N.B. to run his examples in emacs, you'll want these aliases:
(defalias 'plus #'+) (defalias 'quotient #'/) (defalias 'times #'*) (defalias 'difference #'-)
Lisp: Atoms and Lists
IN previous columns I have written quite often about the field of artificial intelligence - the search for ways to program computers so that they might come to behave with flexibility, common sense, insight, creativity, self awareness, humor, and so on. The quest for AI started in earnest over two decades ago, and since then has bifurcated many times, so that today it is a very active and multifaceted research area. In the United States there are perhaps a couple of thousand people professionally involved in AI, and there are a similar number abroad. Although there is among these workers a considerable divergence of opinion concerning the best route to AI, one thing that is nearly unanimous is the choice of programming language. Most AI research efforts are carried out in a language called "Lisp". (The name is not quite an acronym; it stands for "list processing".)
Why is most AI work done in Lisp? There are many reasons, most of which are somewhat technical, but one of the best is quite simple: Lisp is crisp. Or as Marilyn Monroe said in The Seven-Year Itch, "I think it's jus-teeegant!" Every computer language has arbitrary features, and most languages are in fact overloaded with them. A few, however, such as Lisp and Algol, are built around a kernel that seems as natural as a branch of mathematics. The kernel of Lisp has a crystalline purity that not only appeals to the esthetic sense, but also makes Lisp a far more flexible language than most others. Because of Lisp's beauty and centrality in this important area of modern science, then, I have decided to devote a trio of columns to some of the basic ideas of Lisp.
The deep roots of Lisp lie principally in mathematical logic. Mathematical pioneers such as Thoralf Skolem, Kurt Godel, and Alonzo Church contributed seminal ideas to logic in the 1920's and 1930's that were incorporated decades later into Lisp. Computer programming in earnest began in the 1940's, but so-called "higher-level" programming languages (of which Lisp is one) came into existence only in the 1950's. The earliest list-processing language was not Lisp but IPL ("Information Processing Language"), developed in the mid-1950's by Herbert Simon, Allen Newell, and J. C. Shaw. In the years 1956-58, John McCarthy, drawing on all these previous sources, came up with an elegant algebraic list-processing language he called Lisp. It caught on quickly with the young crowd around him at the newly-formed MIT Artificial Intelligence Project, was implemented on the IBM 704, spread to other AI groups, infected them, and has stayed around all these years. Many dialects now exist, but all of them share that central elegant kernel.
Let us now move on to the way Lisp really works. One of the most appealing features of Lisp is that it is interactive, as contrasted with most other higher-level languages, which are noninteractive. What this means is the following. When you want to program in Lisp, you sit down at a terminal connected to a computer and you type the word "lisp" (or words to that effect). The next thing you will see on your screen is a so-called "prompt" - a characteristic symbol such as an arrow or asterisk. I like to think of this prompt as a greeting spoken by a special "Lisp genie", bowing low and saying to you, "Your wish is my command - and now, what is your next wish?" The genie then waits for you to type something to it. This genie is usually referred to as the Lisp interpreter, and it will do anything you want but you have to take great care in expressing your desires precisely, otherwise you may reap some disastrous effects. Shown below is the prompt, showing that the Lisp genie is ready to do your bidding:
The genie is asking us for our heart's desire, so let us type in a simple expression:
> (plus 2 2)
and then a carriage return. (By the way, all Lisp expressions and words will be printed in Helvetica in this and the following two chapters.) Even non-Lispers can probably anticipate that the Lisp genie will print in return. the value 4. Then it will also print a fresh prompt, so that the screen will now appear this way:
> (plus 2 2) 4 >
The genie is now ready to carry out our next command - or, more politely stated, our next wish - should we have one. The carrying-out of a wish expressed as a Lisp statement is called evaluation of that statement. The preceding short interchange between human and computer exemplifies the behavior of the Lisp interpreter: it reads a statement, evaluates it, prints the appropriate value, and then signals its readiness to read a new statement. For this reason, the central activity of the Lisp interpreter is referred to as the read-eval-print loop.
