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
.