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Combinatory logic. / Volume II by [edited by] Haskell B. Curry, J. Roger Hindley, Jonathan P.

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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Y ; (3) also x =y (4 -+y = x. Thus if (0)is added to the rules for 2 , the latter relation becomes weak equality. and we have a formulation of the X o of volume I. In Chapter 5 we sometimes abbreviated a statement of the form xx l . . x,, where x l r. ,x, are variables which do not occur in X or Y, as x (4) Although this is sometimes awkward, it is occasionally useful. The substitution prefix [ M / x ] is defined as in 9 6D, and has the properties with respect to weak reduction and weak equality which were stated in Theorems 6D1, 6D5, and 6D6.

The possibility of carrying out a definitional reduction by a Markov algorithm was worked out in detail in [TEA]. This gives another proof of a theorem (Detlovs [NAR], [ENA]) that a partial recursive numerical function is algorithmic; another proof is in Chernyavskii [KNA]. B. 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).

The adjoined indeterminates will also, on occasion, be called ouriubles. 3According to the conventions of Q 2D this means that we have the following axiom schemes and rules : (el (2) x 2 x, x Z y & y 2 z + x 5 z, (11) x 2 y -+ zx 2 zy, (4 x 2y -+ xz 1 yz. 2. The combinator I is taken as primitive although one could define it as SKK, or, in fact, any SKX. This is because of the needs of strong reduction. In that theory we have SKK )-I, but, if I is taken as primitive, not the converse; so that I is not definable in that theory even though we have I = SKK.

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