Binary combinatory logic

Binary combinatory logic

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

yntax

Backus–Naur form:
* ::= 00 | 01 | 1

emantics

The denotational semantics of BCL may be specified as follows:
* [ 00 ] = "K"
* [ 01 ] = "S"
* [ 1 ] = "(" [] [] ")" 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 ightarrow x
* 11101xyz ightarrow 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.)

ee also

* Iota and Jot

External links

* [http://homepages.cwi.nl/~tromp/cl/cl.html John's Lambda Calculus and Combinatory Logic Playground]


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