The '''regular expression''' notation was invented by Kleene in the 1950's for
specifying a regular language.  This page is about regular expressions in
general, not necessarily Tcl [regular Expressions] in Tcl.



** Resources **

   [http://regexplorer.sourceforge.net/%|%RegExplorer]:   



** Description **

A '''regular expression''' uses a particular notation to describe a
''pattern'' that a string of characters might match.

Some algorithms are easier coded without regular expressions.  Tcl's [string]
command is versatile, and often simplifies problems many programmers hit with
the RE hammer.

To begin with, here are some definitions:

   '''atom''':   a grouping of 0 or more characters which are to be treated as one entity by subsequent metacharacters, etc.  If more than one character is to be included in an atom, then some sort of grouping syntax is needed.  This is by analogy with the physical meaning of ''atom'':  an indivisible building block that underlies all other structures.  Each atom is an individual regular expression, and can also be composed into larger expressions.

   '''bracket expression''':   also called a character class, is a set of characters enlosed in brackets.  It matches any of the characters in the brackets.  For example, `[[a-z]` matches can be any lower case alphabetic character from `a` to `z`.  `[[^a-z]` matches any ASCII character ''except'' a lower case alphabetic character.  The bracket expression can contain various ranges of characters.

   '''branch''':   `ABC|DEF` matches either `ABC` or `DEF`.

   '''escape''':   an escape is one of a small set of characters preceeded by \

   '''metasyntax''':   a sequence of `(?`metachars`)` where ''metachars'' is one or more alphabetic characters affecting the rest of the regular expressions.  ''[[metasyntax is one place where a variety of examples would be useful!]]''

   '''quantifier''':   A metacharacter appended to a regular expression atom to indicate the number of times that atom may exist.


A regular expression atom is made of either a '''literal character''' or a
'''metacharacter'''.

A literal character is the simplest regular expression possible.  For example,
`a` is a one-character regular expression.  It can be used to match a portion
of any string which contains the letter `a`.  Search for a pattern matching the
regular expression `a` in the string `abc`, and you get a match.  Search for a
pattern matching the regular expression `a` in the string `xyz` and you do not
get a match.

A metacharacter, also known as an '''interpreted character''', is intepreted as
some non-literal pattern.  For instance, the metacharacter `.` means 'match any
character'.  Search for the regular expression `.` in any string one character
or longer and you get a match.

Another metacharacter is the `\` (backslash).  which indicates that the next
character should be treated literally, even if it would normally be a
metacharacter.  This comes in most helpfully when attempting to describe
patterns containing metacharacters.


A regular expression looks like a summary of the various forms that the set of
strings it describes may take.  `|` usually delimits  alternative branches of
the expression. `*` (the ''Kleene star'') indicates zero or more occurances of
the previous character, and use parentheses to group sub-expressions. All of
these contsructs appear in the following example:

======none
A((b|cc)a)*
======

The following strings are from the infinite set of matches for this expression:

======none
A Aba Acca Ababa Abacca Accaba Accacca Abababa
======

An important factor for the popularity of regular expressions is their
linear-time complexity: when a given string is to be matched against a regular
expression, it is possible to do it so that every character in the string is
only looked at once. This is attained by compiling the regular expression into
a [finite automaton] — potentially a big chunk of work, but one that only needs
to be done once for each regular expression — and then running the automaton
with the string as input.

Another important factor is that regular expressions can be used for
efficiently ''searching'' through a large body of text. A direct implementation
of the above would produce an algorithm for ''matching'' a string against a
regular expression, but most RE implementations, including [[`[regexp]`], play
a few tricks internally that make them operate in search mode instead. In order
to get matching behaviour (often useful with [switch] -regexp), one uses the
''anchors'' `^` and `$` to require that the particular position in the regular
expression must correspond to the beginning and end of the string respectively
(caveat: sometimes it is beginning and end of line instead; [ARE]s have `\A`
and `\Z` as alternatives).

Other common extensions to the regular expression syntax, which however doesn't
make them any more powerful than the basic set described above, are:

   One-or-more quantifier (`+`):   Similar to `*`, but excluding the case of no repetition. The RE “`(`''re''`)+`” is equivalent to “`(`''re''`)(`''re''`)*`” and “`(`''re''`)*(`''re''`)`”.

