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Regular expression

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A regular expression (abbreviated as regexp, regex, or regxp, with plural forms regexps, regexes, or regexen) is a string that describes or matches a set of strings, according to certain syntax rules. Regular expressions are used by many text editors and utilities to search and manipulate bodies of text based on certain patterns. Many programming languages support regular expressions for string manipulation. For example, Perl and Tcl have a powerful regular expression engine built directly into their syntax. The set of utilities (including the editor sed and the filter grep) provided by Unix distributions were the first to popularize the concept of regular expressions.

Basic concepts

A regular expression, often called a pattern, is an expression that describes a set of strings. They are usually used to give a concise description of a set, without having to list all elements. For example, the set containing the three strings Handel, Händel, and Haendel can be described by the pattern "H(ä|ae?)ndel" (or alternatively, it is said that the pattern matches each of the three strings). As a side note, there are usually multiple different patterns describing any given set. Most formalisms provide the following operations to construct regular expressions.

alternation
A vertical bar separates alternatives. For example, "gray|grey" matches gray or grey.
grouping
Parentheses are used to define the scope and precedence of the operators. For example, "gray|grey" and "gr(a|e)y" are different patterns, but they both describe the set containing gray and grey.
quantification
A quantifier after a character or group specifies how often that preceding expression is allowed to occur. The most common quantifiers are ?, *, and +:
?
The question mark indicates there is 0 or 1 of the previous expression. For example, "colou?r" matches both color and colour.
*
The asterisk indicates there are 0, 1 or any number of the previous expression. For example, "go*gle" matches ggle, gogle, google, etc.
+
The plus sign indicates that there is at least 1 of the previous expression. For example, "go+gle" matches gogle, google, etc. (but not ggle).

These constructions can be combined to form arbitrarily complex expressions, very much like one can construct arithmetical expressions from the numbers and the operations +, -, * and /.

So "H(ae?|ä)ndel" and "H(a|ae|ä)ndel" are valid patterns, and furthermore, they both match the same strings as the example from the beginning of the article. The pattern "((great )*grand )?(father|mother)" matches any ancestor: father, mother, grand father, grand mother, great grand father, great grand mother, great great grand father, great great grand mother, great great great grand father, great great great grand mother and so on.

The precise syntax for regular expressions varies among tools and application areas; more detail is given in the Syntax section.

History

The origins of regular expressions lies in automata theory and formal language theory (both part of theoretical computer science). These fields study models of computation (automata) and ways to describe and classify formal languages. In the 1940s, Warren McCulloch and Walter Pitts described the nervous system by modelling neurons as small simple automata. The mathematician Stephen Kleene later described these models using his mathematical notation called regular sets. Ken Thompson built this notation into the editor QED, and then into the Unix editor ed, which eventually led to grep's use of regular expressions. Ever since that time, regular expressions have been widely used in Unix and Unix-like utilities such as: expr, awk, Emacs, vi, lex, and Perl.

Perl and Tcl regular expressions were derived from regex written by Henry Spencer. Philip Hazel developed pcre (Perl Compatible Regular Expressions) which attempts to closely mimic Perl's regular expression functionality, and is used by many modern tools such as PHP, and Apache.

The integration of regular expressions in most computer languages is still very poor and, even though Perl's regular expression integration is one of the best around, part of the effort in the design of the future Perl6 is improving this integration. This is the subject of Apocalypse 5.

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:

  • (empty set) ∅ denoting the set ∅
  • (empty string) ε denoting the set {ε}
  • (literal character) a in Σ denoting the set {a}

and the following operations:

  • (concatenation) RS denoting the set { αβ | α in R and β in S }. For example {"ab", "c"}{"d", "ef"} = {"abd", "abef", "cd", "cef"}.
  • (alternation) R|S denoting the set union of R and S.
  • (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", ... }.

The above constants and operators form a Kleene algebra.

Many textbooks use the symbols ∪, +, or ∨ for alternation instead of the vertical bar.

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|(b(c*)) can be written as a|bc*.

Examples:

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

The formal definition of regular expressions is purposefully parsimonious and avoids defining the redundant quantifiers ? and +, which can be expressed as follows: a+= aa*, and a? = (ε|a). Sometimes the complement operator ~ is added; ~R denotes the set of all strings over Σ that are not in R. The complement operator is redundant: it can always be expressed by only using the other operators.

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 length of 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 essentially equal, 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.

