Computer Science Logo Style volume 2: Advanced Techniques 2/e Copyright (C) 1997 MIT

Iteration, Control Structures, Extensibility

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Brian Harvey
University of California, Berkeley

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In this chapter we're taking a tour "behind the scenes" of Berkeley Logo. Many of the built-in Logo procedures that we've been using all along are not, strictly speaking, primitive; they're written in Logo itself. When you invoke a procedure, if the Logo interpreter does not already know a procedure by that name, it automatically looks in a library of predefined procedures. For example, in Chapter 6 I used an operation called gensym that outputs a new, unique word each time it's invoked. If you start up a fresh copy of Logo you can try these experiments:

? po "gensym
I don't know how  to gensym
? show gensym
? po "gensym
to gensym
if not namep "gensym.number [make "gensym.number 0]
make "gensym.number :gensym.number + 1
output word "g :gensym.number

The first interaction shows that gensym is not really a Logo primitive; the error message indicates that there is no such procedure. Then I invoked gensym, which made Berkeley Logo read its definition automatically from the library. Finally, once Logo has read the definition, I can print it out.

In particular, most of the tools we've used to carry out a computation repeatedly are not true Logo primitives: for for numeric iteration, foreach for list iteration, and while for predicate-based iteration are all library procedures. (The word iteration just means "repetition.") The only iteration mechanisms that are truly primitive in Logo are repeat and recursion.

Computers are good at doing things over and over again. They don't get bored or tired. That's why, in the real world, people use computers for things like sending out payroll checks and telephone bills. The first Logo instruction I showed you, in the first volume, was

repeat 50 [setcursor list random 75 random 20 type "Hi]

When you were first introduced to turtle graphics, you probably used an instruction like

repeat 360 [forward 1 right 1]

to draw a circle.

Recursion as Iteration

The trouble with repeat is that it always does exactly the same thing repeatedly. In a real application, like those payroll checks, you want the computer to do almost the same thing each time but with a different person's name on each check. The usual way to program an almost-repeat in Logo is to use a recursive procedure, like this:

to polyspi :side :angle :number
if :number=0 [stop]
forward :side
right :angle
polyspi :side+1 :angle :number-1

This is a well-known procedure to draw a spiral. What makes it different from

repeat :number [forward :side right :angle]

is that the first input in the recursive invocation is :side+1 instead of just :side. We've used a similar technique for almost-repetition in procedures like this one:

to multiply :letters :number
if equalp :number 0 [stop]
print :letters
multiply (word :letters first :letters) :number-1

Since recursion can express any repetitive computation, why bother inventing other iteration tools? The answer is that they can make programs easier to read. Recursion is such a versatile mechanism that the intention of any particular use of recursion may be hard to see. Which of the following is easier to read?

to fivesay.with.repeat :text
repeat 5 [print :text]


to fivesay.with.recursion :text
fivesay1 5 :text

to fivesay1 :times :text
if :times=0 [stop]
print :text
fivesay1 :times-1 :text

The version using repeat makes it obvious at a glance what the program wants to do; the version using recursion takes some thought. It can be useful to invent mechanisms that are intermediate in flexibility between repeat and recursion.

As a simple example, suppose that Logo did not include repeat as a primitive command. Here's how we could implement it using recursion:

to rep :count :instr
if :count=0 [stop]
run :instr
rep :count-1 :instr

(I've used the name rep instead of repeat to avoid conflict with the primitive version.) The use of run to carry out the given instructions is at the core of the techniques we'll use throughout this chapter.

Numeric Iteration

Polyspi is an example of an iteration in which the value of a numeric variable changes in a uniform way. The instruction

polyspi 50 60 4

is equivalent to the series of instructions

forward 50 right 60
forward 51 right 60
forward 52 right 60
forward 53 right 60

As you know, we can represent the same instructions this way:

for [side 50 53] [forward :side right 60]

The for command takes two inputs, very much like repeat. The second input, like repeat's second input, is a list of Logo instructions. The first input to for is different, though. It is a list whose first member is the name of a variable; the second member of the list must be a number (or a Logo expression whose value is a number), which will be the initial value of that variable; and the third member must be another number (or numeric expression), which will be the final value of the variable. In the example above, the variable name is side, the initial value is 50, and the final value is 53. If there is a fourth member in the list, it's the amount to add to the named variable on each iteration; if there is no fourth member, as in the example above, then the step amount is either 1 or -1, depending on whether the final value is greater than or less than the initial value.

