David C Keenan, 27-Aug-1996
116 Bowman Parade, Bardon QLD
4065, Australia
d.keenan@uq.net.au, http://uq.net.au/~zzdkeena
Taken from http://www.uq.net.au/~zzdkeena/Lambda/.
The lambda calculus, and the closely related theory of combinators, are important in the foundations of mathematics, logic and computer science. This paper provides an informal and entertaining introduction by means of an animated graphical notation.
In the 1930s and 40s, around the birth of the "automatic computer",
mathematicians wanted to formalise what we mean when we say some result or some
function is "effectively computable", whether by machine or human. A "computer",
originally, was a person who performed arithmetic calculations. The
"effectively" part is included to indicate that we are not concerned with the
time any particular computer might take to produce the result, so long as it
would get there eventually. They wanted to find the simplest possible system
that could be said to compute.
Several such systems were invented and for
the most part looked entirely unlike each other. Remarkably, they were all
eventually shown to be equivalent in the sense that any one could be made to
behave like the others. Today, the best known of these are the Turing Machine,
of the British mathematician Alan Turing (not to be confused with his touring
machine, the bicycle of which he was so fond), and the Lambda Calculus, of the
American logician Alonzo Church (1941). The Turing machine is reflected in the
Von Neumann machine which describes the general form of most computing hardware
today. The Lambda calculus is reflected in the programming language Lisp and its
relatives which are called "functional" or "applicative" languages.
The
Turing machine is based on the idea of a tape of unbounded length, with a head
which can move left or right along the tape reading and writing symbols. These
symbols can be interpreted as instructions (to move left or right or read or
write symbols) but may also be interpreted as data, particularly when it comes
time to read the result.
The lambda calculus is based on the more
abstract notion of "applying a function". For example we may write
y := 2x+3 to describe how to obtain y
given x, but suppose we want to describe, in the abstract, what it
is we are doing to x, so that we can do it to other things. We
might make up a name for it, say "DoubleAndAddThree" or just "f",
and write f(a) = 2a+3 to describe what we
mean. But this is all rather indirect and it would be a nuisance to have to keep
making up new names and remembering what they meant. Church used the greek
letter lambda and a dot and showed that we could simply write la.2a+3 as a name for the function.
This operation is called function abstraction. We could, of course,
have used any symbol we like in place of a. We can then
apply this function to anything we like, for example
z2; (la.2a+3)(z2)
simplifies to 2z2+3. We see that function abstraction
and function application are inverses. Note that we must now be careful to use
an explicit multiplication sign where juxtaposition might be wrongly interpreted
as function application.
We might want to express functions of more than
one argument, such as addition lab.a+b, but we can
always express these as functions of one argument which return another function,
e.g. la.(lb.a+b).This
transformation is called Currying, after Haskell Curry. If we apply
this to the number 5 for example, we obtain lb.5+b which may then be applied to
a second number. We can think of lab.a as
shorthand for la.(lb.a)
You may say, "So what?" All this
seems to be making things more complicated not less. For example, why is
lab.a+b any better than plain old
+?
What is remarkable about the lambda calculus is that one
doesn't need anything as crude, gross or substantial as the digits 0 and 1 or
operators like + and in order to do arithmetic and logic, or indeed any kind of
computation.
All we need is function abstraction and
application. "Abstracted from what and applied to what?", you may ask. Other
such functions! Actually when they are not applied to anything except each other
we refer to them as combinators rather than functions. This is
called the pure lambda calculus.
Since there is no confusion
with multiplication in the pure lambda calculus, we can omit some parentheses
and f(x) may be written simply as fx.
Application is taken to be left associative so that
(f(x))(y) or
(fx)y may be written as just
fxy.
We can write lambda expressions like la.a the identity combinator, or
la.aa the combinator that
applies any combinator to itself, or lab.ba the combinator that reverses
the order of application of two combinators.
When I first understood the
lambda calculus I felt that those arbitrary bound variables, like
a,b,f and x above, just
served to obscure what was really going on. They are only necessitated by the
constraint of writing everything as a one-dimensional string of symbols. So I am
going to introduce you to some of the strange characters that inhabit the world
of combinators by using a two-dimensional notation that I developed. This
notation started from the idea that, instead of the bound variables we could
draw arrows connecting each lambda symbol to blank spaces in the expression
where its variable would have appeared.
