Binary combinatory logic

Binary combinatory logic (BCL) is a formulation of combinatory logic using only the symbols 0 and 1.[1] BCL has applications in the theory of program-size complexity (Kolmogorov complexity).[1][2]

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.)

gollark: I only write for my own systems and whoever is unfortunate enough to try and clone my poorly documented repos so I can use whatever.
gollark: I also don't see why you need a 10D array over just a long 1D one.
gollark: It's probably going to be worse than AES and actual standards like that.
gollark: Please do *not* invent your own cryptography for any serious purposes.
gollark: Anyway, you could just write code for doing so for a 1D array, and then code for filling 10 N-1-dimensional arrays and merging them into a N-dimensional array

See also

References
  1. 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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