As an example in which expressions are used instead of constant numeric values, here's the polyspi procedure using for:

to polyspi :start :angle :number
for [side :start [:start+:number-1]] [forward :side right :angle]

Most of the work in writing for is in evaluating the expressions that make up the first input list. Here is the program:

to for :values :instr
localmake "var first :values
local :var
localmake "initial run first butfirst :values
localmake "final run item 3 :values
localmake "step forstep
localmake "tester ~
          ifelse :step < 0 [[:value < :final]] [[:value > :final]]
forloop :initial

to forstep
if (count :values)=4 [output run last :values]
if :initial > :final [output -1]
output 1

to forloop :value
make :var :value
if run :tester [stop]
run :instr
forloop :value+:step

One slightly tricky part of this program is the instruction

local :var

near the beginning of for. The effect of this instruction is to make whatever variable is named by the first member of the first input local to for. As it turns out, this variable isn't given a value in for itself but only in its subprocedure forloop. (A loop, by the way, is a part of a program that is invoked repeatedly.) But I'm thinking of these three procedures as a unit and making the variable local to that whole unit. The virtue of this local instruction is that a program that uses for can invent variable names for for freely, without having to declare them local and without cluttering up the workspace with global variables. Also, it means that a procedure can invoke another procedure in the instruction list of a for without worrying about whether that procedure uses for itself. Here's the case I'm thinking of:

to a
for [i 1 5] [b]

to b
for [i 1 3] [print "foo]

Invoking A should print the word foo fifteen times: three times for each of the five invocations of B. If for didn't make I a local variable, the invocation of for within B would mess up the value of I in the outer for invoked by A. Got that?

Notice that the make instruction in forloop has :var as its first input, not "var. This instruction does not assign a new value to the variable var! Instead, it assigns a new value to the variable whose name is the value of var.

The version of for actually used in the Berkeley Logo library is a little more complicated than this one. The one shown here works fine as long as the instruction list input doesn't include stop or output, but it won't work for an example like the following. To check whether or not a number is prime, we must see if it is divisible by anything greater than 1 and smaller than the number itself:

to primep :num
for [trial 2 [:num-1]] [if divisiblep :num :trial [output "false]]
output "true

to divisiblep :big :small
output equalp remainder :big :small 0

This example will work in the Berkeley Logo for, but not in the version I've written in this chapter. The trouble is that the instruction

run :instr

in forloop will make forloop output false if a divisor is found, whereas we really want primep to output false! We'll see in Chapter 12 how to solve this problem.

Logo: an Extensible Language

There are two ways to look at a program like for. You can take it apart, as I've been doing in these last few paragraphs, to see how it works inside. Or you can just think of it as an extension to Logo, an iteration command that you can use as you'd use repeat, without thinking about how it works. I think both of these perspectives will be valuable to you. As a programming project, for demonstrates some rather advanced Logo techniques. But you don't have to think about those techniques each time you use for. Instead you can think of it as a primitive, as we've been doing prior to this chapter.

The fact that you can extend Logo's vocabulary this way, adding a new way to control iteration that looks just like the primitive repeat, is an important way in which Logo is more powerful than toy programming languages like C++ or Pascal. C++ has several iteration commands built in, including one like for, but if you think of a new one, there's no way you can add it to the language. In Logo this kind of language extension is easy. For example, here is a preview of a programming project I'm going to develop later in this chapter. Suppose you're playing with spirals, and you'd like to see what happens if you change the line length and the turning angle at the same time. That is, you'd like to be able to say

multifor [[size 50 100 5] [angle 50 100 10]] [forward :size right :angle]

and have that be equivalent to the series of instructions

forward 50 right 50
forward 55 right 60
forward 60 right 70
forward 65 right 80
forward 70 right 90
forward 75 right 100

Multifor should step each of its variables each time around, stopping whenever any of them hits the final value. This tool strikes me as too specialized and complicated to provide in the Logo library, but it seems very appropriate for certain kinds of project. It's nice to have a language in which I can write it if I need it.

