2 Regular expressions in formal language theory

Regular expressions consist of constants and operators that denote sets of strings and operations over these sets, respectively. Given a finite alphabet Σ the following constants are defined:

1. (empty set) ∅ denoting the set ∅
2. (empty string) ε denoting the set {ε}
3. ( literal character) a in Σ denoting the set {"a"}

and the following operations:

1. (concatenation) RS denoting the set { αβ | α in R and β in S }. For example {"ab", "c"}{"d", "ef"} = {"abd", "abef", "cd", "cef"}.
2. (set union) R U S denoting the set union of R and S.
3. ( Kleene star) R* denoting the smallest superset of R that contains ε and is closed under string concatenation. This is the set of all strings that can be made by concatenating zero or more strings in R. For example, {"ab", "c"}* = {ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", ... }.

To avoid brackets it is assumed that the Kleene star has the highest priority, then concatenation and then set union. If there is no ambiguity then brackets may be omitted. For example, (ab)c is written as abc and a U (b(c*)) can be written as a U bc*.

Sometimes the complement operator ~ is added; ~R denotes the set of all strings over Σ that are not in R. In that case the resulting operators form a Kleene algebra. The complement operator is redundant: it can always be expressed by only using the other operators.

Examples:

• a U b* denotes {"a", ε, "b", "bb", "bbb", ...}
• (a U b)* denotes the set of all strings consisting of 'a's and 'b's, including the empty string
• b*(ab*)* the same
• ab*(c U ε) denotes the set of strings starting with 'a', then zero or more 'b's and finally optionally a 'c'.
• ((a U ba(aa)*b)(b(aa)*b)*(a U ba(aa)*b) U b(aa)*b)*(a U ba(aa)*b)(b(aa)*b)* denotes the set of all strings which contain an even number of 'b's and an odd number of 'a's. Note that this regular expression is of the form (X Y*X U Y)*X Y* with X = a U ba(aa)*b and Y = b(aa)*b.

Regular expressions in this sense can express exactly the class of languages accepted by finite state automata: the regular languages. There is, however, a significant difference in compactness: some classes of regular languages can only be described by automata that grow exponentially in size, while the required regular expressions only grow linearly. Regular expressions correspond to the type 3 grammars of the Chomsky hierarchy and may be used to describe a regular language.

We can also study expressive power within the formalism. As the example shows, different regular expressions can express the same language: the formalism is redundant.

It is possible to write an algorithm which for two given regular expressions decides whether the described languages are equal - essentially, it reduces each expression to a minimal deterministic finite state automaton and determines whether they are isomorphic (equivalent).

To what extent can this redundancy be eliminated? Can we find an interesting subset of regular expressions that is still fully expressive? Kleene star and set union are obviously required, but perhaps we can restrict their use. This turns out to be a surprisingly difficult problem. As simple as the regular expressions are, it turns out there is no method to systematically rewrite them to some normal form. They are not finitely axiomatizable. So we have to resort to other methods. This leads to the star height problem.

3 Regular expression syntaxes

3.1 Traditional Unix regexps

The "basic" Unix regexp syntax is now defined as obsolete by POSIX, but is still widely used for the purposes of backwards compatibility. Most Unix utilities (grep, sed...) use it by default.

In this syntax, most characters are treated as literals---they match only themselves ("a" matches "a", "(bc" matches "(bc", etc). The exceptions are called metacharacters:

.	Matches any single character
[ ]	Matches a single character that is contained within the brackets. For example, [abc] matches "a", "b", or "c". [a-z] matches any lowercase letter. These can be mixed: [abcq-z] matches a, b, c, q, r, s, t, u, v, w, x, y, z, and so does [a-cq-z].
[^ ]	Matches a single character that is not contained within the brackets. For example, [^abc] matches any character other than "a", "b", or "c". [^a-z] matches any single character that isn't a lowercase letter. As above, these can be mixed.
^	Matches the start of the line (or any line, when applied in multiline mode)
$	Matches the end of the line (or any line, when applied in multiline mode)
\( \)	Define a "marked subexpression". What the enclosed expression matched can be recalled later. See the next entry, \n. Note that a "marked subexpression" is also a "block"
\n	Where n is a digit from 1 to 9; matches what the nth marked subexpression matched. This construct is theoretically irregular and has not been adopted in the extended regular expression syntax.
*	A single character expression followed by "*" matches zero or more copies of the expression. For example, "[xyz]*" matches "", "x", "y", "zx", "zyx", and so on. \n*, where n is a digit from 1 to 9, matches zero or more iterations of what the nth marked subexpression matched. For example, "\(a?\)c\1*" matches "abcab" and "abcaba" but not "abcac". An expression enclosed in "\(" and "\)" followed by "*" is deemed to be invalid. In some cases (e.g. /usr/bin/xpg4/grep of SunOS 5.8), it matches zero or more iterations of the string that the enclosed expression matches. In other cases (e.g. /usr/bin/grep of SunOS 5.8), it matches what the enclosed expression matches, followed by a literal "*".
\{x,y\}	Match the last "block" at least x and not more than y times. For example, "a\{3,5\}" matches "aaa", "aaaa" or "aaaaa". Note that this is not found in some instances of regex.

