Intuitively, (a) and (b) express the application of the same function to the argument
(\x. \z. z x) y
(\x. \y. y x) y
One can't just rename variables freely. (a) and (b) are different than what's expressed by:
(\z. (\z. z z) y
y into the body of (a)
(\x. \z. z x) is unproblematic:
(\x. \z. z x) y ~~> \z. z y
However, with (b) we have to be more careful. If we just substituted blindly, then we might take the result to be
\y. y y. But this is the self-application function, not the function which accepts an arbitrary argument and applies that argument to the free variable
y. In fact, the self-application function is what (c) reduces to. So if we took (b) to reduce to
\y. y y, we'd wrongly be counting (b) to be equivalent to (c), instead of (a).
To reduce (b), then, we need to be careful to that no free variables in what we're substituting in get captured by binding λs that they shouldn't be captured by.
In practical terms, you'd just replace (b) with (a) and do the unproblematic substitution into (a).
How should we think about the explanation and justification for that practical procedure?
One way to think about things here is to identify expressions of the lambda calculus with particular alphabetic sequences. Then (a) and (b) would be distinct expressions, and we'd have to have an explicit rule permitting us to do the kind of variable-renaming that takes us from (a) to (b) (or vice versa). This kind of renaming is called "alpha-conversion." Look in the standard treatments of the lambda calculus for detailed discussion of this.
Another way to think of it is to identify expressions not with particular alphabetic sequences, but rather with classes of alphabetic sequences, which stand to each other in the way that (a) and (b) do. That's the way we'll talk. We say that (a) and (b) are just typographically different notations for a single lambda formula. As we'll say, the lambda formula written with (a) and the lambda formula written with (b) are literally syntactically identical.
A third way to think is to identify the lambda formula not with classes of alphabetic sequences, but rather with abstract structures that we might draw like this:
(λ. λ. _ _) y ^ ^ | | | |__| | |_______|
Here there are no bound variables, but there are bound positions. We can regard formula like (a) and (b) as just helpfully readable ways to designate these abstract structures.
A version of this last approach is known as de Bruijn notation for the lambda calculus.
It doesn't seem to matter which of these approaches one takes; the logical properties of the systems are exactly the same. It just affects the particulars of how one states the rules for substitution, and so on. And whether one talks about expressions being literally "syntactically identical," or whether one instead counts them as "equivalent modulu alpha-conversion."
(Linguistic trivia: however, some linguistic discussions do suppose that alphabetic variance has important linguistic consequences; see Ivan Sag's dissertation.)
In a bit, we'll discuss other systems that lack variables. Those systems will not just lack variables in the sense that de Bruijn notation does; they will furthermore lack any notion of a bound position.
Define N to be
(\x. x y) z. Then N and
(\x. x y) z are syntactically equal, and we're counting them as syntactically equal to
(\z. z y) z as well, which we will write as:
N ≡ (\x. x y) z ≡ (\z. z y) z
N ~~> z y
means that N beta-reduces to
z y. This:
M <~~> N
means that M and N are beta-convertible, that is, that there's something they both reduce to in zero or more steps.
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 K to our definition of
get-first was our function for extracting the first element of an ordered pair:
\fst snd. fst. Compare this to K and
get-second was our function for extracting the second element of an ordered pair:
\fst snd. snd. 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 arguments in swapped order.)
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
It's possible to build a logical system equally powerful as the lambda calculus (and readily intertranslatable with it) using just combinators, considered as atomic operations. Such a language doesn't have any variables in it: not just no free variables, but no variables at all.
One can do that with a very spare set of basic combinators. These days the standard base is just three combinators: K and I from above, and also one more, S, which behaves the same as the lambda expression
\f g x. f x (g x). behaves. But it's possible to be even more minimalistic, and get by with only a single combinator. (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 argues that reflexive pronouns are argument duplicators.
Notice that the semantic value of himself is exactly
The reflexive pronoun in direct object position combines with the transitive verb. The result is an intransitive verb phrase that takes a subject argument, duplicates that argument, and feeds the two copies to the transitive verb meaning.
W <~~> S(CI):
S(CI) ≡ S((\fxy.fyx)(\x.x)) ~~> S(\xy.(\x.x)yx) ~~> S(\xy.yx) ≡ (\fgx.fx(gx))(\xy.yx) ~~> \gx.(\xy.yx)x(gx) ~~> \gx.(gx)x ≡ W
Ok, here comes a shift in thinking. Instead of defining combinators as equivalent to certain lambda terms, we can define combinators by 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. In pictures,
IX ~~> X
Thinking of this as a reduction rule, we can perform the following computation
II(IX) ~~> IIX ~~> 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 ≡ \fgx.fx(gx)
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 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 is equivalent to a certain crafty combination of Ss and Ks:
SKKX ~~> KX(KX) ~~> X
So the combinator
SKK is equivalent to the combinator I.
