By [edited by] Haskell B. Curry, J. Roger Hindley, Jonathan P. Seldin.

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Extra resources for Combinatory logic. / Volume II

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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.