As pointed out by Raymond
Smullyan the eminent logician and author of several puzzle books, the theory of
combinators is an abstract science dealing with objects whose only important
property is how they act upon each other. We are free to choose other properties
of these objects in any way we like. In his delightful book To mock a
mockingbird, Smullyan (1985) chooses birds for his
combinators, in memory of Haskell Curry, an early pioneer in the theory of
combinators (1958) and an avid bird-watcher.
The story
begins,
"A certain enchanted forest is inhabited by talking birds.
Given any birds A and B, if you call out the name of B to A, then A will respond
by calling out the name of some bird to you; this bird we designate AB. Thus AB
is the bird named by A upon hearing the name of B."
Smullyan also
notes in his preface,
"This remarkable subject is currently playing an
important role in computer science and artificial intelligence Despite the
profundity of the subject, it is no more difficult to learn than high school
algebra or geometry."
In the hope of making it even easier I
introduce the following graphical notation by extending Smullyan's bird
metaphor. We will go inside the birds' heads and see how to draw diagrams of the
internal plumbing which connects their ear to their throat in order to produce
the correct song in response to each song that they hear. In other words we will
draw maps of their brains.
This notation was developed as part of an
attempt to place the lambda calculus on an even deeper foundation. Both the
attempt (so far unsuccessful) and the notation were inspired by George
Spencer-Brown's controversial book Laws of Form
(1969).
"The theme of this book is that a universe comes into being
when a space is severed or taken apart. The act is itself already remembered,
even if unconsciously, as our first attempt to distinguish different things in a
world where, in the first place, the boundaries can be drawn anywhere we please.
At this stage the universe cannot be distinguished from how we act upon it, and
the world may seem like shifting sand beneath our feet."
It is a
wonderfully bizarre fact that each song of a combinatory bird is not merely the
name of another bird but is actually a complete description of the
internal plumbing of that other bird. That is, each song is actually a brain map
of some bird. Since a song is a complete description of how some bird will
respond when it hears another bird, and the only important thing
about a combinatory bird is how it responds when it hears another bird, we see
that songs and singers are interchangeable. So we can say that the birds sing
birds to each other, or we can equally say that what we have is a
bunch of songs that sing songs to each other! Combinatory birds
exist at an almost mystical level. Their language has no distinction between
verbs and nouns. A description of action can equally well be a name. To
emphasise this, in future we will call our diagrams song
maps.
Unfortunately, although we have these diagrams, and the
purely textual notation of the lambda calculus, we no longer know how to turn
them into music. However I am confident that some day soon some brilliant
mathematical musician or musical mathematician (or mathical musimatician?) will
rediscover this ability. In the meantime we must be content to appreciate the
rhyme and rhythm of their graphical forms. It is something of a mystery as to
why the textual notation is called the lambda calculus, since surely any other
letter of the Greek alphabet would have done just as well. One suggestion is
that these songs are actually quite suitable for dancing and that "lambda" is in
fact a corruption of "lambada".
Carol Hindley (1986) has given some
marvellous drawings of the outsides of several well known
combinators in her hilarious note "Care of Your Pet Combinator". Here we find
that they bear somewhat more resemblance to insects and reptiles than to
conventional birds.
The simplest of these birds is called the Identity bird since its response to
hearing the name of any bird is that same (identical) bird. Smullyan remarks
that superficially, the Identity bird appears to have no intelligence at all,
and has been referred to as the Idiot bird. However the real reason for its
apparently unimaginative behaviour is that it has a big heart and is
fond of every bird. So when you call x to the Identity
bird, the reason it responds by calling back x is not that it can't
think of anything else; it's just that it wants you to know that it is fond of
x.
It turns out to be an important fact that every
combinatory bird is fond of at least one bird (speaking technically, we say
every bird has a fixed point). That is, for every bird there is
some song which you can call to it and receive the same song in
response.
Figure 1 shows the song map of the Identity bird. We won't be
as kind as Smullyan. We will refer to it as the Idiot bird from now on since
that's more fun and easier to say.
