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A basis result in combinatory logic

Published online by Cambridge University Press:  12 March 2014

Remi Legrand
Remi Legrand*
Affiliation:
Université Pierre Et Marie Curie, 75231 Paris, France
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Extract

The aim of this article is to show that a basis for combinatory logic [2] must contain at least one combinator with rank strictly greater than two. We use notation of [1].

Let Q be a primitive combinator given by its reduction rule Qx 1 … xn → C, where C is a pure combination of the variables x 1,…, xn . n is called the rank of the combinator.

A set {Q 1,…,Qn } of combinators is a basis for combinatory logic if for every finite set {x 1,…,xm } of variables and every pure combination C of these variables, there exists a pure combinator Q of Q 1,…,Qn such that Qx 1…xm ↠C.

Property. The Church-Rosser theorem and the (quasi-)normalization theorem are valid for the combinatory reduction system under consideration.

Proof. See [4], [5], and [6].

Any basis must contain at least one combinator with rank strictly greater than two.

Let us assume a basis B with combinators of rank strictly less than 3; then there exists a pure combination X of the combinators in B such that: XABC ↠ CAB.

(*) First of all, we remark that if XABC ↠ M ↠ CAB, M must contain at least one occurrence of each of the variables A, B and C.

Notation. We denote by E[X 1,…,Xn ] expressions that contain at least one occurrence of every term X 1,…,Xn .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 4 , December 1988 , pp. 1224 - 1226
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCE

[1] Barendregt, H. P., The lambda-calculus: its syntax and semantics, North-Holland, Amsterdam, 1981.Google Scholar
[2] Bellot, P., A new proof for Craig's theorem, this Journal, vol. 50 (1985), pp. 395–396.Google Scholar
[3] Hindley, J. R. and Seldin, J. P., Introduction to combinators and lambda-calculus, Cambridge University Press, Cambridge, 1986.Google Scholar
[4] Huet, G., Confluent reductions: abstract properties and applications to term rewriting systems, Journal of the Association for Computing Machinery, vol. 27 (1980), pp. 797–821.CrossRefGoogle Scholar
[5] Klop, J. W., Combinatory reduction systems, Mathematical Centre Tracts, vol. 127, Mathematisch Centrum, Amsterdam, 1980.Google Scholar
[6] O'Donnell, M. J., Computing in systems described by equations, Lecture Notes in Computer Science, vol. 58, Springer-Verlag, Berlin, 1977.Google Scholar