Sun, 10 Dec 2006
I came across the FourFours problem recently. Stated succinctly, it asks: what are the ways to calculate each of the integers from 1 to 100 with formulas which use the digit four exactly four times? (No digits other than four can be used at all.)
People have solved it in C# (138 lines, 380ms), Python (119 lines, 20s), Haskell (73 lines, 900ms), and Perl (one (really long) line, 45s). I thought it would be interesting to try it in Common Lisp, so I did.
For one thing, I found that my code was slower than the C# and Haskell solutions; I consistently get about 2.5 seconds for my solution. That seems mostly to be a result of both the dynamic nature of Lisp and the particularly-dynamic approach I took (more on that in a bit). I also took more lines (148 significant lines; excludes comments, doc strings, empty lines, and a couple of ancillary definitions).
In my defense, I did things somewhat more generally than the others; I don't like hardcoded data, so my code generates a lot of stuff on the fly. One notable example is the ordering of operations on the four values: the other solutions I've seen all hardcode the orderings, but my code generates the code structure (just a binary tree) at runtime based on the number of elements. I then use the code/data duality of Lisp to generate both functions and the printed representation of the operations from those trees. Likewise, my code will figure out the possible values to use in the calculations from a supplied digit.
All that flexibility let me play around a lot more with the structure of the problem. Although everyone else used five operations (addition, subtraction, multiplication, division, and exponentiation), I found that only four were necessary (addition, subtraction, multiplication, and division). (I checked smaller numbers of operations, too. I considered all of the functions in Common Lisp that would return ratios or integers when given a pair of ratios or integers as parameters. I was unable to get all the way from 1 to 100 with fewer than four distinct functions.)
With the given parameters (four fours, with 4, 4!, sqrt(4), .4, .4bar, and sqrt(.4bar)) and operators (+, -, *, /), you can actually go all the way up to 102. (Adding exponentiation does not allow you to go any further.)
I can calculate the consecutive chains for arbitrary base digits:
1: 1
1: 2 / 2
1: 3! / (3 + 3)
2: (3 + 3) / 3
3: (3 + 3) - 3
4: 3 + (3 / 3)
5: (3! / 3) + 3
6: (3 * 3) - 3
7: 3! + (3 / 3)
8: 3! + (3! / 3)
9: 3 + (3 + 3)
10: (3! - 3) / .3
11: (.3 + 3) / .3
12: 3 + (3 * 3)
13: (3 / .3) + 3
14: (3! / .3) - 3!
15: 3! + (3 * 3)
16: (3 / .3) + 3!
17: (3! / .3) - 3
18: 3 * (3 + 3)
19: (3! - .3) / .3
20: (3 + 3) / .3
21: (3! * 3) + 3
1: (4 + 4) / (4 + 4)
2: (4 * 4) / (4 + 4)
3: (4 + (4 + 4)) / 4
4: 4 + (4 * (4 - 4))
5: (4 + (4 * 4)) / 4
6: 4 + ((4 + 4) / 4)
7: (4 + 4) - (4 / 4)
8: (4 + (4 + 4)) - 4
9: (4 / 4) + (4 + 4)
10: (4! + (4 * 4)) / 4
11: ((4! + 4) / 4) + 4
12: 4 * (4 - (4 / 4))
13: (4! + (4! + 4)) / 4
14: (4! / 4) + (4 + 4)
15: (4 * 4) - (4 / 4)
16: 4 + (4 + (4 + 4))
17: (4 / 4) + (4 * 4)
18: ((4! * 4) - 4!) / 4
19: 4! - (4 + (4 / 4))
20: 4 * (4 + (4 / 4))
21: (4 / 4) + (4! - 4)
22: 4! - ((4 + 4) / 4)
23: ((4! * 4) - 4) / 4
24: (4 * 4) + (4 + 4)
25: ((4! * 4) + 4) / 4
26: 4! + ((4 + 4) / 4)
27: (4! + 4) - (4 / 4)
28: (4 * (4 + 4)) - 4
29: (4 / 4) + (4! + 4)
30: (4! + (4! * 4)) / 4
31: 4! + ((4! + 4) / 4)
32: (4 * 4) + (4 * 4)
33: ((4 - .4) / .4) + 4!
34: (4! / 4) + (4! + 4)
35: ((4 / .4) + 4) / .4
36: 4 + (4 * (4 + 4))
37: ((.4 + 4!) / .4) - 4!
38: (4 / .4) + (4! + 4)
39: 4! + (4! / (.4 * 4))
40: (4 * (4 * 4)) - 4!