No Perfect Control Structures

Among enthusiasts of the Fortran family of programming languages (that is, all the languages in which you have to say ahead of time whether or not the value of some numeric variable will be an exact integer), there are fierce debates about the "best" control structure. (A control structure is a way of grouping instructions together, just as a data structure is a way of grouping data together. A list is a data structure. A procedure is a control structure. Things like if, repeat, and for are special control structures that group instructions in particular ways, so that a group of instructions can be evaluated conditionally or repeatedly.)

For example, all of the Fortran-derived languages have a control structure for numeric iteration, like my for procedure. But they differ in details. In some languages the iteration variable must be stepped by 1. In others the step value can be either 1 or -1. Still others allow any step value, as for does. Each of these choices has its defenders as the "best."

Sometimes the arguments are even sillier. When Fortran was first invented, its designers failed to make explicit what should happen if the initial value of an iteration variable is greater than the final value. That is, they left open the interpretation of a Fortran do statement (that's what its numeric iteration structure is called) equivalent to this for instruction:

for [var 10 5 1] [print :var]

In this instruction I've specified a positive step (the only kind allowed in the Fortran do statement), but the initial value is greater than the final value. (What will for do in this situation?) Well, the first Fortran compiler, the program that translates a Fortran program into the "native" language of a particular computer, implemented do so that the statements controlled by the do were carried out once before the computer noticed that the variable's value was already too large. Years later a bunch of computer scientists decided that that behavior is "wrong"; if the initial value is greater than the final value, the statements shouldn't be carried out at all. This proposal for a "zero trip do loop" was fiercely resisted by old-timers who had by then written hundreds of programs that relied on the original behavior of do. Dozens of journal articles and letters to the editor carried on the battle.

The real moral of this story is that there is no right answer. The right control structure for you to use is the one that best solves your immediate problem. But only an extensible language like Logo allows you the luxury of accepting this moral. The Fortran people had to fight out their battle because they're stuck with whatever the standardization committee decides.

In the remainder of this chapter I'll present various kinds of control structures, each reflecting a different way of looking at the general idea of iteration.

Iteration Over a List

Numeric iteration is useful if the problem you want to solve is about numbers, as in the primep example, or if some arbitrary number is part of the rules of a game, like the seven stacks of cards in solitaire. But in most Logo projects, it's more common to want to carry out a computation for each member of a list, and for that purpose we have the foreach control structure:

? foreach [chocolate [rum raisin] pumpkin] [print sentence [I like] ?]
I like chocolate
I like rum raisin
I like pumpkin

In comparing foreach with for, one thing you might notice is the use of the question mark to represent the varying datum in foreach, while for requires a user-specified variable name for that purpose. There's no vital reason why I used these different mechanisms. In fact, we can easily implement a version of foreach that takes a variable name as an additional input. Its structure will then look similar to that of for:

to named.foreach :var :data :instr
local :var
if emptyp :data [stop]
make :var first :data
run :instr
named.foreach :var (butfirst :data) :instr

? named.foreach "flavor [lychee [root beer swirl]] ~
                        [print sentence [I like] :flavor]
I like lychee
I like root beer swirl

Just as in the implementation of for, there is a recursive invocation for each member of the data input. We assign that member as the value of the variable named in the first input, and then we run the instructions in the third input.

In order to implement the version of foreach that uses question marks instead of named variables, we need a more advanced version of run that says "run these instructions, but using this value wherever you see a question mark." Berkeley Logo has this capability as a primitive procedure called apply. It takes two inputs, a template (an instruction list with question marks) and a list of values. The reason that the second input is a list of values, rather than a single value, is that apply can handle templates with more than one slot for values.

? apply [print ?+3] [5]
? apply [print word first ?1 first ?2] [Peter Dickinson]

It's possible to write apply in terms of run, and I'll do that shortly. But first, let's just take advantage of Berkeley Logo's built-in apply to write a simple version of foreach:

to foreach :list :template
if emptyp :list [stop]
apply :template (list first :list)
foreach (butfirst :list) :template

Apply, like run, can be either a command or an operation depending on whether its template contains complete Logo instructions or a Logo expression. In this case, we are using apply as a command.