The existence of this Lisp genie (the Lisp interpreter) is what makes Lisp interactive. You get immediate feedback as soon as you have typed a "wish" - a complete statement - to Lisp. And the way to get a bunch of wishes carried out is to type one, then ask the genie to carry it out, then type another, ask the genie again, and so on.
By contrast, in many higher-level computer languages you must write out an entire program consisting of a vast number of wishes to be carried out in some specified order. What's worse is that later wishes usually depend strongly on the consequences of earlier wishes - and of course, you don't get to try them out one by one. The execution of such a program may, needless to say, lead to many unexpected results, because so many wishes have to mesh perfectly together. If you've made the slightest conceptual error in designing your wish list, then a total foul-up is likely - in fact, almost inevitable. Running a program of this sort is like launching a new space probe, untested: you can't possibly have anticipated all the things that might go wrong, and so all you can do is sit back and watch, hoping that it will work. If it fails, you go back and correct the one thing the failure revealed, and then try another launch. Such a gawky, indirect, expensive way of programming is in marked contrast to the direct, interactive, one-wish-at-atime style of Lisp, which allows "incremental" program development and debugging. This is another major reason for the popularity of Lisp.
What sorts of wishes can you type to the Lisp genie for evaluation,
and what sorts of things will it print back to you? Well, to begin
with, you can type arithmetical expressions expressed in a rather
strange way, such as
(times (plus 6 3) (difference 6 3)). The answer
to this is 27, since
(plus 6 3) evaluates to
(difference 6 3)
evaluates to 3, and their product is 27. This notation, in which each
operation is placed to the left of its operands, was invented by the
Polish logician Jan Lukasiewicz before computers existed.
Unfortunately for Lukasiewicz, his name was too formidable-looking for
most speakers of English, and so this type of notation came to be
called Polish notation. Here is a simple problem in this notation for
you, in which you are to play the part of the Lisp genie:
> (quotient (plus 2113) (difference 23 (times 2 (difference 7 (plus 2 2)))))
Perhaps you have noticed that statements of Lisp involve parentheses. A profusion of parentheses is one of the hallmarks of Lisp. It is not uncommon to see an expression that terminates in a dozen right parentheses! This makes many people shudder at first - and yet once you get used to their characteristic appearance, Lisp expressions become remarkably intuitive, even, charming, to the eye, especially when pretty printed, which means that a careful indentation scheme is followed that reveals their logical structure. All of the expressions in displays in this article have been pretty-printed.
The heart of Lisp is its manipulable structures. All programs in Lisp work by creating, modifying, and destroying structures. Structures come in two types: atomic and composite, or, as they are usually called, atoms and lists. Thus, every Lisp object is either an atom or a list (but not both). The only exception is the special object called nil, which is both an atom and a list. More about nil in a moment. What are some other typical Lisp atoms? Here are a few:
hydrogen, helium, j-s-bach, 1729, 3.14159, pi, arf, foo, bar, baz, buttons-&-bows
Lists are the flexible data structures of Lisp. A list is pretty much what it sounds like: a collection of some parts in a specific order. The parts of a list are usually called its elements or members. What can these members be? Well, not surprisingly, lists can have atoms as members. But just as easily, lists can contain lists as members, and those lists can in turn contain other lists as members, and so on, recursively. Oops! I jumped the gun with that word. But no harm done. You certainly understood what I meant, and it will prepare you for a more technical definition of the term to come later.
A list printed on your screen is recognizable by its parentheses. In
Lisp, anything bounded by matching parentheses constitutes a list. So,
(zonk blee strill (croak flonk)) is a four-element list
whose last element is itself a two-element list. Another short list is
(plus 2 2), illustrating the fact that Lisp statements themselves are
lists. This is important because it means that the Lisp genie, by
manipulating lists and atoms, can actually construct new wishes by
itself. Thus the object of a wish can be the construction - and
subsequent evaluation - of a new wish!