41: (.4 + (4 * 4)) / .4
42: (4! + 4!) - (4! / 4)
43: (4! + 4!) - (sqrt(4) / .4)
44: (4 * 4) + (4! + 4)
45: (4! - (4! / 4)) / .4
46: ((4! - 4) / .4) - 4
47: (4! + 4!) - (4 / 4)
48: 4 * (4 + (4 + 4))
49: (4 / 4) + (4! + 4!)
50: (4 + (4 * 4)) / .4
51: ((.4 + 4!) - 4) / .4
52: (4! / .4) - (4 + 4)
53: (sqrt(4) / .4) + (4! + 4!)
54: (4! / 4) + (4! + 4!)
55: ((4! - .4) / .4) - 4
56: 4! + (4 * (4 + 4))
57: ((.4 + 4!) / .4) - 4
58: (4 / .4) + (4! + 4!)
59: (4! / .4) - (4 / 4)
60: (4 * (4 * 4)) - 4
61: (4! / .4) + (4 / 4)
62: (4 * (4 * 4)) - sqrt(4)
63: ((4! - .4) / .4) + 4
64: (4 + 4) * (4 + 4)
65: ((.4 + 4!) / .4) + 4
66: ((4! + 4) / .4) - 4
67: ((4! + 4) / .4bar) + 4
68: 4 + (4 * (4 * 4))
69: ((4! + 4) - .4) / .4
70: (4! / .4) + (4 / .4)
71: (.4 + (4! + 4)) / .4
72: 4! * (4 - (4 / 4))
73: (sqrt(.4bar) + (4! + 4!)) / sqrt(.4bar)
74: ((4! + 4) / .4) + 4
75: (4! + (4! / 4)) / .4
76: ((4! - 4) * 4) - 4
77: ((4! - .4bar) / .4bar) + 4!
78: ((4! - 4) * 4) - sqrt(4)
79: ((4! - sqrt(4)) / .4) + 4!
80: 4 * (4 + (4 * 4))
81: ((4! / .4) - 4!) / .4bar
82: sqrt(4) + ((4! - 4) * 4)
83: ((4! - .4) / .4) + 4!
84: ((4! - 4) * 4) + 4
85: ((4 / .4) + 4!) / .4
86: (4! * 4) - (4 / .4)
87: (4! * 4) - (4 / .4bar)
88: (4! * 4) - (4 + 4)
89: ((sqrt(4) + 4!) / .4) + 4!
90: (4! * 4) - (4! / 4)
91: (4! * 4) - (sqrt(4) / .4)
92: (4! - (4 / 4)) * 4
93: (4! * 4) - (sqrt(4) / sqrt(.4bar))
94: ((4! + 4) / .4) + 4!
95: (4! * 4) - (4 / 4)
96: 4! * ((4 + 4) - 4)
97: (4 / 4) + (4! * 4)
98: .4 + ((.4 + 4!) * 4)
99: ((4! + 4!) - 4) / .4bar
100: (4! + (4 / 4)) * 4
101: (sqrt(4) / .4) + (4! * 4)
102: (4! / 4) + (4! * 4)
1: 5 / (5 + (5 * (5 - 5)))
2: ((5 + (5 + 5)) - 5) / 5
3: (5 + (5 * 5)) / (5 + 5)
4: (5 + (5 + (5 + 5))) / 5
5: (5 + (5 + 5)) - (5 + 5)
⋮
154: (5! / 5) + (5! + (5 + 5))
155: 5 + (5 * (5 + (5 * 5)))
156: (5! + (5! * (.5 + 5))) / 5
157: (5 / (.5bar * (.5bar - .5))) - 5
158: (((5! / 5) - 5) / .5) + 5!
1: (6 + (6 + 6)) / (6 + (6 + 6))
2: (6 + (6 + (6 + 6))) / (6 + 6)
3: ((6 + (6 + (6 + 6))) - 6) / 6
4: (6 * (6 + 6)) / (6 + (6 + 6))
5: (6 + (6 + (6 + (6 + 6)))) / 6
6: 6 + (6 * ((6 + 6) - (6 + 6)))
⋮
881: 6! + (((.6bar * (6! + 6!)) + 6) / 6)
882: ((6! / 6) + 6) * (6 + (6 / 6))
883: ((.6bar + (6! / (.6bar + 6))) / .6bar) + 6!
884: (6! / (.6 * 6)) + (6! - (6 * 6))
885: (.6 * ((6 * 6) + (6! + 6!))) - .6
886: ((6! / (6 - .6)) * (.6 + 6)) + 6
I'll probably experiment with solutions that involve concatenation of digits next. That'll involve some fairly significant code changes, though. And I'll probably just end up duplicating the work of the definitive Four Fours answer key.
