B,C,K,W System

B,C,K,W System

The B, C, K, W system is a variant of combinatory logic that takes as primitive the combinators B, C, K, and W. This system was discovered by Haskell Curry in his doctoral thesis Grundlagen der kombinatorischen Logik, whose results are set out in Curry (1930).

The combinators are defined as follows:

• B x y z = x (y z)
• C x y z = x z y
• K x y = x
• W x y = x y y

Intuitively,

• B x y is the composition of the arguments x and y;
• C x y z swaps the arguments y and z;
• K x y discards the argument y;
• W x y duplicates the argument y.

In recent decades, the SKI combinator calculus, with only two primitive combinators, K and S, has become the canonical approach to combinatory logic. B, C, and W can be expressed in terms of S and K as follows:

• B = S (K S) K
• C = S (S (K (S (K S) K)) S) (K K)
• K = K
• W = S S (S K)

Going the other direction, SKI can be defined in terms of B,C,K,W as:

• I = W K
• K = K
• S = B (B (B W) C) (B B) = B (B W) (B B C).