Assuming Unicode and case-insensitivity, should the pattern ".." match "FfIsS"?

Question

It sounds like a joke, but I can sort of prove it.

Assumptions:

• Dot matches any single character.
• A case-insensitive pattern matches s if and only if it matches s.toUpperCase().

All of the following is pretty logical and holds in Java:

• "ffi".matches(".") LATIN SMALL LIGATURE FFI (U+FB03) is a character, so it must match
• "ß".matches(".") LATIN SMALL LETTER SHARP S (U+00DF) is a character, so it must match
• "ffi".toUpperCase().equals("FFI") by the Unicode standard (there's no capital ligature FFI)
• "ß".toUpperCase().equals("SS") by the Unicode standard (there's a capital sharp S, but it doesn't get used)
• "FfI".toUpperCase().equals("FFI") obviously
• "sS".toUpperCase.equals("SS") obviously

So assuming the first dot in the regex stands for ffi and the second for ß, the regex must match "FFISS" and because of the case insensivity also "FfIsS".

I really do hope there's something wrong, otherwise regexes would get pretty unusable.

The questions:

• What's wrong with my "proof"?
• What does exactly "case insensitive" mean if my second assumption doesn't hold?

Answer 1

On Case Folding

The answer is no, dot will not match ss case insensitively, although the reasons are slightly esoteric.

However, your puzzle has often been raised by some of those most in the know about such things, because they too feel it leads to contradictions.

There are two forms of case mapping in Unicode. There is simple case mapping in which one code point only ever maps to exactly one other code point. So if length(s) == 1, then you are guaranteed that length(fc(s)) == 1 also, where fc is the Unicode foldcase map. But it also applies to uc, tc, and lc case mappings.

The problem with that it is that you do not get as good results analysing certain kinds of real-world text then you make those sorts of exact 1:1 length guarantees.

In fact, there are quite a few of these. The numbers indicate how many individual BMP code points map to the specified lengths under the four case maps:

length lc == 2          1
length lc == 3          0

length fc == 2         88
length fc == 3         16

length uc == 2         86
length uc == 3         16

length tc == 2         32
length tc == 3         16

In full casing, not the simple casing that Java’s regex uses, you can indeed get things like tschüß and TSCHÜSS to match, even though they are of unequal lengths. Perl and Ruby use full case mapping when doing case insensitive comparisons. This leads to strange paradoxes in negated character classes if you aren’t careful though.

But here’s the rub: case insensitive matching does not perform a transitive operation. In other words, if . matches ß and under case insensitive matching, ß matches SS, that does not mean that via transitivity . matches SS case insensitively. It just does not work that way, although smarter people than me have thought deeply upon the matter.

However, both these code points:

• U+00DF ‭ ß LATIN SMALL LETTER SHARP S
• U+1E9E ‭ ẞ LATIN CAPITAL LETTER SHARP S

do certainly case-insensitively match not only each other but also SS, Ss, sS, and ss under full case mapping. They just don’t do so under simple case mapping.

Unicode does make some guarantees about this. One is that if length(s) == n, that length(fn(s)) <= 3*n where fn is any of the four case maps: lc, fc, uc, and tc.

On Normalization

If you think that’s bad, it actually gets a good deal worse when you consider normalization forms. Here the guarantee is 5× not 3×. So length(NFx(s)) <= 5 * length(s), which as you see is getting expensive.

Here is the equivalent table showing how many code points expand to more than one under each of the four normalization forms:

length NFC  == 2        70
length NFC  == 3         2
length NFC  == 4         0
length NFC  == 5         0

length NFKC == 2       686
length NFKC == 3       404
length NFKC == 4        53
length NFKC == 5        15

length NFD  == 2       762
length NFD  == 3       220
length NFD  == 4        36
length NFD  == 5         0

length NFKD == 2      1345
length NFKD == 3       642
length NFKD == 4       109
length NFKD == 5        16

Isn’t that remarkable? For a while, Unicode wanted to try to build canonical equivalence into its pattern matching. They knew it was expensive for the reasons just stated, but it took them a while to figure out that it was fundamentally impossible due to the necessary canonical reordering of combining characters within one grapheme unit.

For this reason, and many others, the current recommendation if you want to compare things “case-insensitively” or “normalization-insensitively” is to yourself run it through the transform on both sides and then compared the results.

For example, given a suitable == code-point–by–code-point equivalence operator

fc(a) == fc(b)

and similarly for a =~ pattern matching operator (which works in the traditional way of course, not like Java’s broken match method that inappropriately anchors things):

fc(a) =~ fc(b)

The problem with that is that you can no longer turn case insensitivity on or off in particular parts of a pattern, such as

/aaa(?i:xxx)bbb/

and have only the xxx part be done case insensitively.

Full casing is hard, but it can (in most circumstances) be done, as Perl and Ruby have proven. But it is also rather non-intuitive (read: surprising) in places you should you understood. You have to do special things with bracketed character classes, especially with their negations, or it leads to nonsense.

Locale Matching

Finally, to make matters truly complicated, in the real world, you have to do more than either or both of case mapping and normalization. In certain national locales, things are more complicated. For example, in a German phonebook, and vowel with an umlaut counts exactly the same as that same base vowel followed by a the letter e. So there, something like müß would be expected to match MUESS case-insensitively.

To do all this right, you really need to tie in not just to the full case mapping and normalization tables, the DUCET itself, the Default Unicode Collation Element Table, and even the CLDR data (see Bibliography):

#!/usr/bin/perl
use utf8;
use open qw(:utf8 :std);
use Unicode::Collate::Locale;

my $Collator = Unicode::Collate::Locale->new(
    locale        => "de__phonebook",
    level         => 1,
    normalization => undef,
);

my $full = "Ich müß Perl studieren.";
my $sub  = "MUESS";
if (my ($pos,$len) = $Collator->index($full, $sub)) {
    my $match = substr($full, $pos, $len);
    print "Found match of literal ‹$sub› at position $pos in ‹$full› as ‹$match›\n";
}

If you run that, you will discover that it indeed works:

Found match of literal ‹MUESS› at position 4 in ‹Ich müß Perl studieren.› as ‹müß›

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Selected Bibliography

Most of these examples were taken from the 4^th edition of Programming Perl by kind permission of its author. :) I write quite a bit about such Unicode matters there, stuff that is not specific to Perl but general to Unicode overall.

The unichars(1) program that allows me to gather statistics like these:

$ unichars 'length fc == 2' | wc -l
      88

$ unichars 'length NFKD == 4' | wc -l
     109

$ unichars '/ss/i'
U+00DF ‭ ß  LATIN SMALL LETTER SHARP S
U+1E9E ‭ ẞ  LATIN CAPITAL LETTER SHARP S

Is part of the Unicode::Tussle CPAN module suite that Brian Foy has been kind enough to maintain for me.

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For further reading

See also:

• The Unicode Case Folding Table

• Unicode Standard Annex #15: Unicode Normalization Forms

• Unicode Technical Standard #18: Unicode Regular Expressions

• Unicode Technical Standard #10: Unicode Collation Algorithm

• CLDR - Unicode Common Locale Data Repository