# Binary combinatory logic (Ofer Abarbanel online library)

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

Backus–Naur form:

<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  → x
• 11101xyz → 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.)

References

1. ^ Jump up to:ab Tromp, John (2007), “Binary lambda calculus and combinatory logic”, Randomness and complexity (PDF), World Sci. Publ., Hackensack, NJ, pp. 237–260, CiteSeerX 10.1.1.695.3142, doi:10.1142/9789812770837_0014, ISBN 978-981-277-082-0, MR 2427553.
2. ^Devine, Sean (2009), “The insights of algorithmic entropy”, Entropy, 11 (1): 85–110, doi:10.3390/e11010085, MR 2534819

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Ofer Abarbanel online library

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Ofer Abarbanel online library