Effective reduction and conversion strategies for combinators

Abstract

We imagine that we are computing with combinators or lambda terms and that successive terms are related by one-step reduction or expansion. We have a term on our screen and we want to click on a redex to be reduced (or expanded) to obtain the next screenful. The choice of redex is determined by a strategy for achieving our ends. Such a strategy must be effective in the sense of being computable. Memory is a serious constraint, since only one term fits on our screen at a time.

We shall prove the following results concerning effective reduction and conversion strategies for combinators. These results constitute answers to certain questions which have appeared in the literature (with one exception). We believe that these questions are interesting because they get right to the heart of the memory problems inherent in one-step reduction and expansion. These questions appear to be even more subtle for lambda terms where each one-step reduction can create a much more radical change of structure than for combinators.

• (1) There is an effective one-step cofinal reduction strategy (answering a question of Barendregt [2] 13.6.6). This is in Section 2.

• (2) There is no effective confluence function but there is an effective one-step confluence strategy (answering'a question of Isles reported in [1]). This is in Section 3.

• (3) There is an effective one-step enumeration strategy (answering an obvious question). This is in Section 4.

• (4) There is an effective one-step Church-Rosser conversion strategy (“almost” answering a question of Bergstra and Klop [3]). This is in Section 5.

Supported in part by the National Science Foundation, CCR-9624681