The version of foreach in the Berkeley Logo library can take more than one data input along with a multi-input template, like this:

? (foreach [John Paul George Ringo] [rhythm bass lead drums]
           [print (sentence ?1 "played ?2)]
John played rhythm
Paul played bass
George played lead
Ringo played drums

We can implement this feature, using a special notation in the title line of foreach to notify Logo that it accepts a variable number of inputs:

to foreach [:inputs] 2
foreach.loop (butlast :inputs) (last :inputs)

to foreach.loop :lists :template
if emptyp first :lists [stop]
apply :template firsts :lists
foreach.loop (butfirsts :lists) :template

First look at the title line of foreach. It tells Logo that the word inputs is a formal parameter--the name of an input. Because :inputs is inside square brackets, however, it represents not just one input, but any number of inputs in the invocation of foreach. The values of all those inputs are collected as a list, and that list is the value of inputs. Here's a trivial example:

to demo [:stuff]
print sentence [The first input is] first :stuff
print sentence [The others are] butfirst :stuff

? (demo "alpha "beta "gamma)
The first input is alpha
The others are beta gamma

As you know, the Logo procedures that accept a variable number of inputs have a default number that they accept without using parentheses; if you want to use more or fewer than that number, you must enclose the procedure name and its inputs in parentheses, as I've done here with the demo procedure. Most Logo primitives that accept a variable number of inputs have two inputs as their default number (for example, sentence, sum, word) but there are exceptions, such as local, which takes one input if parentheses are not used. When you write your own procedure with a single input name in brackets, its default number of inputs is zero unless you specify another number. Demo, for example, has zero as its default number. If you look again at the title line of foreach, you'll see that it ends with the number 2; that tells Logo that foreach expects two inputs by default.

Foreach uses all but its last input as data lists; the last input is the template to be applied to the members of the data lists. That's why foreach invokes foreach.loop as it does, separating the two kinds of inputs into two variables.

Be careful when reading the definition of foreach.loop; it invokes procedures named firsts and butfirsts. These are not the same as first and butfirst! Each of them takes a list of lists as its input, and outputs a list containing the first members of each sublist, or all but the first members, respectively:

? show firsts [[a b c] [1 2 3] [y w d]]
[a 1 y]
? show butfirsts [[a b c] [1 2 3] [y w d]]
[[b c] [2 3] [w d]]

It would be easy to write firsts and butfirsts in Logo:

to firsts :list.of.lists
output map "first :list.of.lists

to butfirsts :list.of.lists
output map "butfirst :list.of.lists

but in fact Berkeley Logo provides these operations as primitives, because implementing them as primitives makes the iteration tools such as foreach and map (which, as we'll see, also uses them) much faster.

Except for the use of firsts and butfirsts to handle the multiple data inputs, the structure of foreach.loop is exactly like that of the previous version of foreach that only accepts one data list.

Like for, the version of foreach presented here can't handle instruction lists that include stop or output correctly.

Implementing Apply

Berkeley Logo includes apply as a primitive, for efficiency, but we could implement it in Logo if necessary. In this section, so as not to conflict with the primitive version, I'll use the name app for my non-primitive version of apply, and I'll use the percent sign (%) as the placeholder in templates instead of question mark.

Here is a simple version of app that allows only one input to the template:

to app :template :input.value
run :template

to %
output :input.value

This is so simple that it probably seems like magic. App seems to do nothing but run its template as though it were an ordinary instruction list. The trick is that a template is an instruction list. The only unusual thing about a template is that it includes special symbols (? in the real apply, % in app) that represent the given value. We see now that those special symbols are really just ordinary names of procedures. The question mark (?) procedure is a Berkeley Logo primitive; I've defined the analogous % procedure here for use by app.

The % procedure outputs the value of a variable, input.value, that is local to app. If you invoke % in some context other than an app template, you'll get an error message because that variable won't exist. Logo's dynamic scope makes it possible for % to use app's variable.

The real apply accepts a procedure name as argument instead of a template:

? show apply "first [Logo]

We can extend app to accept named procedures, but the definition is somewhat messier:

to app :input.value
ifelse wordp ~
       [run list "%] ~

If the first input is a word, we construct a template by combining that procedure name with a percent sign for its input. However, in the rest of this section I'll simplify the discussion by assuming that app accepts only templates, not procedure names.