Tue, 31 Oct 2006
It often happens in programming that you need to name two things that are
similar but different. This can be a difficult task, but one would expect
that if you were designing a language or its standard library you'd put
extra effort into your naming. Sadly, it sometimes seems that the people
designing Delphi didn't put in that effort. For a simple example,
consider division by zero. If you try to divide by zero, Delphi generates
an exception at runtime, but if you were doing an integer divide the
exception is EDivByZero, whereas floating-point division generates
EZeroDivide.
For a more extreme example, let's look at String comparisons. Delphi has functions for native Delphi strings and C-style null-terminated strings, functions for case-sensitive and case-insensitive comparison, functions that understand multibyte character coding systems and use Windows locale info and functions that don't, and some null-terminated functions that also take a character count (analogous to strncmp).
| Function | String Type | Multibyte-aware | Case-Sensitive | Specify Length |
|---|---|---|---|---|
= |
String |
No | Yes | No |
CompareText |
String |
No | No | No |
CompareStr |
String |
No | Yes | No |
AnsiCompareText |
String |
Yes | No | No |
AnsiCompareStr |
String |
Yes | Yes | No |
StrIComp |
PChar |
No | No | No |
StrLIComp |
PChar |
No | No | Yes |
StrComp |
PChar |
No | Yes | No |
StrLComp |
PChar |
No | Yes | Yes |
AnsiStrIComp |
PChar |
Yes | No | No |
AnsiStrLIComp |
PChar |
Yes | No | Yes |
AnsiStrComp |
PChar |
Yes | Yes | No |
AnsiStrLComp |
PChar |
Yes | Yes | Yes |
Granted, the null-terminated functions are pretty consistent with their Ls
and Is, but I can never remember which of CompareStr and CompareText
is which (not to mention remembering which names are for null-terminated
strings and which are for Delphi strings). And the rule of thumb is to
always use the Ansi functions, because they honor locale orderings.
Presumably the non-Ansi functions are around because they're faster
(they do a straight numeric compare on the ordinal value of each
character), but the naming implies that they should be the default.
Another hairy mess of naming I run into moderately often is the task of
turning a TDateTime into a string. Should I use DateToStr or
TimeToStr, which each take one parameter (the TDateTime variable) and
use several global variables to determine their formatting? Should I use
FormatDateTime, which takes two parameters: the TDateTime variable and
a format string? Should I use DateTimeToStr, which works just like
FormatDateTime, but takes a third parameter, a string variable passed by
reference, into which the result is placed? (DateToStr, TimeToStr,
and FormatDateTime are Delphi functions while DateTimeToStr is a
Delphi procedure.)
I consider Format to be Delphi's analog to sprintf, and use it
exclusively, but Delphi provides FmtStr, Format, and FormatBuf.
Format is a function that returns a formatted Delphi string. FmtStr
is a procedure that gets its result variable passed in by reference.
FormatBuf is the direct analog to snprintf: you must pass in the
result variable and the format string as PChars, as well as the lengths
of those two strings; the result is placed into the passed variable and
the function returns the length of that string.
StrUpper upcases an entire PChar. AnsiStrUpper does the same, but
understands multibyte encodings and honors the Windows locale.
UpperCase and AnsiUpperCase are the same, but for Delphi strings.
UpCase upcases a single character.
Low gives the lowest value of a range type or the lowest index of an
array. Lo gives the low-order byte of an integer. Likewise, High
gives the highest value of a range type or the highest index of an array
while Hi gives the high-order byte of an integer. (More properly, Hi
returns the second-lowest-order byte of an integer; it assumes all
integers are 16-bit.)
Trunc takes a floating point number and truncates it to an integer.
Truncate deletes all data in a file past the current seek point. I use
Trunc just rarely enough that I almost always use Truncate when I mean
to use Trunc.
I suppose all languages have similar issues (I'm reminded of Common Lisp
with aref, svref, bit, sbit, nth, elt, char, schar, and (I
suppose) row-major-aref), but in most other cases, I don't often find
myself trying to decide which of several similarly-named functions is the
right one and which is wrong. Delphi's setup just feels more muddied to
me.
Fri, 19 May 2006
When I was going to Catonsville Community College (now the Catonsville Campus of The Community College of Baltimore County), I took a couple of computer science courses. The first of those was essentially an introduction-to-programming course, with the textbook using C. The professor, however, said that he would accept programs in any language that was cleared with him beforehand. Since I already knew C, I used the class as an opportunity to learn Common Lisp.
