Strong Reduction and Normal Form in Combinatory Logic

possible importance are suggested by an equivalence between the theory of combinators and A-conversion, and the Church-Rosser theorem in A-conversion. This theorem implies that if a formula Xis convertible to a formula X* which cannot be further reduced-is irreducible, or in normalform-then Xis convertible to X* by a reduction alone. Moreover, the reduction may be performed in a certain prescribed order. The reduction of a formula in A-conversion is a sequence of replacements of components having certain stated forms (redexes) by certain corresponding forms (their contracts). The steps in the reduction are not further analyzable in the theory. Strong reduction is not defined as a sequence of replacements; it was an open problem, only recently solved by Hindley [3], whether or not strong reduction could be so described. Curry studied strong reduction in CLg I ([1] and [2] henceforth will be referred to as CLg I and CLg II respectively) by defining yet another kind of reduction, normal reduction, the steps of which are analogous to the steps of a reduction to normal form in A-conversion. Normal reductions are strong reductions in the sense that if X reduces to Y in a normal reduction, then X strongly reduces to Y. -If the normal reduction of X terminates in X*, then X* is said to be in normal form, and to be the normal form of X. Curry showed in CLg I that if X is in normal form, then X is strongly irreducible. The converse of Curry's theorem is the main result of this paper: THEoREM 3.2. If X is strongly irreducible, then X is in normalform. In addition, normal forms will be defined directly, without reference to normal reductions, so that a corollary result will be a characterization of strong irreducibility. The results here are contained in my thesis [4] but not always with the present proofs. In particular the present Lemma 2.5 and Theorem 3.1, and their proofs, are based on unpublished results of Professor H. B. Curry, to whom it is a pleasure to acknowledge my debt and express my gratitude. Curry's forthcoming book CLg II will contain new proofs of the results of ??3, 4, and 5 of this paper as well as