By [edited by] Haskell B. Curry, J. Roger Hindley, Jonathan P. Seldin.
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This article is designed to provide the coed a heritage within the foundations of algebra and research. The algebra of symbolic common sense and the concept that of set are brought early within the textual content in order that the most definitional improvement of the complicated quantity method flows simply from a suite of postulates for the common numbers.
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E. using no properties of the basic combinators except their reduction rules (Q 5C1). Formulated as an equational system this would be precisely the system S oof vol. I; however, it will be more convenient to formulate it in terms of the quasi-ordering relation of weak reduction (Q 6E5). ’ 58. Seldin [NDR] shows that the definition in [FML] is not quite the same as the other two; but that the two definitions are equivalent so far as reduction to an ultimate definiens is concerned. In his [ASF] he discusses the relationship to Kleene’s general recursiveness.
The different occurrences of [ U , ] , . , [U,] in V do not overlap, so if p # q, Rf does not overlap R J . If p = q, RS and RI do not overlap since Ri and Rj d o not overlap in U p . Property (vi) follows from (v) by induction on the length of the given reduction. REMARK 2. x(xa))(AyS). Then x 2 P(Pa), and P(Pa) and Pa are both residuals of S according to the definitions of 8 4B. Property (vii) can be proved thus: If S is one of the R i or does not overlap with any R i , the result follows by (iv).
The terms ‘open combination’ and ‘closed combination’ are due to Sanchis [NCT]. 28 ADDENDA TO PURE COMBINATORY LOGIC [llB where x is a new variable, that x = Y. The term 'new' here indicates that x does not occur in X or Y . ) The notations n(X), d(X), where X is a combination, will denote respectively the number of instances of atomic combinators (more briefly, the number of atomic combinators) in X and the number of atoms other than combinators in X . 2. General properties of weak reduction We now turn to the statement of some definitions and theorems concerning the process of weak reduction.