We make the convention that the filled half-circle on the left of the box
represents the bird's ear (combinatory birds only have one ear) and the one on
the right represents its throat. The dotted box is to enclose the plumbing which
connects the two. We can also label the box. The label is not a necessary part
of the graphical notation but merely to remind us of what we call this bird in
English; "I" for "Identity" or "Idiot". In this case the plumbing is the
simplest possible, a single pipe represented by the solid line. I have put an
arrowhead on the line to indicate the direction of flow of information from ear
to throat, but in future since flow will nearly always be from left to right we
will not clutter our diagrams with arrow heads except to indicate reverse
flow.
Another simple bird is the Mockingbird. It is called a Mockingbird
because its response to any bird is the same as that bird's response to itself.
This means that if you call out x to a Mockingbird you will get the
same response as if you had called out x to
x.
Figure 2 shows the song map of the Mocking bird. We will
see later the reason for using a lower-case omega (w)
as its abbreviation. For now you can think of it as an upside-down
'm'.
I have introduced a new symbol, the filled circle, which you might think
of as the combinatory birds' brain cell. I call it an applicator.
We can see that an applicator receives information from above and from its left
and responds to its right. It is called an applicator because of what it does
with the information it receives; it applies the bird whose
description it receives from above, to the song which it receives from its left,
in order to determine its response. The song arriving at the top is called the
operator and the one from the left, the operand. You
might think of the circle as representing an ear and throat with nothing between
them, but waiting to have a song map introduced between them from
above.
You might also think of the applicator as a sort of universal bird
since all it requires is the description of a bird in order to
become that bird.
It may seem a little inside-out,
or like pulling yourself up by your own bootstraps, to define ordinary
combinators as arrangements of universal ones, but such are the foundations of
mathematics.
The strips of movie film in figures 3 and 4 show
the sequence of what happens when a Mockingbird hears the Idiot song and when it
hears the Mockingbird song (i.e. the song describing itself).
It is unfortunate that the primary form of reduction was named after the
second letter of the Greek alphabet, beta (b). The form of conversion which was given the first letter,
alpha (a), turns out to be merely an artifact of the
particular method of representing these songs as one-dimensional strings of
symbols in the textual lambda calculus. In the textual lambda calculus it is not
possible for replication or substitution to occur
separately from beta reduction and so they are not recognised as
separate steps.
Figure 7 describes the allowable transformations by using
a meta-notation. The cloud with an italic character stands for any
expression (arrangement). The fanned-out lines with the ellipsis '...' between
them stands for any number of pipes originating from the same point. The box
with the italic character stands for a box containing any
expression.
It would be a very boring forest if the only birds were the Idiot bird and
Mockingbird and the birds derivable from them. I now introduce the simplest of
the two layer birds, the Kestrel and the Kite. Like the Idiot bird,
these birds contain no applicator and yet they turn out to be amazingly useful.
See figure 8.
The Kestrel's response to hearing a song x is (a song which
describes) a bird which responds with x no matter what it hears. We
say the Kestrels response to x is the constant-x or
Kx bird (In German, the word for "constant" starts with a 'k'). The
Kite ignores what it hears and always responds with the Idiot song, so it is the
KI bird.
Note that these birds have an inner box with an inner ear and
throat. Pipes may pass freely into inner boxes from outer ears without having to
pass through an ear. Combinatory birds can have ears which aren't connected, so
long as something connects to every throat.
Figures 9 and 10 show movies
of the Kestrel hearing the Idiot bird and the Kite hearing the
Mockingbird.
So the Kestrel's response to the Idiot song is in fact the Kite. In the
Kite movie, notice that it wouldn't have mattered what song the Kite heard, it
would still have responded with the Idiot song. The Kestrel and Kite are
sometimes called the true and false birds because they are
commonly interpreted as such when combinatory birds are made to do logic and
arithmetic. See Appendix A.
Note that between frames 3 and 9 of the
Kestrel movie we perform a simple relocation which must not be treated as a beta
reduction. No box is annihilated because no song passes through an ear. Figure
11 describes the general situation in the meta-notation.
Next we introduce the two-layer birds having one applicator. There are
four of these as shown in figure 12, the Idiot bird once removed, the Thrush,
the Konstant Mocker and the Crossed Konstant Mocker. Notice that we've included
the equivalent textual lambda notation, "lab.ab" etc.,
underneath each song map.
