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 computation and also as a basis for the design of functional programming languages. It is based on combinators, which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions—and to remove any mention of variables—particularly in predicate logic. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments.

In mathematics

Combinatory logic was originally intended as a 'pre-logic' that would clarify the role of quantified variables in logic, essentially by eliminating them. Another way of eliminating quantified variables is Quine's predicate functor logic. While the expressive power of combinatory logic typically exceeds that of first-order logic, the expressive power of predicate functor logic is identical to that of first order logic.
The original inventor of combinatory logic, Moses Schönfinkel, published nothing on combinatory logic after his original 1924 paper. Haskell Curry rediscovered the combinators while working as an instructor at Princeton University in late 1927. In the late 1930s, Alonzo Church and his students at Princeton invented a rival formalism for functional abstraction, the lambda calculus, which proved more popular than combinatory logic. The upshot of these historical contingencies was that until theoretical computer science began taking an interest in combinatory logic in the 1960s and 1970s, nearly all work on the subject was by Haskell Curry and his students, or by Robert Feys in Belgium. Curry and Feys, and Curry et al. survey the early history of combinatory logic. For a more modern treatment of combinatory logic and the lambda calculus together, see the book by Barendregt, which reviews the models Dana Scott devised for combinatory logic in the 1960s and 1970s.

In computing

In computer science, combinatory logic is used as a simplified model of computation, used in computability theory and proof theory. Despite its simplicity, combinatory logic captures many essential features of computation.
Combinatory logic can be viewed as a variant of the lambda calculus, in which lambda expressions are replaced by a limited set of combinators, primitive functions without free variables. It is easy to transform lambda expressions into combinator expressions, and combinator reduction is much simpler than lambda reduction. Hence combinatory logic has been used to model some non-strict functional programming languages and hardware. The purest form of this view is the programming language Unlambda, whose sole primitives are the S and K combinators augmented with character input/output. Although not a practical programming language, Unlambda is of some theoretical interest.
Combinatory logic can be given a variety of interpretations. Many early papers by Curry showed how to translate axiom sets for conventional logic into combinatory logic equations. Dana Scott in the 1960s and 1970s showed how to marry model theory and combinatory logic.

Summary of lambda calculus

Lambda calculus is concerned with objects called lambda-terms, which can be represented by the following three forms of strings:

where is a variable name drawn from a predefined infinite set of variable names, and and are lambda-terms.
Terms of the form are called abstractions. The variable v is called the formal parameter of the abstraction, and is the body of the abstraction. The term represents the function which, applied to an argument, binds the formal parameter v to the argument and then computes the resulting value of — that is, it returns, with every occurrence of v replaced by the argument.
Terms of the form are called applications. Applications model function invocation or execution: the function represented by is to be invoked, with as its argument, and the result is computed. If is an abstraction, the term may be reduced:, the argument, may be substituted into the body of in place of the formal parameter of, and the result is a new lambda term which is equivalent to the old one. If a lambda term contains no subterms of the form then it cannot be reduced, and is said to be in normal form.
The expression represents the result of taking the term and replacing all free occurrences of in it with. Thus we write
By convention, we take as shorthand for .
The motivation for this definition of reduction is that it captures the essential behavior of all mathematical functions. For example, consider the function that computes the square of a number. We might write
x here is the formal parameter of the function. To evaluate the square for a particular argument, say 3, we insert it into the definition in place of the formal parameter:
To evaluate the resulting expression, we would have to resort to our knowledge of multiplication and the number 3. Since any computation is simply a composition of the evaluation of suitable functions on suitable primitive arguments, this simple substitution principle suffices to capture the essential mechanism of computation.
Moreover, in lambda calculus, notions such as '3' and can be represented without any need for externally defined primitive operators or constants. It is possible to identify terms in lambda calculus, which, when suitably interpreted, behave like the number 3 and like the multiplication operator, q.v. Church encoding.
Lambda calculus is known to be computationally equivalent in power to many other plausible models for computation ; that is, any calculation that can be accomplished in any of these other models can be expressed in lambda calculus, and vice versa. According to the Church–Turing thesis, both models can express any possible computation.
It is perhaps surprising that lambda-calculus can represent any conceivable computation using only the simple notions of function abstraction and application based on simple textual substitution of terms for variables. But even more remarkable is that abstraction is not even required. Combinatory logic'' is a model of computation equivalent to lambda calculus, but without abstraction. The advantage of this is that evaluating expressions in lambda calculus is quite complicated because the semantics of substitution must be specified with great care to avoid variable capture problems. In contrast, evaluating expressions in combinatory logic is much simpler, because there is no notion of substitution.

Combinatory calculi

Since abstraction is the only way to manufacture functions in the lambda calculus, something must replace it in the combinatory calculus. Instead of abstraction, combinatory calculus provides a limited set of primitive functions out of which other functions may be built.

Combinatory terms

A combinatory term has one of the following forms:

Syntax	Name	Description
x	Variable	A character or string representing a combinatory term.
P	Primitive function	One of the combinator symbols I, K, S.
	Application	Applying a function to an argument. M and N are combinatory terms.

The primitive functions are combinators, or functions that, when seen as lambda terms, contain no free variables.
To shorten the notations, a general convention is that, or even, denotes the term. This is the same general convention as for multiple application in lambda calculus.

Reduction in combinatory logic

In combinatory logic, each primitive combinator comes with a reduction rule of the form
where E is a term mentioning only variables from the set. It is in this way that primitive combinators behave as functions.

Examples of combinators

The simplest example of a combinator is I, the identity combinator, defined by
for all terms x. Another simple combinator is K, which manufactures constant functions: is the function which, for any argument, returns x, so we say
for all terms x and y. Or, following the convention for multiple application,
A third combinator is S, which is a generalized version of application:
S applies x to y after first substituting z into
each of them. Or put another way, x is applied to y inside the environment z.
Given S and K, I itself is unnecessary, since it can be built from the other two:
for any term x. Note that although
x) = for any x,
itself is not equal to I. We say the terms are extensionally equal. Extensional equality captures the mathematical notion of the equality of functions: that two functions are equal if they always produce the same results for the same arguments. In contrast, the terms themselves, together with the reduction of primitive combinators, capture the notion of intensional equality of functions: that two functions are equal only if they have identical implementations up to the expansion of primitive combinators. There are many ways to implement an identity function; and I are among these ways. is yet another. We will use the word equivalent to indicate extensional equality, reserving equal for identical combinatorial terms.
A more interesting combinator is the fixed point combinator or Y combinator, which can be used to implement recursion.

Completeness of the S-K basis

S and K can be composed to produce combinators that are extensionally equal to any lambda term, and therefore, by Church's thesis, to any computable function whatsoever. The proof is to present a transformation,, which converts an arbitrary lambda term into an equivalent combinator.
may be defined as follows:

Note that T as given is not a well-typed mathematical function, but rather a term rewriter: Although it eventually yields a combinator, the transformation may generate intermediary expressions that are neither lambda terms nor combinators, via rule.
This process is also known as abstraction elimination. This definition is exhaustive: any lambda expression will be subject to exactly one of these rules.
It is related to the process of bracket abstraction, which takes an expression E built from variables and application and produces a combinator expression E in which the variable x is not free, such that E x = E holds. A very simple algorithm for bracket abstraction is defined by induction on the structure of expressions as follows:

y := K y
x := I
:= S

Bracket abstraction induces a translation from lambda terms to combinator expressions, by interpreting lambda-abstractions using the bracket abstraction algorithm.