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.
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, imagine what a text editor does: it transforms 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
\fxy.x, and replace S with
\fgx.fx(gx). 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 this mapping, 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).
Assume that for any lambda term T, [T] is the equivalent combinatory logic term. The we can define the [.] mapping as follows:
1. [a] a 2. [(M N)] ([M][N]) 3. [\a.a] I 4. [\a.M] KM assumption: a does not occur free in M 5. [\a.(M N)] S[\a.M][\a.N] 6. [\a\b.M] [\a[\b.M]]
It's easy to understand these rules based on what S, K and I do. The first rule says
that variables are mapped to themselves.
The second rule says that the way to translate an application is to translate the
first element and the second element separately.
The third rule should be obvious.
The fourth rule should also be fairly self-evident: since what a lambda term such as
\x.y does it throw away its first argument and return
y, that's exactly what the combinatory logic translation should do. And indeed,
Ky is a function that throws away its argument and returns
The fifth rule deals with an abstract whose body is an application: the S combinator takes its next argument (which will fill the role of the original variable a) and copies it, feeding one copy to the translation of \a.M, and the other copy to the translation of \a.N. This ensures that any free occurrences of a inside M or N will end up taking on the appropriate value. Finally, the last rule says that if the body of an abstract is itself an abstract, translate the inner abstract first, and then do the outermost. (Since the translation of [\b.M] will not have any lambdas in it, we can be sure that we won't end up applying rule 6 again in an infinite loop.)
[Fussy notes: if the original lambda term has free variables in it, so will the combinatory logic translation. Feel free to worry about this, though you should be confident that it makes sense. You should also convince yourself that if the original lambda term contains no free variables---i.e., is a combinator---then the translation will consist only of S, K, and I (plus parentheses). One other detail: this translation algorithm builds expressions that combine lambdas with combinators. For instance, the translation of our boolean false
[\x[\y.y]] = [\x.I] = KI. In the intermediate stage, we have
\x.I, which mixes combinators in the body of a lambda abstract. It's possible to avoid this if you want to, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.]
[Various, slightly differing translation schemes from combinatorial logic to the lambda calculus are also possible. These generate different metatheoretical correspondences between the two calculii. Consult Hindley and 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.]
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. The goal is to establish that combinators can reverse order, so we use the T combinator, where
T ≡ \x\y.yx:
[\x\y.yx] = [\x[\y.yx]] = [\x.S[\y.y][\y.x]] = [\x.(SI)(Kx)] = S[\x.SI][\x.Kx] = S(K(SI))(S[\x.K][\x.x]) = S(K(SI))(S(KK)I)
We can test this translation by seeing if it behaves like the original lambda term does. The orginal lambda term lifts its first argument (think of it 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 (KX) 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 worries that Jim was talking about in the first class in which using non-referring expressions such as Santa Claus might commit one to believing in non-existant 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 has also developed a variable-free approach of some philosophical and linguistic interest in two books in the 1990's.
A final linguistic application: Steedman's Combinatory Categorial Grammar, where the "Combinatory" is
from combinatory logic (see especially his 2000 book, The Syntactic Processs). Steedman attempts to build
a syntax/semantics interface using a small number of combinators, including T ≡
\xy.yx, B ≡
and our friend S. Steedman used Smullyan's fanciful bird
names for the 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 a single primitive operation!
Here's more to read about combinatorial 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
In the assignment we asked you to reduce various expressions until it wasn't possible to reduce them any further. For two of those expressions, this was impossible to do. One of them was this:
(\x. x x) (\x. x x)
As we saw above, each of the halves of this formula are the combinator
ω; so this can also be written:
This compound expression---the self-application of
ω---is named Ω. It has the form of an application of an abstract (
ω) to an argument (which also happens to be
ω), so it's a redex and can be reduced. But when we reduce it, we get
ω ω again. So there's no stage at which this expression has been reduced to a point where it can't be reduced any further. In other words, evaluation of this expression "never terminates." (This is the standard language, however it has the unfortunate connotation that evaluation is a process or operation that is performed in time. You shouldn't think of it like that. Evaluation of this expression "never terminates" in the way that the decimal expansion of π never terminates. These are static, atemporal facts about their mathematical properties.)
