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This article is about a topic in mathematical logic and theoretical computer science, not to be confused with combinational logic, sometimes known as combinatorial logic, a topic in digital electronics. Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It 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 are higher-order functions that solely use function application and possibly other, earlier defined combinators for defining a result from their arguments. Combinatory logic 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 Willard Quine's predicate functors; see Bacon (1986). The original inventor of combinatory logic, Schönfinkel, largely ceased to publish after Stalin consolidated his power in 1929. Curry rediscovered the combinators while a PhD student at the University of Göttingen in the late 1920s, but that epochal department fell apart when Hitler came to power in 1933. In the latter 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 70s, 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 parallel treatment of combinatory logic and the lambda calculus, see Barendregt (1984), who also reviews the models Dana Scott devised for combinatory logic in the 1960s and 70s. Combinatory logic 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 from which free variables are absent. 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 and character IO. 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 (Hindley and Meredith 1990). Dana Scott in the 1960s and 70s showed how to marry model theory and combinatory logic. Summary of the lambda calculus The lambda calculus is concerned with objects called lambda-terms, which are strings of symbols of one of the following forms: where v is a variable name drawn from a predefined infinite set of variable names, and E1 and E2 are lambda-terms. Terms of the form λv.E1 are called abstractions. The variable v is called the formal parameter of the abstraction, and E1 is the body of the abstraction. The term λv.E1 represents the function which, applied to an argument, binds the formal parameter v to the argument and then computes the resulting value of E1---that is, it returns E1, with every occurrence of v replaced by the argument. Terms of the form (E1 E2) are called applications. Applications model function invocation or execution: the function represented by E1 is to be invoked, with E2 as its argument, and the result is computed. If E1 (sometimes called the applicand) is an abstraction, the term may be reduced: E2, the argument, may be substituted into the body of E1 in place of the formal parameter of E1, 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 (λv.E1 E2) then it cannot be reduced, and is said to be in normal form. The expression E''v'' := ''a'' represents the result of taking the term E and replacing all free occurrences of v with a. Thus we write (λv.E a) => E''v'' := ''a'' By convention, we take (a b c d ... z) as short for (...(((a b) c) d) ... z). (i.e., application is left associative.) 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 The square of x is x (Using " formal parameter: The square of 3 is 3 To evaluate the resulting expression 3 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 the lambda calculus, notions such as '3' and ' operators or constants. It is possible to identify terms in the lambda calculus, which, when suitably interpreted, behave like the number 3 and like the multiplication operator. The lambda calculus is known to be computationally equivalent in power to many other plausible models for computation (including Turing machines); that is, any calculation that can be accomplished in any of these other models can be expressed in the 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 the lambda calculus, but without abstraction. 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: Examples of combinators The simplest example of a combinator is I, the identity combinator, defined by (I x) = x for all terms x. Another simple combinator is K, which manufactures constant functions: (K x) is the function which, for any argument, returns x, so we say ((K x) y) = x for all terms x and y. Or, following the same convention for multiple application as in the lambda-calculus, (K x y) = x A third combinator is S, which is a generalized version of application: (S x y z) = (x z (y z)) S applies x to y after first substituting z into each of them. Given S and K, I itself is unnecessary, since it can be built from the other two: ((S K K) x) = (S K K x) = (K x (K x)) = x for any term x. Note that although ((S K K) x) = (I x) for any x, (S K K) 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 capture the notion of intensional equality of functions: that two functions are 'equal' only if they have identical implementations. There are many ways to implement an identity function; (S K K) and I are among these ways. (S K S) 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 It is a perhaps astonishing fact that 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, T , which converts an arbitrary lambda term into an equivalent combinator. T may be defined as follows: Conversion of a lambda term to an equivalent combinatorial term For example, we will convert the lambda term λx.λy.(y x) to a combinator: T''λx''.''λy''.(''y'' ''x'') = T = T''λx''.('''S''' ''T''''λy''.''y'' ''T''''λy''.''x'') (by 6) = T''λx''.