So far, app takes only one value as input; the real apply takes a list of values. I'll extend app to match:

to app :template :input.values
run :template

to % [:index 1]
output item :index :input.values

No change is needed to app, but % has been changed to use another new notation in its title line. Index is the name of an optional input. Although this notation also uses square brackets, it's different from the notation used in foreach because the brackets include a default value as well as the name for the input. This version of % accepts either no inputs or one input. If % is invoked with one input, then the value of that input will be associated with the name index, just as for ordinary inputs. If % is invoked with no inputs, then index will be given the value 1 (its default value).

? app [print word first (% 1) first (% 2)] [Paul Goodman]

A percent sign with a number as input selects an input value by its position within the list of values. A percent sign by itself is equivalent to (% 1).

The notation (% 1) isn't as elegant as the ?1 used in the real apply. You can solve that problem by defining several extra procedures:

to %1                to %2                to %3
output (% 1)         output (% 2)         output (% 3)
end                  end                  end

Berkeley Logo recognizes the notation ?2 and automatically translates it to (? 2), as you can see by this experiment:

? show runparse [print word first ?1 first ?2]
[print word first ( ? 1 ) first ( ? 2 )]

(The primitive operation runparse takes a list as input and outputs the list as it would be modified by Logo when it is about to be run. That's a handwavy description, but the internal workings of the Logo interpreter are too arcane to explore here.)

Unlike the primitive apply, this version of app works only as a command, not as an operation. It's easy to write a separate version for use as an operation:

to app.oper :template :input.values
output run :template

It's not so easy in non-Berkeley versions of Logo to write a single procedure that can serve both as a command and as an operation. Here's one solution that works in versions with catch:

to app :template :input.values
catch "error [output run :template]
ignore error

This isn't an ideal solution, though, because it doesn't report errors other than "run didn't output to output." It could be improved by testing the error message more carefully instead of just ignoring it.

Berkeley Logo includes a mechanism that solves the problem more directly, but it's not very pretty:

to app :template :input.values
.maybeoutput run :template

The primitive command .maybeoutput is followed by a Logo expression that may or may not produce a value. If so, that value is output, just as it would be by the ordinary output command; the difference is that it's not considered an error if no value is produced.

From now on I'll use the primitive apply. I showed you app for two reasons. First, I think you'll understand apply better by seeing how it can be implemented. Second, this implementation may be useful if you ever work in a non-Berkeley Logo.


So far the iteration tools we've created apply only to commands. As you know, we also have the operation map, which is similar to foreach except that its template is an expression (producing a value) rather than an instruction, and it accumulates the values produced for each member of the input.

? show map [?*?] [1 2 3 4]
[1 4 9 16]
? show map [first ?] [every good boy does fine]
[e g b d f]

When implementing an iteration tool, one way to figure out how to write the program is to start with a specific example and generalize it. For example, here's how I'd write the example about squaring the numbers in a list without using map:

to squares :numbers
if emptyp :numbers [output []]
output fput ((first :numbers) * (first :numbers)) ~
            (squares butfirst :numbers)

Map is very similar, except that it applies a template to each datum instead of squaring it:

to map :template :values
if emptyp :values [output []]
output fput (apply :template (list first :values)) ~
            (map :template butfirst :values)

You may be wondering why I used fput rather than sentence in these procedures. Either would be just as good in the example about squares of numbers, because each datum is a single word (a number) and each result value is also a single word. But it's important to use fput in an example such as this one:

to swap :pair
output list last :pair first :pair

? show map [swap ?] [[Sherlock Holmes] [James Pibble] [Nero Wolfe]]
[[Holmes Sherlock] [Pibble James] [Wolfe Nero]]

? show [swap ?] [[Sherlock Holmes] [James Pibble] [Nero Wolfe]]
[Holmes Sherlock Pibble James Wolfe Nero]

Berkeley Logo does provide an operation in which sentence is used as the combiner; sometimes that's what you want, but not, as you can see, in this example. (A third possibility that might occur to you is to use list as the combiner, but that never turns out to be the right thing; try writing a map.list and see what results it gives!)