The meaning of the term "once removed" will become clearer when we see
some more once removed relatives. You should see what is meant by "crossed" if I
tell you that the Thrush could be called "the Crossed Idiot-bird-once-removed"
(CI*). We can think of the 'T' in 'Thrush' as also standing for 'transposed'. We
see that the Konstant Mocker will respond with the Mockingbird no matter what it
hears. The Crossed Konstant Mocker is of no particular interest but is shown to
complete the family.
Note that the crossed pipes in the Thrush are not to
be considered as connecting together but merely passing each other. This is one
of the limitations of only having two-dimensions in which to draw our song maps.
However, we can always tell the difference between a joint and a crossover
because joints can not occur as fan-ins 
r , only fan-outs 
a . For a crossing to be interpreted as a joint, it would
have to involve an illegal fan-in followed by a fan-out.
There are
sixteen two-layer birds that have two applicators and not all of them have
English names. I show a few of the most popular in figure 13, the Lark, Owl,
Warbler and Crossed Warbler. 
Although the Warbler could have been abbreviated as w* since it is the Mockingbird-once-removed, historically the
use of W goes back further than that of w and in fact
the Mockingbird was referred to only as WI for a long time. This is another
reason for the choice of w rather than M for the
mockingbird. Note that LI = OI = WI = w.
Now that we have birds with more than one layer and more than one
applicator we can distinguish two major sections of any song map, the rhyme and
the rhythm. The rhyme is the section at the leftmost end of a box, where all the
fan-outs and crossings occur. For example the Lark and the Warbler have the same
rhyme. If we assign the letter 'a' to the outermost ear and 'b' to the inner one
and follow the pipes to the right past the sloping sections until they run
parallel again, we can write the Lark and Warbler's rhyme as "abb" corresponding
to the downward direction on the song map. The Crossed Warbler then has "baa"
and the Owl has "bab". For birds with two layers and two applicators there are
eight possible rhyme schemes. The general rule is for birds with l
layers and a applicators there are l a+1
rhyme schemes.
The rhythm section is the section immediately to the right
of the rhyme, containing the applicators. Notice that there are no fan-outs or
crossings in the rhythm section, only pipes being reduced in pairs, via
applicators, until only one pipe remains. This pipe then connects to the throat.
You can see that the Lark and the Owl have the same rhythm but this is different
from the rhythm shared by the Warbler and the Crossed Warbler. The first we
write as "(-(- -))" and the second "((- -) -)", although in future we can omit
the outer brackets without ambiguity. These are the only possible rhythms for a
bird with two applicators. It could be that the duration of the notes is halved
each time we enter parentheses and doubled again when we leave. Figure 14 shows
a Lark and a Crossed Warbler annotated to show the correspondence between
textual and graphical notations.
You may have noticed that a rhythm
section corresponds to a binary tree. I believe, although I have not worked out
a proof, that the number of rhythms using a applicators can be
found by taking the number of rhythms using a-1 applicators and
multiplying by (4a-2)/(a + 1). Of course there is only
one rhythm with zero applicators. The series starts, 1, 1, 2, 5, 14, 42, 132,
429, 1430, 4862, 16796, 58786, ... .
So we see that the ultimate question to
the answer to life the universe and everything, from Douglas Adams' The
hitchhikers guide to the galaxy, could in fact have been "How many
combinatory rhythms can you make with five applicators?".
We can
determine the number of possible birds with a given number of layers and
applicators by multiplying these two numbers together (rhymes x rhythms). We can
combine the textual notations used above for rhyme and rhythm to give song
schemes like "(ab)b" for the Warbler and "b(ab)" for the owl. However, this is
not enough to uniquely describe a bird textually since for example the Idiot
bird and the Kestrel would both be "a". We must also indicate how many layers
the bird has. We do this by prefixing the rhyme and rhythm with a complete list
of the layers involved from outside to inside, preceded traditionally by a
lambda (l) and followed by a dot. So the Idiot bird is
"la.a", the Mockingbird is "la.aa", the Kestrel is "lab.a" and
the Kite is "lab.b".
