Binary combinatory logic

Binary combinatory logic (BCL) is a computer programming language that uses binary terms 0 and 1 to create a complete formulation of combinatory logic using only the symbols 0 and 1. Using the S and K combinators, complex boolean algebra functions can be made. BCL has applications in the theory of program-size complexity (Kolmogorov complexity).

S-K Basis
Utilizing K and S combinators of the Combinatory logic, logical functions can be represented in as functions of combinators:

Syntax
Backus–Naur form:

Semantics
The denotational semantics of BCL may be specified as follows: where " " abbreviates "the meaning of ". Here  and   are the KS-basis combinators, and   is the application operation, of combinatory logic. (The prefix  corresponds to a left parenthesis, right parentheses being unnecessary for disambiguation.)

Thus there are four equivalent formulations of BCL, depending on the manner of encoding the triplet (K, S, left parenthesis). These are  (as in the present version), ,  , and.

The operational semantics of BCL, apart from eta-reduction (which is not required for Turing completeness), may be very compactly specified by the following rewriting rules for subterms of a given term, parsing from the left: where,  , and   are arbitrary subterms. (Note, for example, that because parsing is from the left,  is not a subterm of  .) BCL can be used to replicate algorithms like Turing machines and Cellular automata, BCL is Turing complete.