('''S''' '''I''' ''T''''λy''.''x'') (by 4) = T''λx''.('''S''' '''I''' ('''K''' ''x'')) (by 3) = (S T''λx''.('''S''' '''I''') T''λx''.('''K''' ''x'')) (by 6) = (S (K (S I)) T''λx''.('''K''' ''x'')) (by 3) = (S (K (S I)) (S T''λx''.'''K''' T''λx''.''x'')) (by 6) = (S (K (S I)) (S (K K) T''λx''.''x'')) (by 3) = (S (K (S I)) (S (K K) I)) (by 4) If we apply this combinator to any two terms x and y, it reduces as follows: (S (K (S I)) (S (K K) I) x y) = (K (S I) x (S (K K) I x) y) = (S I (S (K K) I x) y) = (I y (S (K K) I x y)) = (y (S (K K) I x y)) = (y (K K x (I x) y)) = (y (K (I x) y)) = (y (I x)) = (y x) The combinatory representation, (S (K (S I)) (S (K K) I)) is much longer than the representation as a lambda term, λx.λy.(y x). This is typical. In general, the T construction may expand a lambda term of length n to a combinatorial term of length Θ(3n). Explanation of the T[ ] transformation The T transformation is motivated by a desire to eliminate abstraction. Two special cases, rules 3 and 4, are trivial: λx.x is clearly equivalent to I, and λx.E is clearly equivalent to (K E) if x does not appear free in E. The first two rules are also simple: Variables convert to themselves, and applications, which are allowed in combinatory terms, are converted to combinators simply by converting the applicand and the argument to combinators. It's rules 5 and 6 that are of interest. Rule 5 simply says that to convert a complex abstraction to a combinator, we must first convert its body to a combinator, and then eliminate the abstraction. Rule 6 actually eliminates the abstraction. λx.(E1 E2) is a function which takes an argument, say a, and substitutes it into the lambda term (E1 E2) in place of x, yielding (E1 E2)''x'' : = ''a''. But substituting a into (E1 E2) in place of x is just the same as substituting it into both E1 and E2, so (E1 E2)''x'' := ''a'' = (E1''x'' := ''a'' E2''x'' := ''a'') (λx.(E1 E2) a) = ((λx.E1 a) (λx.E2 a)) = (S λx.E1 λx.E2 a) = ((S λx.E1 λx.E2) a) By extensional equality, λx.(E1 E2) = (S λx.E1 λx.E2) Therefore, to find a combinator equivalent to λx.(E1 E2), it is sufficient to find a combinator equivalent to (S λx.E1 λx.E2), and (S T''λx''.''E1'' T''λx''.''E2'') evidently fits the bill. E1 and E2 each contain strictly fewer applications than (E1 E2), so the recursion must terminate in a lambda term with no applications at all—either a variable, or a term of the form λx.E. η-reduction The combinators generated by the T transformation can be made smaller if we take into account the η-reduction rule: T''λx''.(''E'' ''x'') = T''E'' (if x is not free in E) ''λx''.(''E'' x) is the function which takes an argument, x, and applies the function E to it; this is extensionally equal to the function E itself. It is therefore sufficient to convert E to combinatorial form. Taking this simplification into account, the example above becomes: T''λx''.''λy''.(''y'' ''x'') = ... = (S (K (S I)) T''λx''.('''K''' ''x'')) = (S (K (S I)) K) (by η-reduction) This combinator is equivalent to the earlier, longer one: (S (K (S I)) K x y) = (K (S I) x (K x) y) = (S I (K x) y) = (I y (K x y)) = (y (K x y)) = (y x) Similarly, the original version of the T transformation transformed the identity function λf.λx.(f x) into (S (S (K S) (S (K K) I)) (K I)). With the η-reduction rule, λf.λx.(f x) is transformed into I. One point basis There are one point bases from which every combinator can be composed extentionally equal to any lambda term. The simplest example of such a basis is where: X ≡ λx.(((x K)S)K) It is not difficult to verify that: (X X) X =ηß K and X (X X) =ηß S. Since is a basis, it follows that is a basis too. Combinators B, C of Curry The combinators S and K already figure in the work of Schönfinkel (although symbol C was used instead of present K), Curry introduced the use of B, C, W (and K), already in his doctoral thesis of 1930 (see B,C,K,W System). In Combinatory Logic V. I, we return to S, K, but (via Markov's algorithms) he uses B and C to simplify many reductions. This seems to have been used, much later, by David Turner, whose name has been bound to its computational use. Two new combinators are introduced: (C a b c) = (a c b) (B a b c) = (a (b c)) Using these combinators, we can extend the rules for the transformation as follows: Using B and C combinators, the transformation of λx.λy.(y x) looks like this: T''λx''.''λy''.(''y'' ''x'') = T = T''λx''.('''C''' ''T''''λy''.''y'' ''x'') (by rule 7) = T''λx''.('''C''' '''I''' ''x'') = (C I) (η-reduction) = C X= I'(traditional canonical notation: X' = C X) And indeed, (C I x y) does reduce to (y x): (C I x y) = (I y x) = (y x) The motivation here is that B and C are limited versions of S. Whereas S takes a value and substitutes it into both the applicand and its argument before performing the application, C performs the substitution only in the applicand, and B only in the argument. CL A distinction must be made between the CLK as described in this article and the CLI calculus. The distinction corresponds to that between the λK and the λI calculus. Unlike the λK calculus, the λI calculus restricts abstractions to: λv.E1 where v has at least one free occurrence in E1. As a consequence, combinator K is not present in the λI calculus nor in the CLI calculus. The constants of CLI are: I, B, C and S, which form a basis from which all CLI terms can be composed (modulo equality) Together, B and C simulate K. Every λI term can be converted into an equal CLI combinator according to rules similar to those presented above for the conversion of λK terms into CLK combinators. See chapter 9 in Barendregt (1984). Reverse conversion The conversion L from combinatorial terms to lambda terms is trivial: L'''I''' = λx.x L'''K''' = λx.