[Solutions for exercises 4,5 were originally posted together on 5 Dec 91]
Date: Thu 5 Dec 91 10:26:58-EST
From: Michael Downes
Subject: `Around the bend' #2 solutions (4,5)
To: info-tex@shsu.edu
Answers to exercises 4 and 5 of `Around the bend' #2. Discussion of E6
will follow in a separate post because it is rather lengthy. Discussion
of E7 will follow in another couple of weeks (I'm going to be on
vacation next week.)
"***********************************************************************
"*** Exercise 4 (essay):
"
"What should `best' mean when comparing solutions to an `Around the
"bend' exercise? What qualities of a good solution are most important?
"Why? How can they be objectively measured? (Or can they?) On the
"negative side, what qualities indicate an inferior solution?
Peter Schmitt writes:
> What is to be rated as `best' clearly depends on the function used to
> measure quality. And therefore the question makes sense only with
> respect to some particular rating function. Seemingly nothing is gained
> by this statement: Instead of discussing what qualities are required
> for a good solution one has to discuss how the rating system should be
> defined. But nevertheless this shifted point of view has an important
> an important advantage. It makes clear that there is no unique answer:
> Quality is not an absolute notion but a notion relative to some
> (agreed) measure. This measure is not independent of the context ---
> under different conditions different rating functions may be used.
> One further important point must not be forgotten: If matters of
> personal taste are to be excluded than the measuring function has to be
> precisely defined --- demanding simplicity, without giving this notion
> a precise (formal) meaning, is not sufficient.
>
> Therefore I would like to split the original question into two seperate
> questions:
>
> (a) What (formal and informal) rating functions are likely to be
> useful, and under what circumstances?
>
> (b) With respect to some formal rating function, is there always a best
> solution?
>
> Some answers to the first questions are the following (no completeness
> claimed or even intended):
>
> (1) the first solution:
>
> If some special effect is needed for a single application then the
> best solution is the first solution (the solution that can be
> realized with the least effort). This is, however, a purely
> individual criterion that cannot be formalized.
>
> (2) the most economic (in some sense) solution:
>
> Economic considerations are important if a code is used frequently,
> Depending on the nature of the applications running time, memory
> usage, and others, may be relevant. But the time spent for finding
> a good solution still cannot be neglected in a real world
> situation. Of course, for theoretical investigations the time spent
> for research does not matter.
>
> (3) the more robust solution:
>
> If some set of macros is used by a large number of people who not
> always know how to use them correctly (or even do not care to know)
> then it is certainly an advantage if they are robust, i.e. work in
> as many cases (even strange ones) as possible. But again, one has
> to decide what price (in terms of resources) is acceptable for this
> robustness. (In many cases the item (4) below will be more
> important.)
>
> (4) ease-of-use:
>
> If a set of macros is used frequently (by one or more persons) then
> ease-of-use is certainly a mark of quality: easy to remember
> syntax, short commands, natural and good readable embedding into
> the surrounding text, and similar criteria, decide about this.
>
> (5) simplicity:
>
> Simple solutions certainly have a strong appeal --- but what is a
> simple solution? Again this is hard to formalize, since simplicity
> basically is an aesthetic value, closely related to the concepts of
> elegance and beauty. (This is similar to the situation in
> mathematics.) But be careful: Simple is not equivalent to short!
>
> (6) the shortest solution:
>
> This may seem to be an easy rating function, but is it? Should
> length be measured by the number of characters (probably not!), or
> by the number of tokens, or by the number of control sequences? Or
> by something else?
>
> Most of the measures mentioned are difficult to formalize, or cannot be
> formalized at all. Only the resources used (in (2)) and the length of a
> code (in (6)) can be precisely defined. Therefore, with respect to one
> of these cases two solutions of the same problem can be compared.
> Furthermore, in many cases it will be possible to proof that an optimal
> solution exists. (For instance, since the length of a code (in any
> interpretation) is a positive integer, there must exist one or more
> solutions with minimal length, provided there is at least one
> solution.) But unfortunately this does not imply that one is able to
> construct an optimal solution, or to decide whether a given piece of
> code is an optimal solution (or at least near to one). And in some
> cases it may happen that no optimal solution exists, e.g. if to every
> solution there is better --- but longer! --- one.
