Combinators and Combinatory Logic

Combinatory logic is of interest here in part because it provides a useful computational system that is equivalent to the Lambda Calculus, but different from it. In addition, Combinatory Logic has a number of applications in natural language semantics. Exploring Combinatory Logic will involve defining a notion of reduction different from the one we have been using for the Lambda Calculus. This will provide us with a second parallel example when we're thinking through topics such as evaluation strategies and recursion.

Lambda expressions that have no free variables are known as combinators. Here are some common ones:

I is defined to be \x x

K is defined to be \x y. x. That is, it throws away its second argument. So K x is a constant function from any (further) argument to x. ("K" for "constant".) Compare K to our definition of true.

S is defined to be \f g x. f x (g x). This is a more complicated operation, but is extremely versatile and useful (see below): it copies its third argument and distributes it over the first two arguments.

fst was our function for extracting the first element of an ordered pair: \a b. a. Compare this to K and true as well.

snd was our function for extracting the second element of an ordered pair: \a b. b. Compare this to our definition of false.

B is defined to be: \f g x. f (g x). (So B f g is the composition \x. f (g x) of f and g.)

C is defined to be: \f x y. f y x. (So C f is a function like f except it expects its first two (curried) arguments in flipped order.)

T is defined to be: \x y. y x. (So C and T both reorder arguments, just in different ways.)

W is defined to be: \f x . f x x. (So W f accepts one argument and gives it to f twice. What is the meaning of W multiply?)

ω (that is, lower-case omega) is defined to be: \x. x x. Sometimes this combinator is called M. It and W both duplicate arguments, just in different ways.

It's possible to build a logical system equally powerful as the Lambda Calculus (and readily intertranslatable with it) using just combinators, considered as primitive operations. (That is, we refrain from defining them in terms of lambda expressions, as we did above.) Such a language doesn't have any variables in it: not just no free variables, but no variables (or "bound positions") at all.

One can do that with a very spare set of basic combinators. These days the standard base is just three combinators: S, K, and I. (Though we'll see shortly that the behavior of I can be exactly simulated by a combination of S's and K's.) But it's possible to be even more minimalistic, and get by with only a single combinator (see links below for details). (And there are different single-combinator bases you can choose.)

There are some well-known linguistic applications of Combinatory Logic, due to Anna Szabolcsi, Mark Steedman, and Pauline Jacobson. They claim that natural language semantics is a combinatory system: that every natural language denotation is a combinator.

For instance, Szabolcsi 1987 argues that reflexive pronouns are argument duplicators.

everyone   hit           himself
S/(S!NP)   (S!NP)/NP     (S!NP)!((S!NP)/NP)
\f∀x[fx]   \y\z[HIT y z] \h\u[huu]
           --------------------------------- here "hit" is an argument to "himself"
                  S!NP     \u[HIT u u]
-------------------------------------------- here "hit himself" is an argument to "everyone"
                  S        ∀x[HIT x x]

Notice that the semantic value of himself is exactly W. The reflexive pronoun in direct object position combines with the transitive verb "hit". The result is an intransitive verb phrase "hit himself" that takes a subject argument u, duplicates that argument, and feeds the two copies to the transitive verb meaning.

Note that W <~~> S(CI):

S(CI) ≡
S ((\f x y. f y x) (\x x)) ~~>
S (\x y. (\x x) y x) ~~>
S (\x y. y x) ≡
(\f g x. f x (g x)) (\x y. y x) ~~>
\g x. (\x y. y x) x (g x) ~~>
\g x. (g x) x ≡

A different set of reduction rules

Instead of defining combinators in terms of antecedently understood lambda terms, we want to consider the view that takes the combinators as primitive, and understands them in terms of what they do. If we have the I combinator followed by any expression X, I will take that expression as its argument and return that same expression as the result. Diagrammatically:

IX ~~> X

That is, asume that X stands in for any expression. Then if X happens to be the expression I, this schematic pattern guarantees that II ~~> I; if X happens to be the expression SK, the pattern guarantees that I(SK) ~~> SK; and so on. That is, X here is a metavariable over expressions.