   Optional quantifier (`?`):   Also called the zero-or-one quantifier. The RE “`(`''re''`)?`” is equivalent to “`(`''re''`|)`”.

   Bracket expression (`[[`''chars''`]`):   Effectively a shorthand — e.g. `[[abcd]` is equivalent to `(a|b|c|d)` — but often far more compact. Unicode character classes often include thousands of character, so enumerating all of them would be unfeasible.

   Counted quantifiers (`{`''n''`}` or `{`''m''`,`''n''`}`):   Exactly ''n'' occurrencies, or at least ''m'' and at most ''n'' occurrencies, respectively. Can as `?` be reduced to combinations of parentheses and `|`, but requires repeating the core RE the quantifier is applied to at least ''n'' times.

   AND:   Boolean "and", like `|` is boolean "or". Uncommon in regexp engines, and not easily reduced to the fundamental operations, but nonetheless the intersection of two regular languages is again a regular language. The [grammar_fa] package uses `&` to denote this.

   NOT:   Boolean negation. Uncommon in regexp engines, and not easily reduced to the fundamental operations, but nonetheless the complement of a regular language is also a regular language. The [grammar_fa] package uses `!` to denote this.

   Reversal:   Change the direction in which the string is being matched; this may be useful to implement searching backwards in a text editor. There is no common syntax for this, but by hand transforming a regular expression accordingly is typically straightforward.

A somewhat intriguing class of such extensions are the ''constraints'', which
only match the empty string but refuse to do so unless the material surrounding
the position of this match satisfies some condition. Here we find:

   Positive lookahead (`(?=`''re''`)`):   The regular expression ''re'' must match the text that follows after this position. This is similar to AND, but different in that ''re1'' and ''re2'' in `((?=`''re1''`)`''re2''`)` are not required to match the same characters; ''re1'' can match a prefix of what ''re2'' matches, or vice versa.

   Negative lookhead (`(?!`''re''`)`):   The regular expression ''re'' must ''not'' match the text that follows after this position.

   Positive lookbehind (`(?<=`''re''`)`):   The regular expression ''re'' must match the text that comes before this position. This is not available in [ARE]s.

   Negative lookbehind (`(?<!`''re''`)`):   The regular expression ''re'' must not match the text that comes before this position. This is not available in [ARE]s.

   Beginning of word (`\m`), end of word (`\M`):   These are obvious combinations of lookhead and lookbehind constraints, where one checks e.g. that the next character is a word character and the previous character was not.

   Beginning or end of word (`\y`), not beginning or end of word (`\Y`):   Slightly more (but only marginally so) complicated combinations of lookahead and lookbehind.

   Beginning of line (`^`), end of line (`$`):   Similar to beginning and end of word.

The Tcl regexp engine handles lookaheads by compiling and running the
constraint RE separately; in this way it is a "hybrid" RE engine.

Another set of usual extensions to the syntax concern submatch extraction and
greediness. These mainly become meaningful when regular expressions are used
for ''searching'', as there in that case often are several substrings that
match a particular RE, and it matters what match is reported.

   Non-greedy quantifiers:   Conventionally one lets quantifiers default to being greedy (match as much as possible), and introduce non-greedy quantifiers (match as little as possible) as variants; usual syntax is that a greedy quantifier followed by a `?` becomes the corresponding non-greedy quantifier. Implementation-wise, the algorithm for non-greedy searching is easier than that for greedy searching.

   Noncapturing parentheses:   Usual notation is `(?:`''re''`)`. The same as ordinary parentheses for searching and matching purposes, but tells the RE engine that it doesn't have to keep track of what range of a match corresponds to this parenthesis. Again, it is more work to keep track of this than to ignore it, but tradition dictates that parentheses should default to being capturing.

Many "regular expression" engines also support extensions to the syntax which
allow them to go ''beyond'' the realm of regular languages. This is most common
in backtracking RE engines (such as [PCRE] and the [Perl] engine) which ignores
the finite automaton theory and instead uses trial and error to find a match,
but some have escaped into more general standards.