As in the case of foreach, the program gets a little more complicated when we extend it to handle multiple data inputs. Another complication that wasn't relevant to foreach is that when we use a word, rather than a list, as the data input to map, we must use word as the combiner instead of fput. Here's the complete version:

to map :map.template [:template.lists] 2
op map1 :template.lists 1

to map1 :template.lists :template.number
if emptyp first :template.lists [output first :template.lists]
output combine (apply :map.template firsts :template.lists) ~
               (map1 bfs :template.lists :template.number+1)

to combine :this :those
if wordp :those [output word :this :those]
output fput :this :those

This is the actual program in the Berkeley Logo library. One feature I haven't discussed until now is the variable template.number used as an input to map1. Its purpose is to allow the use of the number sign character # in a template to represent the position of each datum within its list:

? show map [list ? #] [a b c]
[[a 1] [b 2] [c 3]]

The implementation is similar to that of ? in templates:

to #
output :template.number

It's also worth noting the base case in map1. When the data input is empty, we must output either the empty word or the empty list, and the easiest way to choose correctly is to return the empty input itself.

Mapping as a Metaphor

In this chapter, we got to the idea of mapping by this route: iteration, numeric iteration, other kinds of iteration, iteration on a list, iterative commands, iterative operations, mapping. In other words, we started thinking about the mapping tool as a particular kind of repetition in a computer program.

But when I first introduced map as a primitive operation, I thought about it in a different way. Never mind the fact that it's implemented through repetition. Instead think of it as extending the power of the idea of a list. When we started thinking about lists, we thought of the list as one complete entity. For example, consider this simple interaction with Logo:

? print count [how now brown cow]

Count is a primitive operation. It takes a list as input, and it outputs a number that is a property of the entire list, namely the number of members in the list. There is no need to think of count as embodying any sort of repetitive control structure. Instead it's one kind of handle on the data structure called a list.

There are other operations that manipulate lists, like equalp and memberp. You're probably in the habit of thinking of these operations as "happening all at once," not as examples of iteration. And that's a good way to think of them, even though it's also possible to think of them as iterative. For example, how does Logo know the count of a list? How would you find out the number of members of a list? One way would be to count them on your fingers. That's an iteration. Logo actually does the same thing, counting off the list members one at a time, as it would if we implemented count recursively:

to cnt :list
if emptyp :list [output 0]
output 1+cnt butfirst :list

I'm showing you that the "all at once" Logo primitives can be considered as iterative because, in the case of map, I want to shift your point of view in the opposite direction. We started thinking of map as iterative; now I'd like you to think of it as happening all at once.

Wouldn't it be nice if we could say

? show 1+[5 10 15]
[6 11 16]

That is, I'd like to be able to "add 1 to a list." I want to think about it that way, not as "add 1 to each member of a list." The metaphor is that we're doing something to the entire list at once. Well, we can't quite do it that way, but we can say

? show map [1+?] [5 10 15]
[6 11 16]

Instead of thinking "Well, first we add 1 to 5, which gives us 6; then we add..." you should think "we started with a list of three numbers, and we've transformed it into another list of three numbers using the operation add-one."

Other Higher Order Functions

Along with map, you learned about the higher order functions reduce, which combines all of the members of a list into a single result, and filter, which selects some of the members of a list. They, too, are implemented by combining recursion with apply. Here's the Berkeley Logo library version of reduce:

to reduce :reduce.function :reduce.list
if emptyp butfirst :reduce.list [output first :reduce.list]
output apply :reduce.function (list (first :reduce.list)
                                    (reduce :reduce.function
                                            butfirst :reduce.list))

If there is only one member, output it. Otherwise, recursively reduce the butfirst of the data, and apply the template to two values, the first datum and the result from the recursive call.

The Berkeley Logo implementation of filter is a little more complicated, for some of the same reasons as that of map: the ability to accept either a word or a list, and the # feature in templates. So I'll start with a simpler one:

to filter :template :data
if emptyp :data [output []]
if apply :template (list first :data) ~
   [output fput (first :data)
                (filter :template butfirst :data)]
output filter :template butfirst :data

If you understand that, you should be able to see the fundamentally similar structure of the library version despite its extra details:

to filter :filter.template :template.list [:template.number 1]
localmake "template.lists (list :template.list)
if emptyp :template.list [output :template.list]
if apply :filter.template (list first :template.list) ~
   [output combine (first :template.list)
                   (filter :filter.template (butfirst :template.list)
output (filter :filter.template (butfirst :template.list) 

Where map used a helper procedure map1 to handle the extra input template.number, filter uses an alternate technique, in which template.number is declared as an optional input to filter itself. When you invoke filter you always give it the default two inputs, but it invokes itself recursively with three.