Thinking of this as a reduction rule, we can perform the following computation:

II(IX) ~~> I(IX) ~~> IX ~~> X

The reduction rule for K is also straightforward:

KXY ~~> X

That is, K throws away its second argument. The reduction rule for S can be constructed by examining the defining lambda term:

S ≡ \f g x. f x (g x)

S takes three arguments, duplicates the third argument, and feeds one copy to the first argument and the second copy to the second argument. So:


If the meaning of a function is nothing more than how it behaves with respect to its arguments, these reduction rules capture the behavior of the combinators S, K, and I completely. We can use these rules to compute without resorting to beta reduction.

For instance, we can show how the I combinator's behavior is simulated by a certain crafty combination of Ss and Ks:

SKKX ~~> KX(KX) ~~> X

So the combinator SKK is equivalent to the combinator I. (Really, it could be SKY for any Y. Hindley & Seldin p. 26 points to discussion later in their book of why it's theoretically more elegant to keep I around, anyway.)

These reduction rule have the same status with respect to Combinatory Logic as beta-reduction and eta-reduction have with respect to the Lambda Calculus: they are purely syntactic rules for transforming one sequence of symbols (e.g., a redex) into another (a reduced form). It's worth noting that the reduction rules for Combinatory Logic are considerably more simple than, say, beta reduction. Also, since there are no variables in Combinatory Logic, there is no need to worry about variables colliding when we substitute.

Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. As we said, the most common system uses S, K, and I as defined here.

The equivalence of the untyped Lambda Calculus and Combinatory Logic

We've claimed that Combinatory Logic is "equivalent to" the Lambda Calculus. If that's so, then S, K, and I must be enough to accomplish any computational task imaginable. Actually, S and K must suffice, since we've just seen that we can simulate I using only S and K. In order to get an intuition about what it takes to be Turing Complete, recall our discussion of the Lambda Calculus in terms of a text editor. A text editor has the power to transform any arbitrary text into any other arbitrary text. The way it does this is by deleting, copying, and reordering characters. We've already seen that K deletes its second argument, so we have deletion covered. S duplicates and reorders, so we have some reason to hope that S and K are enough to define arbitrary functions.

We've already established that the behavior of combinatory terms can be perfectly mimicked by lambda terms: just replace each combinator with its equivalent lambda term, i.e., replace I with \x. x, replace K with \x y. x, and replace S with \f g x. f x (g x). So the behavior of any combination of combinators in Combinatory Logic can be exactly reproduced by a lambda term.

How about the other direction? Here is a method for converting an arbitrary lambda term into an equivalent Combinatory Logic term using only S, K, and I. Besides the intrinsic beauty of such mappings, and the importance of what it says about the nature of binding and computation, it is possible to hear an echo of computing with continuations in this conversion strategy (though you wouldn't be able to hear these echos until we've covered a considerable portion of the rest of the course). In addition, there is a direct linguistic application of this mapping in chapter 17 of Barker and Shan 2014, where it is used to establish a correspondence between two natural language grammars, one of which is based on lambda-like abstraction, the other of which is based on Combinatory Logic-like manipulations.

(Warning This is a different mapping from the Lambda Calculus to Combinatory Logic than we presented in class (and was posted here earlier). It now matches the presentation in Barendregt 1984, and in Hankin Chapter 4 (esp. pp. 61, 65) and in Hindley & Seldin Chapter 2 (esp. p. 26). In some ways this translation is cleaner and more elegant, which is why we're presenting it.)

In order to establish the correspondence, we need to get a bit more official about what counts as an expression in CL. Of course, we count the primitive combinators S, K, and I as expressions in CL. But we will also endow CL with an infinite stock of variable symbols, just like the lambda calculus, including x, y, and z. Finally, (XY) is in CL for any CL expressions X and Y. So examples of CL expressions include x, (xy), Sx, SK, (x(SK)), (K(IS)), and so on. When we omit parentheses, we assume left associativity, so XYZ ≡ ((XY)Z).

It may seem weird to allow variables in CL. The reason this is necessary is because we're trying to show that every lambda term can be translated into an equivalent CL term. Since some lambda terms contain free variables, we need to provide a translation in CL for those free variables. As you might expect, it will turn out that whenever the lambda term in question contains no free variables (i.e., is a Lambda Calculus combinator), its translation in CL will also contain no variables, but will instead just be made up of primitive combinators and parentheses.

