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 (Quine 1960, 1966, 1976).
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 (1958), and Curry et al. (1972) 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 (representing functional abstraction) 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 (sometimes called the applicand) 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