Then there is the empty list - the list with no elements at all. How is this written down? You might think that an empty pair of parentheses - () - would work. Indeed, it will work - but there is a second way of indicating the empty list, and that is by writing nil. The two notations are synonymous, although nil is more commonly written than () is. The empty list, nil, is a key concept of Lisp; in the universe of lists, it is what zero is in the universe of numbers. To use another metaphor for nil, it is like the earth in which all structures are rooted. But for you to understand what this means, you will have to wait a bit.
The most commonly exploited feature of an atom is that it has (or can be given) a value. Some atoms have permanent values, while others are variables. As you might expect, the value of the atom 1729 is the integer 1729, and this is permanent. (I am distinguishing here between the atom whose print name or pname is the four-digit string 1729, and the eternal Platonic essence that happens to be the sum of two cubes in two different ways - i.e., the number 1729.) The value of nil is also permanent, and it is - nil! Only one other atom has itself as its permanent value, and that is the special atom t.
Aside from t, nil, and atoms whose names are numerals, atoms are
generally variables, which means that you can assign values to them
and later change their values at will. How is this done? Well, to
assign the value 4 to the atom pie, you can type to the Lisp genie
(setq pie 4). Or you could just as well type
(setq pie (plus 2 2)) -
(setq pie (plus 1 1 1 1)). In any of these cases, as soon as
you type your carriage return, pie's value will become 4, and so it
will remain forevermore - or at least until you do another setq
operation on the atom pie.
Lisp would not be crisp if the only values atoms could have were
numbers. Fortunately, however, an atom's value can be set to any kind
of Lisp object - any atom or list whatsoever. For instance, we might
want to make the value of the atom pi be a list such as
(a b c) or
(plus 2 2) instead of the number 4. To do the latter, we again
use the setq operation. To illustrate, here follows a brief
conversation with the genie:
> (setq pie (plus 2 2)) 4 > (setq pi '(plus 2 2)) (plus 2 2)
Notice the vast difference between the values assigned to the atoms
pie and pi as a result of these two wishes asked of the Lisp genie,
which differ merely in the presence or absence of a small but critical
quote mark in front of the inner list
(plus 2 2). In the first wish,
containing no quote mark, that inner
(plus 2 2) must be evaluated.
This returns 4, which is assigned to the variable pie as its new
value. On the other hand, in the second wish, since the quote mark is
there, the list
(plus 2 2) is never executed as a command, but is
treated merely as an inert lump of Lispstuff, much like meat on a
butcher's shelf. It is ever so close to being "alive", yet it is dead.
So the value of pi in this second case is the list
(plus 2 2), a
fragment of Lisp code. The following interchange with the genie
confirms the values of these atoms.
> pie 4 > pi (plus 2 2) > (eval pi) 4 >
What is this last step? I wanted to show how you can ask the genie to evaluate the value of an expression, rather than simply printing the value of that expression. Ordinarily, the genie automatically performs just one level of evaluation, but by writing eval, you can get a second stage of evaluation carried out. (And of course, by using eval over and over again, you can carry this as far as you like.) This feature often proves invaluable, but it is a little too advanced to discuss further at this stage.
Every list but nil has at least one element. This first element is
called the list's Car. Thus the car of
(eval pi) is the atom eval. The
cars of the lists
(plus 2 2),
(setq x 17),
(eval pi), and
(car pi) are
all names of operations, or, as they are more commonly called in Lisp,
functions. The car of a list need not be the name of a function; it
need not even be an atom. For instance,
((1)(2 2) (3 3 3)) is a
perfectly fine list. Its car is the list
(1), whose car in turn is not
a function name but merely a numeral.
If you were to remove a list's car, what would remain? A shorter list.
This is called the list's cdr, a word that sounds about halfway
between "kidder" and "could'er". (The words "car" and "cdr" are quaint
relics from the first implementation of Lisp on the IBM 704. The
letters in "car" stand for "Contents of the Address part of Register"
and those in "cdr" for "Contents of the Decrement part of Register
referring to specific hardware features of that machine, now long
since irrelevant.) The cdr of
(a b c d) is the list
(b c d), whose cdr
(c d), whose cdr is
(d), whose cdr is nil. And nil has no cdr, just
as it has no car. Attempting to take the car or cdr of nil causes (or
should cause) the Lisp genie to cough out an error message, just as
attempting to divide by zero should evoke an error message.