There is no limit to the
number of applicators a bird can have. It does not depend on the number of
layers. For example, the multiple-mockingbird family consists of all birds with
only one layer. There are two different double-mockingbirds (having two
applicators), five triple-mockingbirds (three applicators), fourteen
quadruple-mockingbirds and so on. Under this scheme, the Idiot bird may be
considered as the zeruple-mockingbird. [Note: This is different to the sense in
which Smullyan uses "double mockingbird"]
Of the 80 two-layer birds which
have three applicators, I show only the Turing bird, named after its discoverer
the logician and computer scientist Alan Turing. See figure 15. I will tell you
more later about the importance of the Turing bird in connection with
fixed-point birds.
lab.b((aa)b)
We now move on to three-layer birds. We start with the most popular
three-layer two-applicator birds, the Cardinal and the Bluebird in figure 16.
These birds, along with the Idiot bird and the Kestrel, were among the first
five combinators described in 1920 (published in 1924) by Moses Schönfinkel.
Incidentally "schön finkel" is German for "beautiful finch".
The Cardinal might have been called the crossing bird because it always
responds with the crossed cousin of whatever bird it hears. For example if it
hears the Warbler song it will respond with the Crossed Warbler and vice versa.
Its response to the Thrush song is the Idiot bird once removed, and vice versa.
The word "converse" is often used instead of "crossed" in this context. If it
hears a song which has more than two layers the Cardinal will cross the
connections from the two outermost layers only. The Cardinal has the effect of
altering the rhyme of whatever bird it is applied to, but it will never alter
the rhythm.
Figure 17 shows a movie of a Cardinal doing its thing with a Thrush. It
shows between successive frames, (1) a beta reduction, (2) a simple relocation,
(3) a substitution, (4) a second beta reduction, (5) relocation, (6)
substitution, (7) a third beta reduction and (8) relocation (9).
Notice
that the second and third beta reductions are slightly different to those we
have seen before. The connection which allows the box to be removed comes not
from the throat of another bird but from the inside of the ear of
another bird. Our meta-notation (cloud picture) for beta reduction should be
understood to allow for either.
While the Cardinal is a rhyme altering
bird which will never alter the rhythm, the Bluebird is a rhythm altering bird
which will never alter the rhyme. The Bluebird is often called the composition
bird (another musical reference?) because when applied to two birds in
succession it produces a single bird which is the composition of
those two birds. By "applied to two birds in succession" we mean that we apply B
to the first bird a then we apply the resulting bird
Ba to the second bird b to obtain the result
(Ba)b. By the composition of two birds
a and b we mean a single bird which when applied to
some bird c has the same effect as first applying b
and then applying a to the result, i.e.
a(bc). Unfortunately "composition" doesn't start with
a 'B' but we can think of the 'B' as standing for bracketing if we think in
terms of the textual rhythm notation "- (- -)" we saw earlier.
The
Bluebird also functions as a once-removal bird for three layer
birds only. You should have guessed by now that once-removal involves inserting
a new applicator into the topmost pipe and supplying its operator from a new
outer layer. Calling out the Idiot song to any once-removed bird will have the
effect of undoing the once-removal.
Figure 18 shows how a Bluebird
composes a Cardinal and a Thrush. I have substantially reduced the amount of
in-between detail to the point where I no longer show relocations, replications
or substitutions separately where they immediately follow a beta reduction. This
coarse level of detail is the best one can do in the textual lambda
calculus!
The resulting bird is called the Vireo, also known as the pairing bird
because of its ability to take two birds and form a single bird from which
either part may be recovered by applying it to either the Kestrel
(true) or the Kite (false) depending on whether the first or
second part is required. This pairing property can be used repeatedly to make
lists or trees. This property also comes in handy for doing logic and arithmetic
as shown in appendix A.
Note that the first frame of the movie in figure
18 has the form
This is different from
The above equivalences follow from the substitution rule given
in figure 7. Figure 19 expresses this non-equivalence as a general non-rule in
the meta-notation. 
It is traditional in the textual form that operator and operand are
simply placed side by side with the operator on the left and the operand on the
right with parentheses to remove any ambiguity. This is opposite to the order
for direct application in the graphical form. It is also traditional in the
textual lambda calculus that we avoid having to write so many brackets by using
a left-associative convention. That is, BCT is the same as (BC)T.