>
> What is the conclusion of all this? That there may be a best solution
> relative to some side conditions. But that there is no globally best
> solution. This statement is, of course, not very satisfying. One
> would rather prefer to have at least some notion (even a tentative one)
> of a best solution than none at all. I propose therefore the following
> informal definition (often subject to personal taste): If some code is
> optimal or near-optimal in more than one category then it is probably
> as near to a globally optimal solution as this is possible.
My comments:
I propose the following list, based on (1) [my interpretation of]
Knuth's ideas about good macro writing as demonstrated in the TeXbook
and plain.tex, (2) various articles in TUGboat, (3) Schmitt's comments,
(4) discussions I've had in the past with other macro writers, and so
forth.
The characteristics of a good solution to an `Around the bend' exercise
are (in order of decreasing importance):
1. Robustness
2. Brevity (= minimal usage of TeX's main memory)
3. Simplicity
4. Ease of use
5. Suitable commentary
6. Speed
7. Minimal hash table load
8. Minimal save stack load
9. Minimal load in other categories of TeX's memory
10. Comprehensive test suite (when applicable)
Schmitt's point about 'first solution' is well taken but does not apply
to `Around the bend' exercises, because of the stated goal of finding a
'best' solution, with the presumption that normally more than one
solution will be found.
Measurement of these qualities is not too difficult, I think,
except for 3 and 5. Here's how I see the measurements:
1. Robustness: A solution is robust if no one who reads it offers a
counterexample that causes it to fail. If two solutions both fail, the
one with more counterexamples is less robust; if two solutions have
different counterexamples, the solution whose counterexample is more
likely to occur in normal use is the less robust solution.
2. Brevity: Of two different solutions, the one that is
briefer/shorter/more compact is the one that uses less of TeX's main
memory as measured by \tracingstats.
3. Simplicity: Of two different solutions, the shorter one (in the
sense of the previous item) is usually the simpler one, but not always.
A solution that condenses all the necessary operations into a dense,
incomprehensible Gordian knot is less simple than a longer solution
that lays out the operations in a series of easily comprehended steps.
A solution that relies on arcane dirty tricks is less simple than a
solution that uses better-known techniques in a straightforward
approach.
4. Ease of use: I believe this will not be extremely hard to measure in
the context of the particular application; it can't sensibly be
discussed out of context.
5. Suitable commentary: The commentary surrounding a solution should
explicitly mention any necessary assumptions. If the code is complex,
the commentary should give an outline or overview of the intended
algorithm. It should explain the operation of any macro if its
operation is not evident from the code. If an unusual construction is
used where a different construction would normally be expected, the
commentary should give the reason.
6. Speed: Of two solutions, the speedier one is the one that runs
faster on common computer systems. If one solution runs faster and
slower than another, depending on the system ... well, let's not cross
that bridge unless it turns out to be real.
7,8,9. Minimal hash table load, save stack load, etc. These can be
measured by \tracingstats.
10. Comprehensive test suite: If two solutions are equal in other
respects, the one whose accompanying test suite covers more distinct
cases than the other's is better by that much.
It may be argued that I have not sufficiently answered the question of
subjectivity. For example, who's to decide what's an 'arcane dirty
trick' and what's not? What does 'suitable' mean in number 5? The
answer is that I will say that something is an 'arcane dirty trick' if
I think so, and anyone else can do the same. In most cases I believe
that there will be general agreement on such a question; if not, and an
ensuing discussion fails to reach a clear settlement, then each of the
solutions in question will be decreed 'subjectively just as good as the
others'.
Other qualities of a good solution can be expressed in terms of the
ones listed above. For example, self-sufficiency may be considered an
aspect of robustness---if a solution is not entirely self-sufficient,
it can easily be shown to be not robust by giving a counterexample that
exploits the assumption that makes the solution non-self-sufficient.
Elegance? If a solution is simple and easy to use, then I say it is
elegant. A solution doesn't necessarily have to be robust in order to
be elegant, nor even short (although of two solutions that are
otherwise equal, the shorter one is undoubtedly more elegant).
[Solution for exercise 5 moved to answer.005]
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35: # number/pound sign; 61: = equals sign;
36: $ dollar sign; 62: > right elbow;
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58: : colon; 126: ~ tilde
Michael Downes mjd@math.ams.com (Internet)