# Binary combinatory logic

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Binary combinatory logic (BCL) is a formulation of combinatory logic using only the symbols 0 and 1. BCL has applications in the theory of program-size complexity (Kolmogorov complexity).

## Definition

### Syntax

• <term> ::= 00 | 01 | 1 <term> <term>

### Semantics

The denotational semantics of BCL may be specified as follows:

• [ 00 ] == K
• [ 01 ] == S
• [ 1 <term1> <term2> ] == ( [<term1>] [<term2>] )

where "[...]" abbreviates "the meaning of ...". Here K and S are the KS-basis combinators, and ( ) is the application operation, of combinatory logic. (The prefix 1 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 (00, 01, 1) (as in the present version), (01, 00, 1), (10, 11, 0), and (11, 10, 0).

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:

•  1100xy  $\rightarrow$ x
• 11101xyz $\rightarrow$ 11xz1yz

where x, y, and z are arbitrary subterms. (Note, for example, that because parsing is from the left, 10000 is not a subterm of 11010000.)