Why does filter need a local variable named template.lists? There was a variable with that name in map because it accepts more than one data input, but filter doesn't, and in fact there is no reference to the value of template.lists within filter. It's there because of another feature of templates that I haven't mentioned: you can use the word ?rest in a template to represent the portion of the data input to the right of the member represented by ? in this iteration:

to remove.duplicates :list
output filter [not memberp ? ?rest] :list

? show remove.duplicates [ob la di ob la da]
[di ob la da]

Since ?rest is allowed in map templates as well as in filter templates, its implementation must be the same for both:

to ?rest [:which 1]
output butfirst item :which :template.lists

Mapping Over Trees

It's time to move beyond the iteration tools in the Logo library and invent our own new ones.

So far, in writing operations on lists, we've ignored any sublist structure within the list. We do something for each top-level member of the input list. It's also possible to take advantage of the complex structures that lists make possible. For example, a list can be used to represent a tree, a data structure in which each branch can lead to further branches. Consider this list:

[[the [quick brown] fox] [[jumped] [over [the [lazy] dog]]]]

My goal here is to represent a sentence in terms of the phrases within it, somewhat like the sentence diagrams you may have been taught in elementary school. This is a list with two members; the first member represents the subject of the sentence and the second represents the predicate. The predicate is further divided into a verb and a prepositional phrase. And so on. (A representation something like this, but more detailed, is used in any computer program that tries to understand "natural language" interaction.)

Suppose we want to convert each word of this sentence to capital letters, using Berkeley Logo's uppercase primitive that takes a word as input. We can't just say

map [uppercase ?] ~
    [[the [quick brown] fox] [[jumped] [over [the [lazy] dog]]]]

because the members of the sentence-list aren't words. What I want is a procedure map.tree that applies a template to each word within the input list but maintains the shape of the list:

? show map.tree [uppercase ?]~
     [[the [quick brown] fox] [[jumped] [over [the [lazy] dog]]]]

After our previous adventures in mapping, this one is relatively easy:

to map.tree :template :tree
if wordp :tree [output apply :template (list :tree)]
if emptyp :tree [output []]
output fput (map.tree :template first :tree) ~
            (map.tree :template butfirst :tree)

This is rather a special-purpose procedure; it's only good for trees whose "leaves" are words. That's sometimes the case but not always. But if you're dealing with sentence trees like the one in my example, you might well find several uses for a tool like this. For now, I've introduced it mainly to make the point that the general idea of iteration can take many different forms, depending on the particular project you're working on. (Technically, this is not an iteration, because it doesn't have a two-part structure in which the first part is to perform one step of a computation and the second part is to perform all the rest of the steps. Map.tree does have a two-part structure, but both parts are recursive calls that might carry out several steps. But map.tree does generalize the broad idea of dividing a large computation into similar individual pieces. We'll go into the nature of iteration more carefully in a moment.)

Iteration and Tail Recursion

If you look back at the introduction to recursion in the first volume, you'll find that some recursive commands seem to be carrying out an iteration, like down, countdown, or one.per.line. (In this chapter we've seen how to implement countdown using for, and you should easily be able to implement one.per.line using foreach. Down isn't exactly covered by either of those tools; can you see why I call it an iterative problem anyway?) Other recursive commands don't seem to be repeating or almost-repeating something, like downup or hanoi. The difference is that these commands don't do something completely, then forget about it and go on to the next repetition. Instead, the first invocation of downup, for example, still has work of its own to do after all the lower-level invocations are finished.

It turns out that a command that is tail recursive is one that can be thought of as carrying out an iteration. A command that invokes itself somewhere before the last instruction is not iterative. But the phrase "tail recursive" doesn't mean "equivalent to an iteration." It just happens to work out, for commands, that the two concepts are equivalent. What "tail recursive" means, really, is "invokes itself just before stopping."

