The SKI combinator calculus is a combinatory logic system and a computational system. It can be thought of as a computer programming language, though it is not convenient for writing software. Instead, it is important in the mathematical theory of algorithms because it is an extremely simple Turing complete language. It can be likened to a reduced version of the untyped lambda calculus. It was introduced by Moses Schönfinkel and Haskell Curry. All operations in lambda calculus can be encoded via abstraction elimination into the SKI calculus as binary trees whose leaves are one of the three symbols S, K, and I (called combinators). I itself is redundant and can be expressed with S and K only, e.g. as SKK, but its use often makes the definitions shorter and easier to grasp.
Notation Although the most formal representation of the objects in this system requires binary trees, for simpler typesetting they are often represented as parenthesized expressions, as a shorthand for the tree they represent. Any subtrees may be parenthesized, but often only the right-side subtrees are parenthesized, with left associativity implied for any unparenthesized applications. For example, ISK means ((IS)K). Using this notation, a tree whose left subtree is the tree KS and whose right subtree is the tree SK can be written as KS(SK). If more explicitness is desired, the implied parentheses can be included as well: ((KS)(SK)).
Informal description Informally, and using programming language jargon, a tree (xy) can be thought of as an application of the function x to an argument y. When evaluated (i.e., when the "function" is "applied" to the argument), the tree "returns a value", i.e., transforms into another tree. The "function", "argument" and the "value" are either combinators or binary trees with application nodes. If they are binary trees, they may be thought of as functions too, if needed. The evaluation operation is defined as follows: (x, y, and z represent expressions made from the combinators S, K, and I, and possibly variables standing for some as yet unspecified SKI expressions): I returns its argument:
Ix = x K, when applied to any argument x, yields a one-argument constant function Kx, which, when applied to any argument y, returns x:
Kxy = x S is a substitution operator. It takes three arguments and then returns the first argument applied to the third, which is then applied to the result of the second argument applied to the third. More clearly:
Sxyz = xz(yz) Example computation: SKSK evaluates to KK(SK) by the S-rule. Then if we evaluate KK(SK), we get K by the K-rule. As no further rule can be applied, the computation halts here. For all trees x and all trees y, SKxy will always evaluate to y in two steps, Ky(xy) = y, so the ultimate result of evaluating SKxy will always be the same as the result of evaluating y. We say that SKx and I are "functionally equivalent" for any x, because they always yield the same result when applied to any y. From these definitions it can be shown that SKI calculus, while being a minimalistic system, can fully perform any computations of the lambda calculus. All occurrences of I in any expression can be replaced by (SKK) or (SKS) or (SK x) for any x, and the resulting expression will yield the same result. So the "I" is merely syntactic sugar. Since I is optional, the system is also referred to as SK calculus or SK combinator calculus. It is possible to define a complete system using only one (improper) combinator. An example is Chris Barker's iota combinator, which can be expressed in terms of S and K as follows:
ιx = xSK = S(λx.xS)(λx.K)x = S(S(λx.x)(λx.S))(KK)x = S(SI(KS))(KK)x It is possible to reconstruct S, K, and I from the iota combinator. Applying ι to itself gives ιι = ιSK = SSKK = SK(KK) which is functionally equivalent to I. K can be constructed by applying ι twice to I (which is equivalent to application of ι to itself): ι(ι(ιι)) = ι(ιιSK) = ι(ISK) = ι(SK) = SKSK = K. Applying ι one more time gives ι(ι(ι(ιι))) = ιK = KSK = S. The simplest possible term forming a basis is X = λf.f (λxyz.x z (y z)) (λxyz.x), which satisfies X X = K, and X (X X) = S.
Formal definition The terms and derivations in this system can also be more formally defined: Terms: The set T of terms is defined recursively by the following rules.
S, K, and I are terms. If τ1 and τ2 are terms, then (τ1τ2) is a term. Nothing is a term if not required to be so by the first two rules. Derivations: A derivation is a finite sequence of terms defined recursively by the following rules (where α and ι are words over the alphabet {S, K, I, (, )} while β, γ and δ are terms):
If Δ is a derivation ending in an expression of the form α(Iβ)ι, then Δ followed by the term αβι is a derivation. If Δ is a derivation ending in an expression of the form α((Kβ)γ)ι, then Δ followed by the term αβι is a derivation. If Δ is a derivation ending in an expression of the form α(((Sβ)γ)δ)ι, then Δ followed by the term α((βδ)(γδ))ι is a derivation. Assuming a sequence is a valid derivation to begin with, it can be extended using these rules. All derivations of length 1 are valid derivations.
Conversion of lambda terms to SKI combinators By extensionality, an expression in the lambda calculus can be converted into a corresponding SKI combinator calculus expression in accordance with the following rules:
λx. x = I λx. c = K c (provided that c does not depend on x) λx. c x = c (provided that c does not depend on x) λx. y z = S (λx.y) (λx.z) Applied to an arbitrary argument, both left and right hand side expressions for each rule produce the same results.
SKI expressions
Self-application and recursion SII is an expression that takes an argument and applies that argument to itself:
SIIα = Iα(Iα) = αα This is also known as U combinator, Ux = xx. One interesting property of it is that its self-application is irreducible:
SII(SII) = I(SII)(I(SII)) = SII(I(SII)) = SII(SII) Or, using the equation Ux = xx as its definition directly, we immediately get U U = U U. Another thing is that it allows one to write a function that applies one thing to the self application of another thing:
(S(Kα)(SII))β = Kαβ(SIIβ) = α(Iβ(Iβ)) = α(ββ) or it can be seen as defining yet another combinator directly, Hxy = x(yy). This function can be used to achieve recursion. If β is the function that applies α to the self application of something else,
β = Hα = S(Kα)(SII) then the self-application of this β is the fixed point of that α:
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