One of the problems from the course that I still remember fondly was the "do your own math" one. We were allowed to have a function that used the language's built-in addition operator to add one to a supplied value, and the same for the subtraction operator. Building on those two functions (i.e. not using any of the language's other math operators or functions), we had to write functions to implement addition, subtraction, multiplication, division, modulus, exponentiation, greatest common divisor, and least common multiple. To simplify things, we only had to make the functions work in the domain of positive integers (so the division would be integer divisions). Because I felt like setting myself a little more of a challenge, I added a couple of my own constraints. I decided that I would make my functions work for all integers, both positive and negative, and that I would match the interfaces for the Common Lisp functions I was replacing (at least as far as integers were concerned).
I found the problem to be quite interesting because it involves breaking down common mathematical operations into their basic forms. It's also rather neat to build up a hierarchy of common routines where you've constructed each layer yourself and you end up with a tower where you've built each layer, aside from some small contributions for the foundation.
I was bored recently and decided to redo the problem, now that I've got more Common Lisp experience under my belt. In particular, I realized that I could make a new package and shadow the functions that I was replacing, so all of my code would look normal, even though I was using my own functions. I also thought it would be interesting to compare my Common Lisp programming from nearly a decade ago with my Common Lisp programming now. Here's my original code and my new code.
Probably my biggest surprise was that my original version of least common multiple gave wrong answers when supplied with multiple arguments. I clearly hadn't tested it enough in school. What's interesting is that I originally tried the same approach with my recent code, but had to fix it when I saw that it wasn't right. My recent code is also more efficient; division is orders of magnitude faster, especially for small divisors, and my old code triggers a stack overflow when asked to multiply large numbers. Structurally speaking, the newer stuff tends to be more compact; helper functions are declared with labels rather than at the toplevel, and I make better use of some Common Lisp functions, most notably reduce.
So now I've gotten even more out of the interesting problem. In addition to the fun of solving it at all, I also get a comparison of changes in my programming style over time. Perhaps I'll address it again in another decade or so.
Fri, 27 Jan 2006
I've recently been playing with Ruby, and I think I've finally found a replacement for my various uses of Perl. For a while now, I've been using one of two languages for my at-home programming: Common Lisp for the more complex or interesting tasks; and Perl for the simpler tasks, standalone programs, and one-line text processing.
I've long had a love/hate relationship with Perl. On the one hand, its
DWIM approach to things make it easy to do simple things, especially
one-off text munging. perl -pe '<stuff>' has long been my friend. It
also has CPAN going for it; many of my problems have been solved with a
CPAN module and a little glue code. On the other hand, Perl tends towards
extreme ugliness, especially when the code starts to get complicated.
I've never liked the OO interface for Perl, for example; not only does it
feel very obviously bolted on to an existing infrastructure, it didn't
even get the sharp, pointy edges sanded down when they did it.
Thus it was that I was impressed with Python when I first met it. It
felt like a language designed with OO from the bottom up. (But it wasn't;
the addition of OO stuff had just been done in a much better way than
Perl's.) It had nice data introspection. It had functions as first-class
objects (so they got the benefit of introspection, as well as the other
benefits of first-classness). I figured I could live with the
whitespace-as-syntax wart. As I got to know it more, Python started
falling down in a few places. For one thing, Guido van Rossum apparently
doesn't like functional programming. Python has some functional
programming constructs, like map and filter, but Guido keeps threatening
to remove them. There's no good way to create anonymous functions,
because Python's lambda is only useful for relatively trivial purposes.
Python is also not terribly useful from the command line, largely because
of the whitespace-as-syntax thing. Again, it works for some trivial
cases, but is not general purpose enough.
So it was that I resisted bothering with Ruby for a while. "Oh," I said, "It's just another fad, like Python." But I kept hearing good things about it, even apart from all the buzz that Ruby on Rails has generated. So I finally resolved to poke at it a bit, and I'm pretty impressed. It's a much cleaner language than Perl, but it's better at functional and command-line usage than Python. It's enough to make me want to switch to Ruby for all my non-Lisp needs.
There are some things that feel like they might bother me, though. While Ruby not only supports anonymous functions but uses anonymous-function-like blocks all over the place, functions in Ruby are not first class objects, which means that you can't quite sling them around as easily as data. The use of special characters to determine variable scope (globals, class variables, instance variables, etc.) seems a bit weird to me, but at least it's not as bad as Perl's variable special characters.
Basically, Python has too much insistence on doing things its own particular, sometimes convoluted way. Ruby stole all the useful things from Perl while otherwise maintaining a very clean language that doesn't spit on my Lisp-influenced thinking.