There are infinitely many formulas in the lambda calculus that have this same property. Ω is the syntactically simplest of them. In our meta-theory, it's common to assign such formulas a special value,
⊥, pronounced "bottom." When we get to discussing types, you'll see that this value is counted as belonging to every type. To say that a formula has the bottom value means that the computation that formula represents never terminates and so doesn't evaluate to any orthodox, computed value.
From a "Fregean" or "weak Kleene" perspective, if any component of an expression fails to be evaluable (to an orthodox, computed value), then the whole expression should be unevaluable as well.
However, in some such cases it seems we could sensibly carry on evaluation. For instance, consider:
(\x. y) (ω ω)
Should we count this as unevaluable, because the reduction of
(ω ω) never terminates? Or should we count it as evaluating to
This question highlights that there are different choices to make about how evaluation or computation proceeds. It's helpful to think of three questions in this neighborhood:
Q1. When arguments are complex, as
(ω ω)is, do we reduce them before substituting them into the abstracts to which they are arguments, or later?
Q2. Are we allowed to reduce inside abstracts? That is, can we reduce:
(\x y. x z) (\x. x)
only this far:
\y. (\x. x) z
or can we continue reducing to:
Q3. Are we allowed to "eta-reduce"? That is, can we reduce expressions of the form:
\x. M x
where x does not occur free in
With regard to Q3, it should be intuitively clear that
\x. M x and
M will behave the same with respect to any arguments they are given. It can also be proven that no other functions can behave differently with respect to them. However, the logical system you get when eta-reduction is added to the proof theory is importantly different from the one where only beta-reduction is permitted.
If we answer Q2 by permitting reduction inside abstracts, and we also permit eta-reduction, then where none of
y1, ..., yn occur free in M, this:
\x y1... yn. M y1... yn
will eta-reduce by n steps to:
When we add eta-reduction to our proof system, we end up reconstruing the meaning of
<~~> and "normal form", all in terms that permit eta-reduction as well. Sometimes these expressions will be annotated to indicate whether only beta-reduction is allowed (
~~>β) or whether both beta- and eta-reduction is allowed (
The logical system you get when eta-reduction is added to the proof system has the following property:
Nare normal forms with no free variables, then
M ≡ Niff
Nbehave the same with respect to every possible sequence of arguments.
This implies that, when
N are (closed normal forms that are) syntactically distinct, there will always be some sequences of arguments
L1, ..., Ln such that:
M L1 ... Ln x y ~~> x N L1 ... Ln x y ~~> y
So closed beta-plus-eta-normal forms will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.
So the proof theory with eta-reduction added is called "extensional," because its notion of normal form makes syntactic identity of closed normal forms coincide with extensional equivalence.
See Hindley and Seldin, Chapters 7-8 and 14, for discussion of what should count as capturing the "extensionality" of these systems, and some outstanding issues.
The evaluation strategy which answers Q1 by saying "reduce arguments first" is known as call-by-value. The evaluation strategy which answers Q1 by saying "substitute arguments in unreduced" is known as call-by-name or call-by-need (the difference between these has to do with efficiency, not semantics).
When one has a call-by-value strategy that also permits reduction to continue inside unapplied abstracts, that's known as "applicative order" reduction. When one has a call-by-name strategy that permits reduction inside abstracts, that's known as "normal order" reduction. Consider an expression of the form:
((A B) (C D)) (E F)
Its syntax has the following tree:
((A B) (C D)) (E F) / \ / \ ((A B) (C D)) \ /\ (E F) / \ /\ / \ E F (A B) (C D) /\ /\ / \ / \ A B C D
Applicative order evaluation does what's called a "post-order traversal" of the tree: that is, we always go down when we can, first to the left, and we process a node only after processing all its children. So
(C D) gets processed before
((A B) (C D)) does, and
(E F) gets processed before
((A B) (C D)) (E F) does.
Normal order evaluation, on the other hand, will substitute the expresion
(C D) into the abstract that
(A B) evaluates to, without first trying to compute what
(C D) evaluates to. That computation may be done later.
With normal-order evaluation (or call-by-name more generally), if we have an expression like:
(\x. y) (C D)
the computation of
(C D) won't ever have to be performed. Instead,
(\x. y) (C D) reduces directly to
y. This is so even if
(C D) is the non-evaluable
Call-by-name evaluation is often called "lazy." Call-by-value evaluation is also often called "eager" or "strict". Some authors say these terms all have subtly different technical meanings, but I haven't been able to figure out what it is. Perhaps the technical meaning of "strict" is what I above called the "Fregean" or "weak Kleene" perspective: if any argument of a function is non-evaluable or non-normalizing, so too is the application of the function to that argument.