Let's say that for any lambda term T, [T] is the equivalent Combinatory Logic term. Then we define the [.] mapping as follows.

 1. [a]          =   a
 2. [(\aX)]      =   @a[X]
 3. [(XY)]       =   ([X][Y])

Wait, what is that @a ... business? Well, that's another operation on (a variable and) a CL expression, that we can define like this:

 4. @aa          =   I
 5. @aX          =   KX           if a is not in X
 6. @a(Xa)       =   X            if a is not in X
 7. @a(XY)       =   S(@aX)(@aY)

Think of @aX as a pseudo-lambda abstract. (Hankin and Barendregt write it as λ*a. X; Hindley & Seldin write it as [a] X.) It is possible to omit line 6, and some presentations do, but Hindley & Seldin observe that this "enormously increases" the length of "most" translations.

It's easy to understand these rules based on what S, K and I do.

Rule (1) says that variables are mapped to themselves. If the original lambda expression had no free variables in it, then any such translations will only be temporary. The variable will later get eliminated by the application of other rules.

Rule (2) says that the way to translate an application is to first translate the body (i.e., [X]), and then prefix a kind of temporary psuedo-lambda built from @ and the original variable.

Rule (3) says that the translation of an application of X to Y is the application of the translation of X to the translation of Y.

As we'll see, the first three rules sweep through the lambda term, changing each lambda to an @.

Rules (4) through (7) tell us how to eliminate all the @'s.

In rule (4), if we have @aa, we need a CL expression that behaves like the lambda term \aa. Obviously, I is the right choice here.

In rule (5), if we're binding into an expression that doesn't contain any variables that need binding, then we need a CL term that behaves the same as \aX would if X didn't contain a as a free variable. Well, how does \aX behave? When \aX occurs in the head position of a redex, then no matter what argument it occurs with, it throws away its argument and returns X. In other words, \aX is a constant function returning X, which is exactly the behavior we get by prefixing K.

Rule (6) should be intuitive; and as we said, we could in principle omit it and just handle such cases under the final rule.

The easiest way to grasp rule (7) is to consider the following claim:

\a(XY) <~~> S(\aX)(\aY) 

To prove it to yourself, just consider what would happen when each term is applied to an argument a. Or substitute \x y a. x a (y a) in for S and reduce.

Persuade yourself that if the original lambda term contains no free variables --- i.e., is a Lambda Calculus combinator --- then the translation will consist only of S, K, and I (plus parentheses).

Various, slightly differing translation schemes from Combinatory Logic to the Lambda Calculus are also possible. These generate different meta-theoretical correspondences between the two calculi. Consult Hindley & Seldin for details.

Also, note that the combinatorial proof theory needs to be strengthened with axioms beyond anything we've here described in order to make [M] convertible with [N] whenever the original lambda-terms M and N are convertible. But then, we've been a bit cavalier about giving the full set of reduction rules for the Lambda Calculus in a similar way.

For instance, one issue we mentioned in the notes on ?Reduction Strategies is whether reduction rules (in either the Lambda Calculus or Combinatory Logic) apply to embedded expressions. Often, we do want that to happen, but making it happen requires adding explicit axioms.

Let's see the translation rules in action. We'll start by translating the combinator we use to represent false:

   [\y (\n n)] 
==  @y [\n n]      rule 2
==  @y (@n n)      rule 2
==  @y I           rule 4
==    KI           rule 5

Let's check that the translation of the false boolean behaves as expected by feeding it two arbitrary arguments:

KIXY ~~> IY ~~> Y

Throws away the first argument, returns the second argument---yep, it works.

Here's a more elaborate example of the translation. Let's say we want to establish that combinators can reverse order, so we set out to translate the T combinator (\x y. y x):

==  @x[\y(yx)]
==  @x(@y[yx])
==  @x(@y([y][x]))
==  @x(@y(yx))
==  @x(S(@yy)(@yx))
==  @x(S I   (@yx))
==  @x(S I    (Kx))
==     S(@x(SI))(@x(Kx))
==     S (K(SI))(S(@xK)(@xx))
==     S (K(SI))(S (KK) I)

By now, you should realize that all rules (1) through (3) do is sweep through the lambda term turning lambdas into @'s.