I've said before that this isn't a very important thing to worry about. The reason I'm coming back to it now is to try to clear up a confusion that has been part of the Logo literature. Logo implementors talk about tail recursion because there is a tricky way to implement tail recursion that takes less memory than the more general kind of recursion. Logo teachers, on the other hand, tend to say "tail recursive" when they really mean "iterative." For example, teachers will ask, "Should we teach tail recursion first and then the general case?" What's behind this question is the idea that iteration is easier to understand than recursion. (By the way, this is a hot issue. Most Logo teachers would say yes; they begin by showing their students an iterative command like poly or polyspi. I generally say no; you may recall that the first recursive procedure I showed you is downup. One reason is that I expect some of my readers have programmed in Pascal or C, and I want to make it as hard as possible for such readers to convince themselves that recursion is just a peculiar way to express the idea of iteration.)

There are two reasons people should stop making a fuss about tail recursion. One is that they're confusing an idea about control structures (iteration) with a Logo implementation strategy (tail recursion). The second is that this way of thinking directs your attention to commands rather than operations. (When people think of iterative procedures as "easier," it's always commands that they have in mind. Tail recursive operations are, if anything, less straightforward than versions that are non-tail recursive.) Operations are more important; they're what gives Logo much of its flexibility. And the best way to think about recursive operations isn't in implementation terms but in terms of data transformation abstractions like mapping, reduction, and filters.

Multiple Inputs to For

Earlier I promised you multifor, a version of for that controls more than one numeric variable at a time. Its structure is very similar to that of the original for, except that we use map or foreach (or firsts or butfirsts, which are implicit uses of map) in almost every instruction to carry out for's algorithm for each of multifor's numeric variables.

to multifor :values.list :instr
localmake "vars firsts :values.list
local :vars
localmake "initials map "run firsts butfirsts :values.list
localmake "finals map [run item 3 ?] :values.list
localmake "steps (map "multiforstep :values.list :initials :finals)
localmake "testers map [ifelse ? < 0 [[?1 < ?2]] [[?1 > ?2]]] :steps
multiforloop :initials

to multiforstep :values :initial :final
if (count :values)=4 [output run last :values]
if :initial > :final [output -1]
output 1

to multiforloop :values
(foreach :vars :values [make ?1 ?2])
(foreach :values :finals :testers [if run ?3 [stop]])
run :instr
multiforloop (map [?1+?2] :values :steps)

This is a very dense program; I wouldn't expect anyone to read and understand it from a cold start. But if you compare it to the implementation of for, you should be able to make sense of how each line is transformed in this version.

Here is an example you can try:

? multifor [[a 10 100 5] [b 100 10 -10]] ~
           [print (sentence :a "+ :b "= (:a + :b))]
10 + 100 = 110
15 + 90 = 105
20 + 80 = 100
25 + 70 = 95
30 + 60 = 90
35 + 50 = 85
40 + 40 = 80
45 + 30 = 75
50 + 20 = 70
55 + 10 = 65

The Evaluation Environment Bug

There's a problem with all of these control structure tools that I haven't talked about. The problem is that each of these tools uses run or apply to evaluate an expression that's provided by the calling procedure, but the expression is evaluated with the tool's local variables active, in addition to those of the calling procedure. This can lead to unexpected results if the name of a variable used in the expression is the same as the name of one of the local variables in the tool. For example, forloop has an input named final. What happens if you try

to grade :final
for [midterm 10 100 10] [print (sum :midterm :final) / 2]

? grade 50

Try this example with the implementation of for in this chapter, not with the Logo library version. You might expect each iteration to add 10 and 50, then 20 and 50, then 30 and 50, and so on. That is, you wanted to add the iteration variable midterm to the input to grade. In fact, though, the variable that contributes to the sum is forloop's final, not grade's final.

The way to avoid this problem is to make sure you don't use variables in superprocedures of these tools with the same names as the ones inside the tools. One way to ensure that is to rewrite all the tool procedures so that their local variables have bizarre names:

to map :template :inputs


to map :map.qqzzqxx.template :map.qqzzqxx.inputs

Of course, you also have to change the names wherever they appear inside the definition, not just on the title line. You can see why I preferred not to present the procedures to you in that form!

It would be a better solution to have a smarter version of run, which would allow explicit control of the evaluation environment--the variable names and values that should be in effect while evaluating run's input. Some versions of Lisp do have such a capability.

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Brian Harvey,