Concept

B, C, K, W system

Summary
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). Definition 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 x and y;
  • C x is x with the flipped arguments order;
  • K x is the "constant x" function, which discards the next argument;
  • W duplicates its second argument for the doubled application to the first. Thus, it "joins" its first argument's two expectations for input into one.
Connection to other combinators 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
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