Here is a little table showing the car and cdr of a few lists, just to make sure the notions are unambiguous.
list car cdr ((a) b (c)) (a) (b (c)) (plus 2 2) plus (2 2) ((car x) (car y)) (car x) ((car y)) (nil nil nil nil) nil (nil nil nil) (nil) nil nil nil **ERROR** **ERROR**
Just as car and cdr are called functions, so the things that they
operate on are called their arguments. Thus in the command
(plus pie 2),
plus is the function name, and the arguments are the atoms pie and
2. In evaluating this command (and most commands), the genie figures
out the values of the arguments, and then applies the function to
those values. Thus, since the value of the atom pie is 4, and the
value of the atom 2 is 2, the genie returns the atom 6.
Suppose you have a list and you'd like to see a list just like it,
only one element longer. For instance, suppose the value of the atom x
(cake cookie) and you'd like to create a new list called y just
like x, except with an extra atom-say pie - at the front. You can then
use the function called cons (short for "construct"), whose effect is
to make a new list out of an old list and a suggested car. Here's a
transcript of such a process:
>(setq x '(cake cookie)) (cake cookie) >(setq y (cons 'pie x)) (pie cake cookie) > x (cake cookie)
Two things are worth noticing here. I asked for the value of x to be printed out after the cons operation, so you could see that x itself was not changed by the cons. The cons operation created a new list and made that list be the value of y, but left x entirely alone. The other noteworthy fact is that I used that quote mark again, in front of the atom pie. What if I had not used it? Here's what would have happened.
> (setq z (cons pie x)) (4 cake cookie)
Remember, after all, that the atom pie still has the value 4, and whenever the genie sees an unquoted atom inside a wish, it will always use the value belonging to that atom, rather than the atom's name. (Always? Well, almost always. I'll explain in a moment. In the meantime, look for an exception - you've already encountered it.)
Now here are a few exercises - some a bit tricky - for you. Watch out
for the quote marks! Oh, one last thing: I use the function reverse,
which produces a list just like its argument, only with its elements
in reverse order. For instance, the genie, upon being told
(reverse '((a b) (c d e))) will write
((c d e) (a b)). The genie's lines in
this dialogue are given afterward.
> (setq w (cons pie '(cdr z))) > (setq v (cons 'pie (cdr z))) > (setq u (reverse v)) > (cdr (cdr u)) > (car (cdr u)) > (cons (car (cdr u)) u) > u > (reverse '(cons (car u) (reverse (cdr u)))) > (reverse (cons (car u) (reverse (cdr u)))) > u > (cons 'cookie (cons 'cake (cons 'pie nil)))
Answers (as printed by the genie):
(4 cdr z) (pie cake cookie) (cookie cake pie) (pie) cake (cake cookie cake pie) (cookie cake pie) ((reverse (cdr u)) (car u) cons) (cake pie cookie) (cookie cake pie) (cookie cake pie)
The last example, featuring repeated use of cons, is often called, in Lisp slang "consing up a list". You start with nil, and then do repeated cons operations. It is analogous to building a positive integer by starting at zero and then performing the successor operation over and over again. However, whereas at any stage in the latter process there is a unique way of performing the successor operation, given any list there are infinitely many different items you can cons onto it, thus giving rise to a vast branching tree of lists instead of the unbranching number line. It is on account of this image of a tree growing out of the ground of nil and containing all possible lists that I earlier likened nil to "the earth in which all structures are rooted".
