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
K is defined to be
\x y. x. That is, it throws away its second argument. So
K xis a constant function from any (further) argument to
x. ("K" for "constant".) Compare
Kto our definition of
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
snd was our function for extracting the second element of an ordered pair:
\a b. b. Compare this to our definition of
B is defined to be:
\f g x. f (g x). (So
B f gis the composition
\x. f (g x)of
C is defined to be:
\f x y. f y x. (So
C fis a function like
fexcept it expects its first two (curried) arguments in flipped order.)
T is defined to be:
\x y. y x. (So
Tboth reorder arguments, just in different ways.)
W is defined to be:
\f x . f x x. (So
W faccepts one argument and gives it to
ftwice. What is the meaning of
ω (that is, lower-case omega) is defined to be:
\x. x x. Sometimes this combinator is called M. It and
Wboth 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:
(Though we'll see shortly that the behavior of
I can be exactly
simulated by a combination of
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
that argument, and feeds the two copies to the transitive verb meaning.
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 ≡ W
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
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
happens to be the expression
I, this schematic pattern guarantees
II ~~> I; if
X happens to be the expression
SK, the pattern
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
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:
SFGX ~~> FX(GX)
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
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
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
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
I must be enough to accomplish any computational task
K must suffice, since we've just seen that we can
I using only
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
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
\x. x, replace
\x y. x,
\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
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
I as expressions in CL. But
we will also endow CL with an infinite stock of variable symbols, just like the lambda
(XY) is in CL for any CL
Y. So examples of CL expressions include
(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)
@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
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
the application of the translation of
X to the translation of
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
In rule (4), if we have
@aa, we need a CL expression that behaves
like the lambda term
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
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
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
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
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 onis 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[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
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) ~~> Y X
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
\x y. y x),
\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
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 ⊃ 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
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:
- Combinatory logic at Wikipedia
- Combinatory logic at the Stanford Encyclopedia of Philosophy
- SKI combinatory calculus
- B,C,K,W system
- Chris Barker's Iota and Jot
- Jeroen Fokker, "The Systematic Construction of a One-combinator Basis for Lambda-Terms" Formal Aspects of Computing 4 (1992), pp. 776-780. http://people.cs.uu.nl/jeroen/article/combinat/combinat.ps