This proves to be rather unfortunate when compared with the graphical form. The
graphical expression which looks most like "BCT" (apart from the left-right
reversal) corresponds in fact to B(CT). If we were to make a
right-associative convention for the textual form, the
correspondence with the graphical form would be more obvious. György Révész also
remarks on how unfortunate is the traditional choice and, in fact, chooses to go
against tradition in his book (1988, p16-17). We will stay with tradition in the
remainder of this paper.
The Starling, shown in figure 20, is the last of Schönfinkel's original
five combinators to be introduced. These are the Idiot bird, Kestrel, Cardinal,
Bluebird and Starling, I, K, C, B, S (although Schönfinkel called some of them
by different letters at the time). These five birds alone can easily generate
all other birds. The Starling is the only one of these birds to perform
replication. Note that SII = w.
While it is remarkable enough that the five birds I, K, C, B, S can be used
to derive all others, it is an even more remarkable fact, shown by Shönfinkel,
that all other birds can be derived from the Starling and Kestrel alone,
although the expressions involved can be enormous. For example:
I =
SKK
B = S(KS)K
C = S(BBS)(KK) = S(S(KS)K(S(KS)K)S)(KK)
K and S have
been likened to Adam and Eve or Yin and Yang. Note that crudely speaking, K
destroys (eliminates) and S creates and changes (replicates and changes both
rhyme and rhythm).
There are deterministic rules which allow us to take
any expression and work backwards to obtain an equivalent in terms of K and S
alone (see Smullyan, 1985; Peyton Jones, 1987). This fact has been used to build
computers called combinator reduction machines, whose basic instructions
correspond to K and S, although for efficiency reasons I, C, B and Y (which we
will meet later) are usually included as well. With regard to the number 42 and
The hitchhikers guide to the galaxy; could it be that
Deep Thought was a combinator reduction machine?
Schönfinkel
dreamed of another bird called the Jay from which the Starling and Kestrel (and
so all birds) could be derived. It seems that the closest we can come to this is
the quite complicated bird shown in figure 21. While not quite what Schönfinkel
had in mind, it has been given the name Jay.
The Jay was discovered in 1935 by J. Barkley Rosser and has the property
that it can be used in association with the Idiot bird to derive all birds
except those which ignore or eliminate one or more of their inputs (such as the
Kestrel and Kite). I will show how we obtain C, B and S. To do this it is
convenient to show certain other birds along the way. We have met the Thrush,
Mockingbird and Warbler before, but the Robin (R) is new.
T = JII
R =
JT = J(JII)
C = RRR = J(JII)(J(JII))(J(JII))
Here the expressions become
too big to continue expanding fully.
B = C(JIC)JI
w = C(C(C(BJT)T)T)T
W = C(BwR)
S = B(BW)(BBC)
To test your understanding of
reduction you might like to work out the normal form of the Robin now by
reducing the graphical version of the expression JT. The Robin is shown in
Appendix A.
Many other sets of primitive combinators are possible and we
could argue forever over which is more fundamental. In fact there is a direct
relationship between bases for the theory of combinators and axiom schemes for
implication logic, via a mapping called "Formulae as Types". See Hindley &
Seldin (1986).
One basis that proves particularly convenient for this
graphical notation is K, w, T, B, in which elimination,
replication, permuting (rhyme-changing) and regrouping (rhythm-changing) are
represented in their simplest form, each by a separate combinator. Given that R
= BBT we can use the derivations above to show that S is derivable from w, T, and B.
I remind you that the bird/song analogy is only an analogy. There is really
no distinction between birds and songs. This corresponds merely to the
operator/operand distinction which is relative to a particular applicator.
If we cause a Turing bird (U) to hear its own song we end up with a bird
called the Theta (Q) bird which is a fixed-point
finding bird or simply a fixed-point bird. The Turing and Theta
birds were discovered by Alan Turing in 1937. The Theta bird does not have a
normal form. Figure 22 shows one possible representation obtained by performing
a single beta reduction on UU and relocating the first U inside the remains of
the second. This version of the Theta bird is at least in weak head normal
form which means only that it has an outer box with an ear.