As I mentioned a moment ago, the genie doesn't always replace
(unquoted) atoms by their values. There are cases where a function
treats its arguments, though unquoted, as if quoted. Did you go back
and find such a case? It's easy. The answer is the function setq. In
particular, in a setq command, the first atom is taken straight-not
evaluated. As a matter of fact, the q in setq stands for "quote",
meaning that the first argument is treated as if quoted. Things can
get quite tricky when you learn about set, a function similar to setq
except that it does evaluate its first argument. Thus, if the value of
the atom x is the atom k, then saying
(set x 7) will not do anything
to x-its value will remain the atom k-but the value of the atom k will
now become 7. So watch closely:
> (setq a 'b) > (setq b 'c) > (setq c 'a) > (set a c) > (set c b)
Now tell me: What are the values of the atoms a, b, and c? Here comes the answer, so don't peek. They are, respectively: a, a, and a. This may seem a bit confusing. You may be reassured to know that in Lisp, set is not very commonly used, and such confusions do not arise that often.
Psychologically, one of the great powers of programming is the ability to define new compound operations in terms of old ones, and to do this over and over again, thus building up a vast repertoire of ever more complex operations. It is quite reminiscent of evolution, in which ever more complex molecules evolve out of less complex ones, in an ever-upward spiral of complexity and creativity. It is also quite reminiscent of the industrial revolution, in which people used very simple early machines to help them build more complex machines, then used those in turn to build even more complex machines, and so on, once again in an ever-upward spiral of complexity and creativity. At each stage, whether in evolution or revolution, the products get more flexible and more intricate, more "intelligent" and yet more vulnerable to delicate "bugs" or breakdowns.
Likewise with programming in Lisp, only here the "molecules" or
"machines" are now Lisp functions defined in terms of previously known
Lisp functions. Suppose, for instance, that you wish to have a
function that will always return the last element of a list, just as
car always returns the first element of a list. Lisp does not come
equipped with such a function, but you can easily create one. Do you
see how? To get the last element of a list called lyst, you simply do
a reverse on lyst and then take the car of that:
(car (reverse lyst)).
To dub this operation with the name rac (car backwards), we use the
def function, as follows:
> (def rac (lambda (lyst) (car (reverse lyst))))
Using def this way creates a function definition. In it, the word
lambda followed by (lyst) indicates that the function we are defining
has only one parameter, or dummy variable, to be called lyst. (It
could have been called anything; I just happen to like the atom lyst.)
In general, the list of parameters (dummy variables) must immediately
follow the word lambda. After this "def wish" has been carried out,
the rac function is as well understood by the genie as is car. Thus
(rac '(your brains)) will yield the atom
brains. And we can use rac
itself in definitions of yet further functions. The whole thing
snowballs rather miraculously, and you can quickly become overwhelmed
by the power you wield.
Here is a simple example. Suppose you have a situation where you know you are going to run into many big long lists and you know it will often be useful to form, for each such long list, a short list that contains just its car and rac. We can define a one-parameter function to do this for you:
> (def readers-digest-condensed-version (lambda (biglonglist) (cons (car biglonglist) (cons (rac biglonglist) nil))))
Thus if we apply our new function readers-digest-condensed-version to
the entire text of James Joyce's Finnegans Wake (treating it as a big
long list of words), we will obtain the shorter list
Unfortunately, reapplying the condensation operator to this new list
will not simplify it any further.
It would be nice as well as useful if we could create an inverse
operation to readers-digest-condensed-version called rejoyce that,
given any two words, would create a novel beginning and ending with
them, respectively - and such that James Joyce would have written it
(had he thought of it). Thus execution of the Lisp statement
(rejoyce 'Stately 'Yes) would result in the Lisp genie generating from
scratch the entire novel Ulysses. Writing this function is left as an
exercise for the reader. To test your program, see what it does with
(rejoyce 'karma 'dharma).
One goal that has seemed to some people to be both desirable and feasible using Lisp and related programming languages is (1) to make every single statement return a value and (2) to have it be through this returned value and only through it that the statement has any effect. The idea of (1) is that values are handed "upward" from the innermost function calls to the outermost ones, until the full statement's value is returned to you. The idea of (2) is that during all these calls, no atom has its value changed at all (unless the atom is a dummy variable). In all dialects of Lisp known to me, (1) is true, but (2) is not necessarily true.