There is no representation of the set union operator in this syntax.

Examples:

".at" matches any three-character string ending with "at"

"[hc]at" matches "hat" and "cat"

"[^b]at" matches any three-character string ending with "at" and not beginning with 'b'.

"^[hc]at" matches "hat" and "cat" but only at the beginning of a line

"[hc]at$" matches "hat" and "cat" but only at the end of a line

3.2 POSIX modern (extended) regexps

The more modern "extended" regexp can often be used with modern Unix utilities by including the command line flag "-E".

POSIX extended regexps are similar in syntax to the traditional Unix regexp, with some exceptions. The following metacharacters are added:

+	Match the last "block" one or more times - "ba+" matches "ba", "baa", "baaa" and so on
?	Match the last "block" zero or one times - "ba?" matches "b" or "ba"
|	The choice (or set union) operator: match either the expression before or the expression after the operator - "abc|def" matches "abc" or "def".

Also, backslashes are removed: \{...\} becomes {...} and \(...\) becomes (...)

Examples:

"[hc]+at" matches with "hat", "cat", "hhat", "chat", "hcat", "ccat" et cetera

"[hc]?at" matches "hat", "cat" and "at"

"([cC]at | [dD]og)" matches "cat", "Cat", "dog" and "Dog"

Since the characters '(', ')', '[', ']', '.', '*', '?', '+', '^' and '$' are used as special symbols they have to be "escaped" somehow if they are meant literally. This is done by preceding them with '\' which therefore also has to be "escaped" this way if meant literally.

Examples:

".\.(\(|\))" matches with the string "a.)"

3.3 Perl-compatible regular expressions (PCRE)

Perl has a much richer syntax than even the extended POSIX regexp. In addition, its syntax is somewhat more predictable

(for example, a backstroke always quotes a non-alphanumeric character). For these reasons, the Perl syntax has been adopted in other utilities and applications -- Tcl and exim, for example. Although still named "regular expressions", the Perl extensions give an expressive power that far exceeds the regular languages.

This extra power comes at a cost. The worst-case complexity of matching a string against a Perl regular expression is exponential in the size of the input. That means the regular expression could take an extremely long time to process under just the right conditions of expression and input string, although it rarely happens in practice.

4 See also

Regular expressions with Perl examples

5 References

• Jeffrey Friedl : Mastering Regular Expressions, O'Reilly, BooksEnthsiast.com
• Tony Stubblebine : Regular Expression, Pocket Reference, O'Reilly, BooksEnthsiast.com
• Ben Forta : Sams Teach Yourself Regular Expressions in 10 Minutes, Sams, BooksEnthsiast.com
• Mehran Habibi : Real World Regular Expressions with Java 1.4, Springer, BooksEnthsiast.com
• Francois Liger , Craig McQueen , Paul Wilton : Visual Basic .NET Text Manipulation Handbook, Wrox Press, BooksEnthsiast.com
• Michael Sipser : Introduction to the Theory of Computation, PWS Publishing Company, BooksEnthsiast.com

6 External links

• DMOZ listing
• Articles/Archives:
  • Regexp Syntax Summary, reference table for different styles of regular expressions
  • online regular expression archive
  • Regular expression information
  • Syntax and Semantics of Regular Expressions by Xerox Research Centre Europe
  • Learning to Use Regular Expressions by David Mertz
  • Mastering Regular Expressions book website
  • Regenechsen Beginners tutorial for using Regular Expressions
  • An incomplete history of the QED text editor
  • xpressive
  • The GRETA Regular Expression Template Archive
• Tools:
  • Regular expression .NET online tester
  • The Regex Coach, a very powerful interactive learning system to hone your regexp skills
  • The Regulator: Advanced and free Regex testing tool
  • RegexBuilder Simple .NET tool to test regular expressions
  • Tcl Regular Expression Visualiser
  • JRegexpTester, free java regexp testing tool.

Formal languages Mathematical logic Computer science

<Hide>