   Back references (`\`''n''):   Matches the exact substring matched by the ''n''th capturing parenthesis in the RE. This is labelled '''the Feature from the Black Lagoon''' in the sources for the Tcl regexp engine. The (only?) non-regular feature found in the regular expressions of the [POSIX] standard. Practical applications include parsing data where one can introduce custom separator strings.

    :   To see the power of back references, one may observe that they can be used to recognise [primes]. `^(oo+?)\1+$` will match a string of ''n'' `o`'s if and only if ''n'' is a product ''ab'' for integers ''a,b'' ≥ 2; ''a'' is the length of submatch 1 and ''b'' is the number of times it is repeated.

   Recursive match (`\R`) and subroutines:   Behave as if a copy of the whole regular expression, or of some "named subexpression", occurred at this point when trying to match. Together with `|`, this is all one needs for a general context-free grammar [http://en.wikipedia.org/wiki/Context-free_grammar] (although the syntax is typically hideous compared to the standard [BNF] notation for these), which conversely means any engine capable of these features have to be ''at least as slow'' (roughly time cubic in input size) as a context-free language parser when handling these, and quite likely far slower for the worst cases. The Tcl regexp engine does not have these features.



** How Regular Expression Features are Implemented using Finite Automata **

This is a partial list of tricks. It also assumes some familiarity with finite
automata theory, such as knowing what distinguishes an NFA from a DFA, how one
runs them, and how one can construct one from the other (all of which is
standard material in relevant computer science courses).



***Search mode regexps***

Given a match-mode regexp engine, as one would get from running a finite
automaton over a string and inspect whether the end state is final, one can run
a regular expression ''re'' in search mode on it by running `.*(`''re''`).*` in
match mode.

[[Also explain what one can do to ''find'' the match searched for, without
going for a full-blown submatch-capable engine.]]


*** Submatch capturing ***

As usually defined, finite automata can only answer "yes" or "no", so there's
no way to get submatch information out of them. 

An extension of the formalism (keeping track of positions within the string
corresponding to positions in the regexp, as well as the basic automaton state)
can be found in [http://laurikari.net/ville/spire2000-tnfa.ps%|%NFAs with
Tagged Transitions, their Conversion to Deterministic Automata and Application
to Regular Expressions]


*** Boolean AND / language intersection ***

This is a classic trick.

Given one (ε-free) automaton ''A1'' for matching the regular expression ''re1''
and another (ε-free) automaton ''A2'' for matching the regular expression
''re2'', it is straightforward to construct an automaton ''A'' for ''re1'' AND
''re2'' as follows:

   1. If the state set of ''A1'' is ''S1'' and the state set of ''A2'' is ''S2'', then the state set of ''A'' will be ''S1'' × ''S2'' ([cartesian product] of the two sets).
   1. There is a transition from (''u1'',''u2'') to (''v1'',''v2'') labelled ''x'' in ''A'' if and only if there is a transition from ''u1'' to ''v1'' labelled ''x'' in ''A1'' and a transition from ''u2'' to ''v2'' labelled ''x'' in ''A2''.
   1. A state (''u'',''v'') is final in ''A'' if and only if ''u'' is final in ''A1'' and ''v'' is final in ''A2''.
   1. Similarly, a state (''u'',''v'') is initial in ''A'' if and only if ''u'' is initial in ''A1'' and ''v'' is initial in ''A2''.

What this means in practice is that running ''A'' is equivalent to running
''A1'' and ''A2'' simultaneously; each ''A'' state is a pair of an ''A1'' state
and an ''A2'' state. ''A'' accepts a string only if both ''A1'' and ''A2''
would do so.


*** Boolean NOT / language complementation ***

Assuming you've got a DFA for matching a regular language, this is very
straightforward: negate the "final" status for every state. States that were
previously accepting will then be non-accepting, and vice versa, so strings
that were previously accepted will now be rejected (and vice versa).


*** Lookahead/-behind constraints ***

The basic idea is the same as for boolean AND. First make an automaton for the
main RE, then make so many copies of this that you can simultaneously keep
track of the state visavi the man RE and the constraint RE, while adjusting the
transitions suitably so that you run both in parallel; the projection onto
either axis will however still be automata for the main and constraint
respectively RE. Finally modify the sets of initial and final states
appropriately for the wanted result.