Thus if x is bound to
(a b c d e) and you say
(car (cdr (reverse x))),
the first thing that happens is that
(reverse x) is calculated; then
this value is handed "up" to the cdr function, which calculates the
cdr of that list; finally, this shorter list is handed to the car
function, which extracts one element-namely the atom d-and returns it.
In the meantime, the atom x has suffered no damage; it is still bound
(a b c d e).
It might seem that an expression such as
(reverse x) would change the
value of x by reversing it, just as carrying out the oral command
"Turn your sweater inside out" will affect the sweater. But actually,
carrying out the wish
(reverse x) no more changes the value of x than
carrying out the wish
(plus 2 2) changes the value of 2. Instead,
(reverse x) causes a new (unnamed) list to come into being,
just like x, only reversed. And that list is the value of the
statement; it is what the statement returns. The value of x itself,
however, is untouched. Similarly, evaluating
(cons 5 pi) will not
change the list named pi in the slightest; it merely returns a new
list with 5 as its car and whatever pi's value is as its cdr.
Such behavior is to be contrasted with that of functions that leave "side effects" in their wake. Such side effects are usually in the form of changed variable bindings, although there are other possibilities, such as causing input or output to take place. A typical "harmful" command is a setq, and proponents of the "applicative" school of programming - the school that says you should never make any side effects whatsoever - are profoundly disturbed by the mere mention of setq. For them, all results must come about purely by the way that functions compute their values and hand them to other functions.
The only bindings that the advocates of the applicative style approve
of are transitory "lambda bindings" - those that arise when a function
is applied to its arguments. Whenever any function is called, that
function's dummy variables temporarily assume "lambda bindings". These
bindings are just like those caused by a setq, except that they are
fleeting. That is, the moment the function is finished computing, they
go away - vanishing without a trace. For example, during the
(rac '(a b c)), the lambda binding of the dummy
variable lyst is the list
(a b c); but as soon as the answer c is
passed along to the function or person that requested the rac, the
value of the atom lyst used in getting that answer is totally
forgotten. The Lisp interpreter will tell you that lyst is an "unbound
atom" if you ask for its value. Applicative programmers much prefer
lambda bindings to ordinary setq bindings.
I personally am not a fanatic about avoiding setq's and other functions that cause side effects. Though I find the applicative style to be jus-telegant, I find it impractical when it comes to the construction of large AI-style programs. Therefore I shall not advocate the applicative style here, though I shall adhere to it when possible. Strictly speaking, in applicative programming, you cannot even define new functions, since a def statement causes a permanent change to take place in the genie's memory - namely, the permanent storage in memory of the function definition. So the ideal applicative approach would have functions, like variable bindings, being created only temporarily, and their definitions would be discarded the moment after they had been used. But this is extreme "applicativism".
For your edification, here are a few more simple function definitions.
> (def rdc (lambda (lyst) (reverse (cdr (reverse lyst))))) > (def snoc (lambda (x lyst) (reverse (cons x (reverse lyst))))) > (def twice (lambda (n) (plus n n)))
The functions rdc and snoc are analogous to cdr and cons, only
backwards. Thus, the rdc of
(a b c d e) is
(a b c d), and if you type
(snoc 5 '(1 2 3 4)), you will get
(1 2 3 4 5) as your answer.
All, of this is mildly interesting so far, but if you want to see the genie do anything truly surprising, you have to allow it to make some decisions based on things that happen along the way. These are sometimes called "conditional wishes". A typical example would be the following:
> (cond ((eq x 1) 'land) ((eq x 2) 'sea))
The value returned by this statement will be the atom land if x has value 1, and the atom sea if x has value 2. Otherwise, the value returned will be nil (i.e., if x is 5). The atom eq (pronounced "eek") is the name of a common Lisp function that returns the atom t (standing for "true") if its two arguments have the same value, and nil (for "no" or "false") if they do not.
A cond statement is a list whose car is the function name cond, followed by any number of cond clauses, each of which is a two-element list. The first element of each clause is called its condition, the second element its result. The clauses' conditions are checked out by the Lisp genie one by one, in order; as soon as it finds a clause whose condition is "true" (meaning that the condition returns anything other than nil!), it begins calculating that clause's result, whose value gets returned as the value of the whole cond statement. None of the further clauses is even so much as glanced at! This may sound more complex than it ought to. The real idea is no more complex than saying that it looks for the first condition that is satisfied, then it returns the corresponding result.
