressively high and the CPU time to obtain the desired information startlingly small. For each fragment we study, we use the kernel strategy to attempt to determine whether the strong or the weak fixed point property holds. WhereA is a given fragment with basisB, the strong fixed point property holds forA if and only if there exists a combinatory such that, for all combinatorsx,yx=x(yx), wherey is expressed purely in terms of elements ofB. The weak fixed point property holds forA if and only if for all combinatorsx there exists a combinatory such thaty=xy, wherey is expressed purely in terms of the elements ofB and the combinatorx. Because the use of the kernel strategy is so effective in addressing questions focusing on either fixed point property, its formulation marks an important advance for combinatory logic. Perhaps of especial interest to logicians is an infinite class of infinite sets of tightly coupled fixed point combinators (presented here), whose unexpected discovery resulted directly from the application of the strategy by an automated reasoning program. We also offer various open questions for possible research and focus on an automated reasoning program and input files that may prove useful for such research.