Binary combinatory logic

Computer programming language

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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.[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).[1][2]

## Definition

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### S-K Basis

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

List of Logical Operations as Binary Combinators[3] | Boolean Algebra  | S-K Basis

|
|

True(1)  | K(KK)
False(0)  | K(K(SK))
AND  | SSK
NOT  | SS(S(S(S(SK))S))(KK)
| OR  | S(SS)S(SK)
| NAND  | S(S(K(S(SS(K(KK)))))))S
| NOR  | S(S(S(SS(K(K(KK)))))(KS))
| XOR  | S(S(S(SS)(S(S(SK)))S))K

### Syntax

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Backus–Naur form:

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

### Semantics

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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`.)

One step of Rule 110 Cellular Automata in SK-Basis(Written in the Wolfram Language).[3]

BCL can be used to replicate algorithms like Turing machines and Cellular automata,[3] BCL is Turing complete.

## See also

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* Iota and Jot

## References

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1. ^ a b 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, Bibcode:2009Entrp..11...85D, doi:10.3390/e11010085, MR 2534819
3. ^ a b c Wolfram, Stephen (2021-12-06). "Combinators: A Centennial View". writings.stephenwolfram.com. Archived from the original on 2020-12-06. Retrieved 2021-02-17.

## Further reading

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* Tromp, John (October 2007). "Binary Lambda Calculus and Combinatory Logic". Randomness and Complexity, from Leibniz to Chaitin: 237–260. doi:10.1142/9789812770837_0014. ISBN 978-981-277-082-0.

## External links

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* John's Lambda Calculus and Combinatory Logic Playground
* A minimal implementation in C
* Lambda Calculus in 383 Bytes
* Brauner, Paul. "Lambda Diagrams YouTube Playlist". YouTube. Archived from the original on 2021-12-21.