Definitions

combinatory

Combinatory logic

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. A combinator is a higher-order function which, for defining a result from its arguments, solely uses function application and earlier defined combinators.

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 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, and largely ceased to publish after Stalin consolidated his power in 1929. Curry rediscovered the combinators while working as an instructor at the University of Princeton in the late 1927. In the latter 1930s, Alonzo Church and his students at Princeton University 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 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 (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:

  • v
  • λv.E1
  • (E1 E2)

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*x

(Using "*" to indicate multiplication.) 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:

The square of 3 is 3*3

To evaluate the resulting expression 3*3, 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 the 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 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. 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:

  • x
  • P
  • (E1 E2)

where x is a variable, P is one of the primitive functions, and (E1 E2) is the application of combinatory terms E1 and E2. The primitive functions themselves are combinators, or functions that, when seen as lambda terms, contain no free variables. To shorten the notations, a general convention is that (E1 E2 E3 ... En), or even E1 E2 E3... En, denotes the term (...((E1 E2) E3)... En). 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

(P x1 ... xn) = E

where E is a term mentioning only variables from the set x1 ... xn. 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

(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 convention for multiple application,

(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. 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:

((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, 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 when these ones are applied to enough arguments. 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:

  1. T[x] => x
  2. T[(E₁ E₂)] => (T[E₁] T[E₂])
  3. T[λx.E] => (K T[E]) (if x is not free in E)
  4. T[λx.x] => I
  5. T[λx.λy.E] => T[λx.T[λy.E]] (if x is free in E)
  6. T[λx.(E₁ E₂)] => (S T[λx.E₁] T[λx.E₂])

This process is also known as abstraction elimination.

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[λx.T[λy.(y x)]] (by 5)
= 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 and 1)
= (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 T[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.

Simplifications of the transformation

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 extensionally equal to any lambda term. The simplest example of such a basis is {X} where:

       Xλx.((xS)K)

It is not difficult to verify that:

       X (X (X X)) =ηβ K and
       X (X (X (X X)))) =ηβ S.

Since {K, S} is a basis, it follows that {X} is a basis too. The Iota programming language uses X as its sole combinator.

Another simple example of a one-point basis is:

       X' ≡ λx.(x K S K) with
       (X' X') X' =β K and
       X' (X' X') =β S

X' does not need η contraction in order to produce K and S. X is a one-point basis in CL+ext but not in CL without extensionality. X' is a one-point basis in both CL and CL+ext.

Combinators B, C

In addition to S and K, Schönfinkel's paper included two combinators which are now called B and C, with the following reductions:

       (C a b c) = (a c b)
       (B a b c) = (a (b c))

He also explains how they in turn can be expressed using only S and K.

These combinators are extremely useful when translating predicate logic or lambda calculus into combinator expressions. They were also used by Curry, and much later by David Turner, whose name has been associated with their computational use. Using them, we can extend the rules for the transformation as follows:

  1. T[x] => x
  2. T[(E₁ E₂)] => (T[E₁] T[E₂])
  3. T[λx.E] => (K T[E]) (if x is not free in E)
  4. T[λx.x] => I
  5. T[λx.λy.E] => T[λx.T[λy.E]] (if x is free in E)
  6. T[λx.(E₁ E₂)] => (S T[λx.E₁] T[λx.E₂]) (if x is free in both E₁ and E₂)
  7. T[λx.(E₁ E₂)] => (C T[λx.E₁] T[E₂]) (if x is free in E₁ but not E₂)
  8. T[λx.(E₁ E₂)] => (B T[E₁] T[λx.E₂]) (if x is free in E₂ but not E₁)

Using B and C combinators, the transformation of λx.λy.(y x) looks like this:

         T[λx.λy.(y x)]
       = T[λx.T[λy.(y x)]]
       = T[λx.(C T[λy.y] x)]     (by rule 7)
       = T[λx.(C I x)]
       = (C I)                   (η-reduction)
       = C*(traditional canonical notation : X* = X I)
       = 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.

The modern names for the combinators come from Haskell Curry's doctoral thesis of 1930 (see B,C,K,W System). In Schönfinkel's original paper, what we now call S, K, I, B and C were called S, C, I, Z, and T respectively.

CLK versus CLI calculus
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:
λx.E where x has at least one free occurrence in E.
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). 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

A normal form is any combinatory term in which the primitive combinators that occur, if any, are not applied to enough arguments to be simplified. 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 represent the conventional Church encodings of true and false, λx.λy.x and λx.λy.y, transformed into combinatory logic. 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. The proof of this theorem is rather simple.