Most programming languages, including Scheme and OCaml, use the call-by-value evaluation strategy. (But they don't permit evaluation to continue inside an unappplied function.) There are techniques for making them model call-by-name evaluation, when necessary. But by default, arguments will always be evaluated before being bound to the parameters (the
\xs) of a function.
For languages like Scheme that permit functions to take more than one argument at a time, a further question arises: whether the multiple arguments are evaluated left-to-right, or right-to-left, or nothing is guaranteed about what order they are evaluated in. Different languages make different choices about this.
Some functional programming languages, such as Haskell, use the call-by-name evaluation strategy.
The lambda calculus can be evaluated either way. You have to decide what the rules shall be.
As we'll see in several weeks, there are techniques for forcing call-by-value evaluation of a computation, and also techniques for forcing call-by-name evaluation. If you liked, you could even have a nested hierarchy, where blocks at each level were forced to be evaluated in alternating ways.
Call-by-value and call-by-name have different pros and cons.
One important advantage of normal-order evaluation in particular is that it can compute orthodox values for:
(\x. y) (ω ω)
\z. (\x. y) (ω ω)
Indeed, it's provable that if there's any reduction path that delivers a value for a given expression, the normal-order evalutation strategy will terminate with that value.
An expression is said to be in normal form when it's not possible to perform any more reductions (not even inside abstracts).
There's a sense in which you can't get anything more out of
ω ω, but it's not in normal form because it still has the form of a redex.
A computational system is said to be confluent, or to have the Church-Rosser or diamond property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.
The untyped lambda calculus is confluent. So long as a computation terminates, it always terminates in the same way. It doesn't matter which order the sub-expressions are evaluated in.
A computational system is said to be strongly normalizing if every permitted evaluation path is guaranteed to terminate. The untyped lambda calculus is not strongly normalizing:
ω ω doesn't terminate by any evaluation path; and
(\x. y) (ω ω) terminates only by some evaluation paths but not by others.
But the untyped lambda calculus enjoys some compensation for this weakness. It's Turing complete! It can represent any computation we know how to describe. (That's the cash value of being Turing complete, not the rigorous definition. There is a rigrous definition. However, we don't know how to rigorously define "any computation we know how to describe.") And in fact, it's been proven that you can't have both. If a computational system is Turing complete, it cannot be strongly normalizing.
A computational system is said to be weakly normalizing if there's always guaranteed to be at least one evaluation path that terminates. The untyped lambda calculus is not weakly normalizing either, as we've seen.
The typed lambda calculus that linguists traditionally work with, on the other hand, is strongly normalizing. (And as a result, is not Turing complete.) It has expressive power (concerning types) that the untyped lambda calculus lacks, but it is also unable to represent some (terminating!) computations that the untyped lambda calculus can represent.
Other more-powerful type systems we'll look at in the course will also fail to be Turing complete, though they will turn out to be pretty powerful.
- Evaluation strategy
- Eager evaluation
- Lazy evaluation
- Strict programming language
- Church-Rosser theorem
- Normalization property
- Turing completeness
The question whether two formulas are syntactically equal is "decidable": we can construct a computation that's guaranteed to always give us the answer.
What about the question whether two formulas are convertible? Well, to answer that, we just need to reduce them to normal form, if possible, and check whether the results are syntactically equal. The crux is that "if possible." Some computations can't be reduced to normal form. Their evaluation paths never terminate. And if we just kept trying blindly to reduce them, our computation of what they're convertible to would also never terminate.
So it'd be handy to have some way to check in advance whether a formula has a normal form: whether there's any evaluation path for it that terminates.
Is it possible to do that? Sure, sometimes. For instance, check whether the formula is syntactically equal to Ω. If it is, it never terminates.
But is there any method for doing this in general---for telling, of any given computation, whether that computation would terminate? Unfortunately, there is not. Church proved this in 1936; Turing also essentially proved it at the same time. Geoff Pullum gives a very reader-friendly outline of the proofs here:
- Scooping the Loop Snooper, a proof of the undecidability of the halting problem in the style of Dr Seuss by Geoffrey K. Pullum
Interestingly, Church also set up an association between the lambda calculus and first-order predicate logic, such that, for arbitrary lambda formulas
N, some formula would be provable in predicate logic iff
N were convertible. So since the right-hand side is not decidable, questions of provability in first-order predicate logic must not be decidable either. This was the first proof of the undecidability of first-order predicate logic.