The Universe of Discourse
Wed, 11 Jan 2012

Where should usage messages go?
Last week John Speno complained about Unix commands which, when used incorrectly, print usage messages to standard error instead of to standard output. The problem here is that if the usage message is long, it might scroll off the screen, and it's a pain when you try to pipe it through a pager with command | pager and discover that the usage output has gone to stderr, missed the pager, and scrolled off the screen anyway.

Countervailing against this, though, is the usual argument for stderr: if you had run the command in a pipeline, and it wrote its error output to stdout instead of to stderr, then the error message would have gotten lost, and would possibly have caused havoc further down the pipeline. I considered this argument to be the controlling one, but I ran a quick and informal survey to see if I was in the minority.

After 15 people had answered the survey, Ron Echeverri pointed out that although it makes sense for the usage message to go to stderr when the command is used erroneously, it also makes sense for it to go to stdout if the message is specifically requested, say by the addition of a --help flag, since in that case the message is not erroneous. So I added a second question to the survey to ask about where the message should go in such a case.

83 people answered the first question, "When a command is misused, should it deliver its usage message to standard output or to standard error?". 62 (75%) agreed that the message should go to stderr; 11 (13%) said it should go to stdout. 10 indicated that they preferred a more complicated policy, of which 4 were essentially (or exactly) what M. Echeverri suggested; this brings the total in favor of stderr to 66 (80%). The others were:

  1. stdout, if it is a tty; stderr otherwise
  2. stdout, if it is a pipe; stderr otherwise
  3. A very long response that suggested syslog.
  4. stderr, unless an empty stdout would cause problems
  5. It depends, but the survey omitted the option of printing directly on the console
  6. It depends
I think #2 must have been trying to articulate #1, but (a) got it backwards and (b) missed. #3 seemed to be answering a different question than the one that was asked; syslog may make sense for general diagnostics, but to use it for usage messages seems peculiar. #5 also seems strange to me, since my idea of "console" is the line printer hardwired to the back of the mainframe down in the machine room; I think the writer might have meant "terminal".

68 people answered the second question, "Where should the command send the output when the user specifically requests usage information?". (15 people took the survey before I added this question.) 50 (74%) said the output should go to stdout, 12 (18%) to the user's default pager and then to stdout, and 5 (7%) to stderr. One person (The same as #5 above) said "it depends".

Thanks to everyone who participated.

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Tue, 10 Jan 2012

Elaborations of Russell's paradox
When Iris was five or six, I told her about Russell's paradox in the following form: in a certain library, some books are catalogs that contain lists of other books. For example, there is a catalog of all the books on the second floor, and a catalog of all the books about birds. Some catalogs might include themselves. For example, the catalog of all the books in the library certainly includes itself. Such catalogs have red covers; the other catalogs, which do not include themselves, such as the catalog of all the plays of Shakespeare, have blue covers. Now is there a catalog of all the catalogs with blue covers?

I wasn't sure she would get this, but it succeeded much better than I expected. After I prompted her to consider what color cover it would have, she thought it out, first ruling out one color, and then, when she got to the second color, she just started laughing.

A couple of days ago she asked me if I could think of anything that was like that but with three different colors. Put on the spot, I suggested she consider what would happen if there could be green catalogs that might or might not include themselves. This is somewhat interesting, because you now can have a catalog of all the blue catalogs; it can have a green cover. But I soon thought of a much better extension.

I gave it to Iris like this: say you have a catalog, let's call it X. If X mentions a catalog that mentions X, it has a gold stripe on the spine. Otherwise, it has a silver stripe. Now:

  1. Could there be a red catalog with a gold stripe?
  2. Could there be a red catalog with a silver stripe?
  3. Could there be a blue catalog with a gold stripe?
  4. Could there be a blue catalog with a silver stripe?
And more interesting:

  1. Is there a catalog of all the catalogs with gold stripes?
  2. Is there a catalog of all the catalogs with silver stripes?
I knew that early 20th century logicians, trying to repair the Russell paradox, first tried a very small patch: since comprehension over the predicate XX causes problems, just forbid that predicate. This unfortunately doesn't solve the problem at all; there are an infinite number of equally problematic predicates. (Whitehead and Russell's theory of types is an attempt to fix this; Quine's New Foundations is a different attempt.) One of these predicates is ¬∃Y.X∈Y and Y∈X. You can't construct the set of all X such that ¬∃Y.X∈Y and Y∈X because there is no such set, for reasons similar to the reason why there's no set of all X such that XX, so that's where I got the silver stripe predicate.