Proof: By reductio ad absurdum. Suppose there is a complete non trivial predicate, say N.
Because N is supposed to be non trivial there are combinators A and B such that
(N A) = T and
(N B) = F.

Define NEGATION ≡ λx.(if (N x) then B else A) ≡ λx.((N x) B A)
Define ABSURDUM ≡ (Y NEGATION)

Fixed point theorem gives: ABSURDUM = (NEGATION ABSURDUM), for
ABSURDUM ≡ (Y NEGATION) = (NEGATION (Y NEGATION)) ≡ (NEGATION ABSURDUM).

Because N is supposed to be complete either:

  1. (N ABSURDUM) = F or
  2. (N ABSURDUM) = T

Case 1: F = (N ABSURDUM) = N (NEGATION ABSURDUM) = (N A) = T, a contradiction.
Case 2: T = (N ABSURDUM) = N (NEGATION ABSURDUM) = (N B) = F, again a contradiction.

Hence (N ABSURDUM) is neither T nor F, which contradicts the presupposition that N would be a complete non trivial predicate. QED.

From this undecidability theorem it immediately follows that there is no complete predicate that can discriminate between terms that have a normal form and terms that do not have a normal form. It also follows that there is no complete predicate, say EQUAL, such that:
(EQUAL A B) = T if A = B and
(EQUAL A B) = F if AB.
If EQUAL would exist, then for all A, λx.(EQUAL x A) would have to be a complete non trivial predicate.

Applications

Compilation of functional languages

Functional programming languages are often based on the simple but universal semantics of the lambda calculus.

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).

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 in proof theory.

The K and S combinators correspond to the axioms

AK: A → (BA),
AS: (A → (BC)) → ((AB) → (AC)),
and function application corresponds to the detachment (modus ponens) rule
MP: from A and AB 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 langle W,subseteqrangle is an intuitionistic Kripke frame, and we define a model Vdash in this frame by
XVdash Aiff Ain X.
This definition obeys the conditions on satisfaction of →: on one hand, if XVdash Ato B, and Yin W is such that Ysupseteq X and YVdash A, then YVdash B by modus ponens. On the other hand, if XnotVdash Ato B, then X,Anotvdash B by the deduction theorem, thus the deductive closure of Xcup{A} is an element Yin W such that Ysupseteq X, YVdash A, and YnotVdash B.

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 XnotVdash A, and A is not intuitionistically valid.

See also

References

Further reading

  • Hendrik Pieter Barendregt, 1984. The Lambda Calculus, Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics, Volume 103, North-Holland. ISBN 0-444-87508-5
  • Curry, Haskell B.; Robert Feys (1958). Combinatory Logic Vol. I. Amsterdam: North Holland.
  • Curry, Haskell B.; J. Roger Hindley and Jonathan P. Seldin (1972). Combinatory Logic Vol. II. Amsterdam: North Holland. ISBN 0-7204-2208-6.
  • Field, Anthony J. and Peter G. Harrison, 1998. Functional Programming. . Addison-Wesley. ISBN 0-201-19249-7
  • Hindley, Roger, and Meredith, 1990, "Principal Type-Schemes and Condensed Detachment," Journal of Symbolic Logic 55: 90-105
  • Paulson, Lawrence C., 1995. Foundations of Functional Programming. University of Cambridge.
  • Quine, W. V., 1960 "Variables explained away", Proceedings of the American Philosophical Society 104:3:343-347 (Jun. 15, 1960) at JSTOR Reprinted as Chapter 23 of Quine's Selected Logic Papers (1966), pp. 227–235
  • Moses Schönfinkel, 1924, "Über die Bausteine der mathematischen Logik," translated as "On the Building Blocks of Mathematical Logic" in From Frege to Gödel: a source book in mathematical logic, 1879-1931, Jean van Heijenoort, ed. Harvard University Press, 1967. ISBN 0-674-32449-8. The article that founded combinatory logic.
  • Sørensen, Morten Heine B. and Paweł Urzyczyn, 1999. Lectures on the Curry-Howard Isomorphism. University of Copenhagen and University of Warsaw, 1999.
  • Smullyan, Raymond, 1985. To Mock a Mockingbird. Knopf. ISBN 0-394-53491-3. A gentle introduction to combinatory logic, presented as a series of recreational puzzles using bird watching metaphors.
  • --------, 1994. Diagonalization and Self-Reference. Oxford Univ. Press. Chpts. 17-20 are a more formal introduction to combinatory logic, with a special emphasis on fixed point results.
  • Wolfengagen, V.E. Combinatory logic in programming. Computations with objects through examples and exercises. -- 2-nd ed. -- M.: "Center JurInfoR" Ltd., 2003. -- x+337 с. ISBN 5-89158-101-9.

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