Often one wants to have a catch-all clause at the end whose condition is sure to be satisfied, so that, if all other conditions fail, at least this one will be true and the accompanying result, rather than nil, will be returned. It is easy as pie to make a condition whose value is non-nil; just choose it to be t for instance, as in the following:
(cond ((eq x 1) 'land) ((eq x 2) 'sea) (t 'air))
Depending on what the value of x is, we will get either land, sea, or air as the value of this cond, but we'll never get nil. Now here are a few sample cond statements for you to play genie to:
> (cond ((eq (oval pi) pie) (oval (snot pie pi))) (t (eval (snoc (rac pi) pi)))) > (cond ((eq 2 2) 2) ((eq 3 3) 3)) > (cond (nil 'no-no-no) ((eq '(car nil) '(cdr nil)) 'hmmm) (t 'yes-yes-yes))
The answers are: 8, 2, and yes-yes-yes. Did you notice that
(cdr nil) were quoted?
I shall close this portion of the column by displaying a patterned family of function definitions, so obvious in their pattern that you would think that the Lisp genie would just sort of "get the hang of it" after seeing the first few... Unfortunately, though, Lisp genies are frustratingly dense (or at least they play at being dense), and they will not jump to any conclusion unless it has been completely spelled out. Look first at the family:
> (def square (lambda (k) (times k k))) > (def cube (lambda (k) (times k (square k)))) > (def 4th-power (lambda (k) (times k (cube k)))) > (def 5th-power (lambda (k) (times k (4th-power k)))) > (def 6th-power (lambda (k) (times k (5th-power k)))) > . > . > . > .
My question for you is this: Can you invent a definition for a two
parameter function that subsumes all of these in one fell swoop? More
concretely, the question is: How would one go about defining a
two-parameter function called power such that, for instance,
(power 9 3)
yields 729 on being evaluated, and
(power 7 4) yields 2,401 ? I
have supplied you, in this column, with all the necessary tools to do
this, provided you exercise some ingenuity.
I thought I would end this column with a newsbreak about a freshly discovered beast - the homely Glazunkian porpuquine, so called because it is found only on the island of Glazunkia (claimed by Upper Bitbo, though it is just off the coast of Burronymede). And what is a porpuquine, you ask? Why, it's a strange breed of porcupine, whose quills - of which, for some reason, there are always exactly nine (in Outer Glazunkia) or seven (in Inner Glazunkia) - are smaller porpuquines. Oho! This would certainly seem to be an infinite regress! But no. It's just that I forgot to mention that there is a smallest size of porpuquine: the zero-inch type, which, amazingly enough, is totally bald of quills. So, quite luckily (or perhaps unluckily, depending on your point of view), that puts a stop to the threatened infinite regress. This remarkable beast is shown in a rare photograph in Figure 17-1.
Students of zoology might be interested to learn that the quills on 5-inch porpuquines are always 4-inch porpuquines, and so on down the line. And students of anthropology might be equally intrigued to know that the residents of Glazunkia (both Outer and Inner) utilize the nose (yes, the nose) of the zero-inch porpuquine as a unit of barter - an odd thing to our minds; but then, who are you and I to question the ancient wisdom of the Outer and Inner Glazunkians? Thus, since a largish porpuquine - say a 3-incher or 4-incher - contains many, many such tiny noses, it is a most valuable commodity. The value of a porpuquine is sometimes referred to as its "buying power", or just "power" for short. For instance, a 2-incher found in Inner Glazunkia is almost twice as powerful as a 2-incher found in Outer Glazunkia. Or did I get it backward? It's rather confusing!
Anyway, why am I telling you all this? Oh, I just thought you'd like to hear about it. Besides, who knows? You just might wind up visiting Glazunkia (Inner or Outer) one of these fine days. And then all of this could come in mighty handy.