Definition
S-K Basis
Utilizing K and S combinators of theSyntax
Semantics
The 00 ''K''
* 01 ''S''
* <term2> ">1 <term1> <term2> ( ">lt;term1> ">lt;term2>)
where " ../code>" abbreviates "the meaning of ...
". Here ''K''
and ''S''
are the ''KS''-basis combinators, and ( )
is the ''application'' operation, of combinatory logic
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of com ...
. (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
Operational semantics is a category of formal programming language semantics in which certain desired properties of a program, such as correctness, safety or security, are verified by constructing proofs from logical statements about its exec ...
of BCL, apart from eta-reduction (which is not required for Turing completeness
In computability theory, a system of data-manipulation rules (such as a model of computation, a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can b ...
), may be very compactly specified by the following rewriting rules for subterms of a given term, parsing
Parsing, syntax analysis, or syntactic analysis is a process of analyzing a String (computer science), string of Symbol (formal), symbols, either in natural language, computer languages or data structures, conforming to the rules of a formal gramm ...
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
.)
BCL can be used to replicate algorithms like Turing machines
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer alg ...
and Cellular automata, BCL is Turing complete.
See also
* Iota and Jot
In formal language theory and computer science, Iota and Jot (from Greek iota ι, Hebrew yodh י, the smallest letters in those two alphabets) are languages, extremely minimalist formal systems, designed to be even simpler than other more popula ...
References
Further reading
*
*
External links
John's Lambda Calculus and Combinatory Logic Playground
Lambda Calculus in 383 Bytes
* {{cbignore
Algorithmic information theory
Combinatory logic