Basically, the projection of the constraint onto the main RE "axis" would be an
ε-transition, but it is not parallel to that axis. Rather, it will take you
from being debt-free (having all instances of the constraint satisfied, and the
string as a whole thus eligible for acceptance) to a debt of 1 match
(preventing the string from being accepted), and in order to become free of
this debt you will have to work the automaton until reaching a
constraint-accepting state again (kind of like an old platform game, where one
might fall down a chute into the underground and then painfully having to work
one's way up to the surface again before being allowed to finish the game).

What makes this more complicated than the boolean AND is that several instances
of the constraint may be relevant at the same time. Consider for example

======none
((?=(...)?a)[abc])*
======

where each iteration of the outer parenthesis eats one letter, but the
constraint is looking for an `a` up to four characters ahead. It seems the only
way to manage that is to run all possibilities in parallel, which means the
"constraint axis" of the composite automaton is labelled not by states of an
automaton for matching the constraint RE, but by ''sets of such states'' (like
in the subset construction — no wonder the Tcl engine prefers to go hybrid
here)!

Also, that construction only takes care of one constraint at a time. In order to remove all constraints, it would be necessary to repeat it as many times as there are constraints in the expression! Luckily, constraint REs encountered in practice (such as those for beginning-of-word, look one character back and forth) tend to require only a very small number of states, so it is in fact feasible to use them, even with a pure automaton engine.



** Regular expression flavours **

[[Old discussion, could do with refactoring.]]

Okay then - feel free to add information here on the other RE flavors available
in Tcl...

[LES] says that there no other RE flavors available in Tcl. Tcl only uses
[ARE]. What I meant is that regular expressions may be construed as any one (or
all) of its several variations, but [Regular Expressions] only discusses Tcl's
[ARE]. I said that because this wiki discusses many things under several
contexts, not necessarily that of Tcl, and I thought it would be good to note
that, at least in this case, '''it is''' restricted to the context of Tcl.
Anyone interested in a different or more ample discussion of Regular
Expressions will have to look elsewhere. E.g. on [PCRE].

** See Also **

   [Regular Expressions]:   describe's Tcl's regular expressions ([ARE%|%advanced regular expressions])

   [http://pubs.opengroup.org/onlinepubs/9699919799/basedefs/V1_chap09.html%|%The Open Group Base Specifications Issue 7, Regular Expressions]:   AKA POSIX 

   [http://www.regular-expressions.info/%|%regularexpressions.info]:   lots of general information, and a thorough tutorial with examples ([CL]: although its imprecisions exasperate).  See the [http://regular-expressions.info/examples.html%|%examples] for an extensive listing of regulare expressions for various tasks.


   [http://en.wikipedia.org/wiki/Regular_language%|%Regular Language, Wikipedia]:   a formal definitions of "regular language".

   [BOOK Mastering Regular Expressions], by [Jeffrey E. F. Friedl]:   considered almost the definitive tome on regular expressions, from the Unix '''grep'''(1) command to Tcl and beyond.

   [http://blog.staffannoteberg.com/2013/01/30/regular-expressions-a-brief-history/%|%Regular Expressions - a brief history], by Staffan Nöteberg, 2013-01-30:   

   [http://www2.imm.dtu.dk/~phbi/files/talks/2012remhsacS.pdf%|%Regular Expression Matching: History, Status, and Challenges], Philip Bille:   

   [http://web.archive.org/web/20120704232743/http://zez.org/article/articleprint/11%|%Regular Expressions explained], Jan Borsodi, 2000-10-30:   

   [http://www.onlamp.com/pub/a/onlamp/2003/08/21/regexp.html%|%Five Habits for Successful Regular Expressions], Tony Stubblebine, 2003-08-21:   

   [http://www.ibm.com/developerworks/aix/library/au-regexp/%|%Know your regular expressions], Michael Stutz, 2007-06-14:   features a flexible RE generator

   [http://www.regexlib.com/%|%regexlib.com]:   a community maintained library of regular expressions




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