λy.x L'''C''' = λx.λy.λz.(x z y) L'''B''' = λx.λy.λz.(x (y z)) L'''S''' = λx.λy.λz.(x z (y z)) L(''E1'' ''E2'') = (L''E1'' L''E2'') Note, however, that this transformation is not the inverse transformation of any of the versions of T that we have seen. Undecidability of combinatorial calculus It is undecidable whether a general combinatory term has a normal form; whether two combinatory terms are equivalent, etc. This is equivalent to the undecidability of the corresponding problems for lambda terms. However, a direct proof is as follows: First, observe that the term Ω = (S I I (S I I)) has no normal form, because it reduces to itself after three steps, as follows: (S I I (S I I)) = (I (S I I) (I (S I I))) = (S I I (I (S I I))) = (S I I (S I I)) and clearly no other reduction order can make the expression shorter. Now, suppose N were a combinator for detecting normal forms, such that (N x) => T, if x has a normal form F, otherwise. (Where T and F the transformations of the conventional lambda-calculus definitions of true and false, λx.λy.x and λx.λy.y. The combinatory versions have T = K and F = (K I).) Now let Z = (C (C (B N (S I I)) Ω) I) now consider the term (S I I Z). Does (S I I Z) have a normal form? It does if and only if the following do also: (S I I Z) = (I Z (I Z)) = (Z (I Z)) = (Z Z) = (C (C (B N (S I I)) Ω) I Z) (definition of Z) = (C (B N (S I I)) Ω Z I) = (B N (S I I) Z Ω I) = (N (S I I Z) Ω I) Now we need to apply N to (S I I Z). Either (S I I Z) has a normal form, or it does not. If it does have a normal form, then the foregoing reduces as follows: (N (S I I Z) Ω I) = (K Ω I) (definition of N) = Ω but Ω does not have a normal form, so we have a contradiction. But if (S I I Z) does not have a normal form, the foregoing reduces as follows: (N (S I I Z) Ω I) = (K I Ω I) (definition of N) = (I I) I which means that the normal form of (S I I Z) is simply I, another contradiction. Therefore, the hypothetical normal-form combinator N cannot exist. The combinatory logic analogue of Rice's theorem says that there is no complete nontrivial predicate. A predicate is a combinator that, when applied, returns either T or F. A predicate N is nontrivial if there are two arguments A and B such that NA=T and NB=F. A combinator N is complete if and only if NM has a normal form for every argument M. The analogue of Rice's theorem then says that every complete predicate is trivial. Combinatory terms as graphs (TO DO: Add details with illustrations; don't forget to discuss the effect of fixed-point combinators.) Compilation of functional languages Functional programming languages are often based on the simple but universal semantics of the lambda calculus. (Add details.) David Turner used his combinators to implement the SASL programming language. Kenneth E. Iverson used primitives based on Curry's combinators in his J programming language, a successor to APL. This enabled what Iverson called tacit programming, that is, programming in functional expressions containing no variables, along with powerful tools for working with such programs. It turns out that tacit programming is possible in a clumsier manner in any APL-like language with user-defined operators (Pure Functions in APL and J). (Discuss strict vs. lazy evaluation semantics. Note implications of graph reduction implementation for lazy evaluation. Point out efficiency problem in lazy semantics: Repeated evaluation of the same expression, in, e.g. (square COMPLICATED) => ( Discuss benefit of graph reduction in this case: when (square COMPLICATED) is evaluated, the representation of COMPLICATED can be shared by both branches of the resulting graph for ( Logic The Curry-Howard isomorphism implies a connection between logic and programming: every proof of a theorem of intuitionistic logic corresponds to a reduction of a typed lambda term, and conversely. Moreover, theorems can be identified with function type signatures. Specifically, a typed combinatory logic corresponds to a Hilbert system (consisting of axioms and rules) in proof theory. The K and S combinators correspond to the axioms AK: A → (B → A), AS: (A → (B → C)) → ((A → B) → (A → C)), and function application corresponds to the detachment (modus ponens) rule MP: from A and A → B infer B. The calculus consisting of AK, AS, and MP is complete for the implicational fragment of the intuitionistic logic, which can be seen as follows. Consider the set W of all deductively closed sets of formulas, ordered by inclusion. Then is an intuitionistic Kripke frame, and we define a model in this frame by This definition obeys the conditions on satisfaction of →: on one hand, if , and is such that and , then by modus ponens. On the other hand, if , then by the deduction theorem, thus the deductive closure of is an element such that , , and . Let A be any formula which is not provable in the calculus. Then A does not belong to the deductive closure X of the empty set, thus , and A is not intuitionistically valid. See also Further reading | |||||||
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