We can test this translation by seeing if it behaves like the original lambda term does. The original lambda term "lifts" its first argument x, in the sense of wrapping it into a "one-tuple" or a package that accepts an operation y as a further argument, and then applies y to x. (Or just think of T as reversing the order of its two arguments.)

S (K(SI)) (S(KK)I) X Y ~~>
(K(SI))X ((S(KK)I) X) Y ~~>
SI ((KK)X (IX)) Y ~~>
SI (K X) Y ~~>
IY (KXY) ~~>

Voilà: the combinator takes any X and Y as arguments, and returns Y applied to X.

One very nice property of Combinatory Logic is that there is no need to worry about alphabetic variance, or variable collision---since there are no (bound) variables, there is no possibility of accidental variable capture, and so reduction can be performed without any fear of variable collision. We haven't mentioned the intricacies of alpha equivalence or safe variable substitution, but they are in fact quite intricate. (The best way to gain an appreciation of that intricacy is to write a program that performs lambda reduction.)

Back to linguistic applications: one consequence of the equivalence between the Lambda Calculus and Combinatory Logic is that anything that can be done by binding variables can just as well be done with combinators. This has given rise to a style of semantic analysis called Variable-Free Semantics (in addition to Szabolcsi's papers, see, for instance, Pauline Jacobson's 1999 Linguistics and Philosophy paper, "Towards a variable-free Semantics").

Somewhat ironically, reading strings of combinators is so difficult that most practitioners of variable-free semantics express their meanings using the Lambda Calculus rather than Combinatory Logic. Perhaps they should call their enterprise Free Variable-Free Semantics.

A philosophical connection: Quine went through a phase in which he developed a variable-free logic.

Quine, Willard. 1960. "Variables explained away" Proceedings of the American Philosophical Society. Volume 104: 343--347. Also in W. V. Quine. 1960. Selected Logical Papers. Random House: New York. 227--235.

The reason this was important to Quine is similar to the worry that using non-referring expressions such as Santa Claus might commit one to believing in non-existent things. Quine's slogan was that "to be is to be the value of a variable." What this was supposed to mean is that if and only if an object could serve as the value of some variable, we are committed to recognizing the existence of that object in our ontology. Obviously, if there are no variables, this slogan has to be rethought.

Quine did not appear to appreciate that Shoenfinkel had already invented Combinatory Logic, though he later wrote an introduction to Shoenfinkel's key paper reprinted in Jean van Heijenoort (ed) 1967 From Frege to Goedel, a source book in mathematical logic, 1879--1931.

Cresswell also developed a variable-free approach of some philosophical and linguistic interest in two books in the 1990s.

A final linguistic application: Steedman's Combinatory Categorial Grammar, where the "Combinatory" is from Combinatory Logic (see especially his 2012 book, Taking Scope). Steedman attempts to build a syntax/semantics interface using a small number of combinators, including T (\x y. y x), B (\f g x. f (g x)), and our friend S. Steedman used Smullyan's fanciful bird names for these combinators: Thrush, Bluebird, and Starling.

Many of these combinatory logics, in particular, the SKI system, are Turing Complete. In other words: every computation we know how to describe can be represented in a logical system consisting of only primitive combinators, even some systems with only a single primitive combinator.

A connection between Combinatory Logic and Sentential Logic

The combinators K and S correspond to two well-known axioms of sentential logic:

AK: A ⊃ (B ⊃ A)
AS: (A ⊃ (B ⊃ C)) ⊃ ((A ⊃ B) ⊃ (A ⊃ C))

When these two axiom schemas are combined with the rule of modus ponens (from A and A ⊃ B, conclude B), the resulting proof system is complete for the "implicational fragment" of intuitionistic logic. (That is, the part of intuitionistic logic you get when is your only connective. To get a complete proof system for classical sentential logic, you need only add one more axiom schema, constraining the behavior of a new connective ¬.) The way we'll favor viewing the relationship between these axioms and the S and K combinators is that the axioms correspond to type schemas for the combinators. This will become more clear once we have a theory of types in view.

Here's more to read about Combinatory Logic. Surely the most entertaining exposition is Smullyan's To Mock a Mockingbird. Other sources include: