Logic

Combinatory Logic in Programming. Computations with Objects by V.E. Wolfengagen

By V.E. Wolfengagen

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These (and many other) of his works gave not only the mathematical foundations, but all the contemporary system of thinking in computer science. Computation theory, semantic of programming languages, object based computations – this is far not complete list of research directions, inspired by this mathematician. S. Kuzichev, [16],[17], [18], [19],[20], [21], [22], [23],[24]). In theses papers a research experience of expressive power of the systems with operators of application and (functional) abstraction is covered.

B–1. a)b = [b/x]a; a=b ; ca = cb a = b; b = c ; (σ) a=c a=b . b=a B–2. xz(yz). xz(yz)) yzw) (β) = x(yw(zw)), ≡S B–3. The schemes (K) and (S) result in: a(bc) = = = = Kac(bc) S(Ka)bc KSa(Ka)bc S(KS)Kabc. (K) (S) (K) (S) Checking. A verification that B = S(KS)K is given below. 53 C HAPTER 2: D ERIVATION OF O BJECT B–1. S(KS)Kabc = KSa(Ka)bc, because in the scheme (S) it can be assumed that y ≡ (KS), z ≡ K, w ≡ a. e. S(KS)Ka = (KS)a(Ka). Omitting the non-significant parentheses results in S(KS)Ka = KSa(Ka).

Task formulation. Using K, S and other predefined objects derive the object with combinatory characteristic: Cabc = acb, by postulates α, β, µ, ν, σ, τ , ξ of λ-calculus. Solution. (C) 54 C HAPTER 2: D ERIVATION OF O BJECT C–1. List the postulates which determine a conversion relation ‘=’ (see task given above). C–2. use: Recall the combinatory characteristics of the objects in (K) Kxy = x, (I) Ix = x, (S) Sxyz = xz(yz), (B) Bxyz = x(yz). 1 on p. 51 is added to already known schemes (K), (S) (I).

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