A fixed-point bird has the amazing property that on hearing any bird
b, it responds with a bird f of which b
is fond. That is, it responds with a fixed-point of b. So if we
want a solution for f in the recursive equation f =
bf we need only write Qb (or
Yb as we shall see later). The proof that fixed-point birds exist,
is a very powerful result since it says that it is meaningful to
define functions by means of recursive equations such as that for
f above. While a combinatory bird may have more than one fixed
point it turns out that the fixed points can always be ordered according to
their definedness. It is beyond the scope of this paper to describe
the precise meaning of definedness in this context, but recall that the Omega
bird is considered to be completely undefined. The thing defined by a recursive
equation is considered to be that fixed point which is least defined, called
simply "the least fixed point". See any of the introductory texts mentioned in
the references at the end of this paper.
The simplest fixed-point bird,
the Why bird, is generally attributed to Haskell Curry around 1942 (see the
notes on p185 of Curry, 1958). Perhaps the name refers to the incredulity of its
discoverer, "Why does it exist?" or "Why does it work the way it
does?". See figure 23.
As with the Omega bird, the Why bird turns out to have a simpler cyclic
description (Simon Peyton Jones, 1987) which better supports our intuition about
how it works. See figure 24.
Note for example that YI = W for both cyclic and
acyclic forms of Y and W. Every combinator is a fixed
point of I but W is the least fixed-point
of I although the meaning of "fixed" must be stretched somewhat to accommodate
this. Some other interesting derivations involving Y are Y = SLL, (where L =
CBw, the Lark) and Q = YO,
(where O = SI, the Owl).
The cyclic textual form shown for
the Why bird in figure 24 can similarly be used to describe the Omega bird as
"
".
In such a fundamental discipline as this, we should not be
surprised to find an object which is apparently disappearing up its own
fundament.
The surprising thing is that Y, Q and W can be written non-cyclically.
It is a
fact of mathematical life (as shown by Kurt Gödel) that if we are to have
something as powerful as a Why bird (a fixed-point bird) we must accept the risk
of producing an Omega bird (a non-terminating bird). Our old friend the
Mockingbird, with its ability to apply a bird to itself, is implicated in
both.
I hope that this paper has shown the potential of a carefully designed animated graphical notation to bring human spatial and temporal intuitions to bear on the study of such an abstract discipline as the lambda calculus. I hope it will be as useful as an intuition amplifier for the lambda calculus and combinators as is Warren Robinett's (1983?) game Rocky's Boots for Boolean algebra and John Conway's game of Life for cellular automata (see Poundstone, 1985).
Some important aspects of the notation are:
While this paper has been an attempt at an informal introduction
to lambda calculus, the graphical notation should be formally described and the
correspondences between it and the well-understood textual lambda calculus
should be elucidated if it is to be widely used. If the drawings are to be made
by machine or with machine assistance, the aesthetic layout rules should also be
made explicit.
I might be convinced that the diagrams should be flipped
horizontally so that the order of applications maps more directly
onto the textual form, however the order of abstractions would no
longer map directly, unless (for example) labc.def were
to be rewritten as def.cbal.
If this exposition
were to be extended, some explanation should be given of normal
order reduction versus unsafe reduction orders, and head normal
form and weak head normal form should be explained fully.
The graphical version of eta (h) reduction
should be introduced to allow for the extensional equivalence of combinators
such as I and I*. Note that these are the only extensionally equivalent pair of
combinators in this paper.
I hope that someone, with more resources than
I, will implement it as a computer 'game'. The user would set up an initial
expression and watch as it evolved according to the combinatory 'laws of
physics' (normal order reduction). It might be more interesting if the metaphor
was changed to one of mythical beasts devouring one another. It should also be
possible to find a mapping which will give a unique musical tune for most
combinators. The tune might correspond to the type of the
combinator when it has one. See Hindley & Seldin (1986).
The notation
might be extended to become a full programming language, or at least a program
reading language. For this purpose I would allow the internal
details of lower-level named abstractions to be suppressed while providing pan,
zoom-in and zoom-out facilities for program browsing and editing. A translator
from an existing textual functional language would be a worthwhile project,
although matching some of their more recent syntactic conveniences could be
difficult.
There are many ways of interpreting lambda calculus for logic
and arithmetic. By far the most popular choices for true and
false are K and KI respectively, although I and KI have some
advantages, and of course any such pair could also be swapped. The former is a
particularly useful choice because the 'if then else ' function is then
simply the Identity combinator (or no combinator at all). Remember that K
(true) returns the first of two arguments and KI (false)
returns the second. Figure 25 is to remind you what these birds look like, along
with the Thrush and two of its descendants, the Vireo and the Robin.