Translating this into barber language is left as an exercise for the reader.

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Wed, 04 Jan 2012

Mental astronomical calculations
As you can see from the following graph, the daylight length starts increasing after the winter solstice (last week) but it does so quite slowly at first, picking up speed, and reaching a maximum rate of increase at the vernal equinox.

The other day I was musing on this, and it is a nice mental calculation to compute the rate of increase.

The day length is given by a sinusoid with amplitude that depends on your latitude (and also on the axial tilt of the Earth, which is a constant that we can disregard for this problem.) That is, it is a function of the form a + k sin 2πt/p, where a is the average day length (12 hours), k is the amplitude, p is the period, which is exactly one year, and t is amount of time since the vernal equinox. For Philadelphia, where I live, k is pretty close to 3 hours because the shortest day is about 3 hours shorter than average, and the longest day is about 3 hours longer than average. So we have:

day length = 12 hours + 3 hours · sin(2πt / 1 year)
Now let's compute the rate of change on the equinox. The derivative of the day length function is:

rate of change = 3h · (2π / 1y) · cos(2πt / 1y)
At the vernal equinox, t=0, and cos(…) = 1, so we have simply:

rate of change = 6πh / 1 year = 18.9 h / 365.25 days
The numerator and the denominator match pretty well. If you're in a hurry, you might say "Well, 360 = 18·20, so 365.25 / 18.9 is probably about 20," and you would be right. If you're in slightly less of a hurry, you might say "Well, 361 = 192, so 365.25 / 18.9 is pretty close to 19, maybe around 19.2." Then you'd be even righter.

So the change in day length around the equinox (in Philadelphia) is around 1/20 or 1/19 of an hour per day—three minutes, in other words.

The exact answer, which I just looked up, is 2m38s. Not too bad. Most of the error came from my estimation of k as 3h. I guessed that the sun had been going down around 4:30, as indeed it had—it had been going down around 4:40, so the correct value is not 3h but only 2h40m. Had I used the correct k, my final result would have been within a couple of seconds of the right answer.

Exercise: The full moon appears about the same size as a U.S. quarter (1 inch diameter circle) held nine feet away (!) and also the same size as the sun, as demonstrated by solar eclipses. The moon is a quarter million miles away and the sun is 93 million miles away. What is the actual diameter of the sun?

[ Addendum 20120104: An earlier version of this article falsely claimed that the full moon appears the same size as a quarter held at arm's length. This was a momentary brain fart, not a calculational error. Thanks to Eric Roode for pointing out this mistake. ]

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Tue, 03 Jan 2012

Eta-reduction in Haskell and English
The other day Iris and I were putting together a model, and she asked what a certain small green part was for. I said "It's a thing for connecting a thing to another thing."

Iris objected that this was a completely unhelpful explanation, but I disagreed. I would have agreed that it was an excessively verbose explanation, but she didn't argue that point.

Later, it occurred to me that Haskell has a syntax for eliding unnecessary variables in cases like this. In Haskell, one can abbreviate the expression

        λx → λy → x + y
to just (+). (Perl users may find it helpful to know that the Perl equivalent of the expression above is sub { my ($x) = @_; return sub { my ($y) = @_; return $x + $y } }.) This is an example of a general transformation called η-reduction. In general, for any function f, λxf x is a function that takes an argument x and returns f x. But that's exactly what f does. So we can replace the longer version with the shorter version, and that's η-reduction, or we can go the other way, which is η-expansion.

Anyway, once I thought of this it occurred to me that, just like the longer expression could be reduced to (+), my original explanation that the small green part was "a thing for connecting a thing to another thing" could be η-reduced to "a connector".

Perhaps if I had said that in the first place Iris would not have complained.

Happy new year, all readers.

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