We have seen the Vireo before as the result of BCT. The Robin may also be
derived from the Thrush as BBT. I now give the usual interpretations of the
logical constants and connectives from Henk Barendregt (1984), which I leave for
you to verify for yourself. Note that all functions are prefix not infix, that
is, we must write "implies a b" rather than
"a implies b". You may like to consider the
expression "Y not" (the least fixed point of the logical negation
function) to determine why Curry referred to the Y combinator as the paradoxical
combinator.
true = K
false = KI
not = V
false true = V(KI)K
implies = R true =
RK
and = R false = R(KI)
or = T true =
TK
equiv = CS not = CS(V(KI)K)
Here are Barendregt's
interpretations for Peano arithmetic (many others are
possible).
zero = I
succ = V false =
V(KI)
pred = T false = T(KI)
isZero = T
true = TK
succ is the successor function such that
succ n = n+1, pred is the predecessor function such
that pred n+1 = n, and isZero returns
true if its argument is zero and false if it is
positive. We don't care what pred zero is and we don't care
what happens when any of the functions are applied to expressions which do not
represent numbers. Expressions which represent numbers are called numerals. Here
are some example numerals. You should test the operation of pred and
isZero on them.
one = succ zero =
V(KI)I
two = succ one = V(KI)(V(KI)I)
Note
that these numerals work by tallying (sometimes called unary although it is not
base one). The numeral n consists of a list of n
falses terminated by an I (as the empty list). It is possible to
produce more efficient numeral systems, such as binary, in the lambda calculus
but the definitions of their succ, pred and isZero
functions are more complicated. For example den Hoed's binary numerals are labc.a, labc.ac, labc.abc, labc.acc, labc.abbc, labc.acbc, , where b and
c act as 0 and 1 and the most significant bit is the rightmost (van der Poel et
al., 1980).
Given any numeral system with succ, pred
and isZero, we can define addition recursively as
follows.
add b c
= if
(c = 0) then b else ((add
b c-1)+1)
= if (isZero
c) then b
else (succ
(add b (pred c)))
=
ifThenElse (isZero c)
b
(succ (add b (pred
c)))
= (isZero c) b
(succ (add b (pred
c)))
= isZero c b
(succ (add b (pred
c)))
Then by abstracting b and c
(abstraction is the inverse of beta-reduction) we can write
add
b c
= (lbc.isZero
c b
(succ (add b (pred c)))) b
c
so
add
= lbc.isZero c b (succ (add b
(pred c)))
Now by abstracting add itself we can
write
add
= (labc.isZero c
b
(succ (a b (pred c)))) add
This is now in
the form of f = bf so we can use a fixed-point
combinator to solve it, and we write
add
= Y (labc.isZero c b
(succ (a b
(pred c))))
This procedure can be continued to define
multiplication and exponentiation and indeed any computable function on the
integers, as proved by Stephen Kleene.
Here's another way of interpreting
the pure lambda calculus for arithmetic, called Church numerals. Church numerals
have a particularly simple form when expressed in the textual
notation.
zero = lfx.x = KI
succ =
lafx.f(afx) = SB
So
one =
lfx.fx
two =
lfx.f(fx)
three
= lfx.f(f(fx))
etc.
add
= labfx.af(bfx)
multiply
= labf.a(bf)
power
= lab.ab
Jeff James
(1993) has developed a system, inspired by Spencer-Brown's work, which does
arithmetic using a topological notation with only two kinds of boundary. These
may be represented textually as brackets () and [], with the interpretation of
the empty expression as zero, () as one, ()() as two etc., addition is simple
juxtaposition ab and multiplication is
([a][b]) and so on. If () is understood as an
exponential function and [] as a logarithmic function, it all suddenly makes
sense, however the arithmetic is actually performed using only four simple
axioms: ([a]) = a, [(a)] =
a, []a = [],
([a][b])x =
([ax][bx]). I mention this because I can't help
feeling that it is somehow related to the lambda calculus and combinators but I
don't see how. Maybe you do.
This work was carried out while the author was supported by an Australian Postgraduate Coursework Award. Thanks go to Lou Kauffman and Dick Shoup for their discussions regarding earlier drafts of this paper.
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