It is worth noting that many real-world "regular expression" engines implement features that cannot be expressed in the regular expression algebra; see below for more on this.

Syntax

Traditional Unix regular expressions

The "basic" Unix regular expression syntax is now defined as obsolete by POSIX, but is still widely used for the purposes of backwards compatibility. Most regular-expression–aware Unix utilities, for example grep and 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].

The '-' character should be literal only if it is the last or the first character within the brackets: [abc-] or [-abc]. To match an '[' or ']' character, the easiest way is to make sure the closing bracket is first in the enclosing square brackets: [][ab] matches ']', '[', 'a' or 'b'.

[^ ] 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 is not 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.

Note that particular implementations of regular expressions interpret backslash differently in front of some of the metacharacters. For example, egrep and Perl interpret unbackslashed parentheses and vertical bars as metacharacters, reserving the backslashed versions to mean the literal characters themselves.

Old versions of grep did not support the alternation operator "|". Examples:

".at" matches any three-character string like hat, cat or bat
"[hc]at" matches hat and cat
"[^b]at" matches all the matched strings from the regex ".at" except bat
"^[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

Since many ranges of characters depends on the chosen locale setting (e.g., in some settings letters are organized as abc..yzABC..YZ while in some others as aAbBcC..yYzZ) the POSIX standard defines some classes or categories of characters as shown in the following table:

POSIX class similar to meaning
[:upper:] [A-Z] uppercase letters
[:lower:] [a-z] lowercase letters
[:alpha:] [A-Za-z] upper- and lowercase letters
[:alnum:] [A-Za-z0-9] digits, upper- and lowercase letters
[:digit:] [0-9] digits
[:xdigit:] [0-9A-Fa-f] hexadecimal digits
[:punct:] [.,!?:...] punctuation
[:blank:] [ \t] space and TAB
[:space:] [ \t\n\r\f\v] blank characters
[:cntrl:] control characters
[:graph:] [^ \t\n\r\f\v] printed characters
[:print:] [^\t\n\r\f\v] printed characters and space

example: [[:upper:]ab] should only return the uppercase letters and lowercase 'a' and 'b'.

(For an ASCII chart color-coded to show the POSIX classes, see http://www.billposer.org/Software/RedetManual/builtincharclass.html.)

POSIX modern (extended) regular expressions

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

POSIX extended regular expressions are similar in syntax to the traditional Unix regular expressions, 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", "ccchat" etc.
"[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 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:

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

Perl-compatible regular expressions (PCRE)

Perl has a richer but more predictable syntax than even the extended POSIX regexp. An example of its predictability is that \ always quotes a non-alphanumeric character. An example of something that you can specify with Perl but not POSIX is whether you want part of the match to be greedy or not. For instance in the pattern /a.*b/, the .* will match as much as it can, while in the pattern /a.*?b/, .*? will match as little. So given the string "a bad dab", the first pattern will match the whole string, and the second will only match "a b".

For these reasons, many other utilities and applications have adopted syntaxes that look a lot like Perl's — Python, exim, and BBEdit, for example. But not all claims to implement Perl regexps are honest: for instance, Tcl tries to both follow the POSIX specification and implement Perl's extensions. Unfortunately this means that Tcl's definition of a non-greedy match is, "Match as little as you can here while still having the overall match be as long as possible." So in the above example, both patterns will match the whole string.

Patterns for irregular languages

Many patterns provide an expressive power that far exceeds the regular languages. For example, the ability to group subexpressions with brackets and recall them in the same expression means that a pattern can match strings of repeated words like "papa" or "WikiWiki", called squares in formal language theory. The pattern for these strings is just "(.*)\1". However, the language of squares is not regular, nor is it context-free. Pattern matching with an unbounded number of back references, as supported by a number of modern tools, is NP-complete.

However, many tools, libraries, and engines that provide such constructions still use the term regular expression for their patterns. This has lead to a nomenclature where the term "regular expression" has different meanings in formal language theory and pattern matching. It has been suggested to use the term regex or simply "pattern" for the latter. Larry Wall (author of Perl) writes in Apocalypse 5:

"'[R]egular expressions' […] are only marginally related to real regular expressions. Nevertheless, the term has grown with the capabilities of our pattern matching engines, so I'm not going to try to fight linguistic necessity here. I will, however, generally call them "regexes" (or "regexen", when I'm in an Anglo-Saxon mood)."

Implementations and running times

There are at least two different algorithms that decide if (and how) a given string matches a regular expression.

The oldest and fastest relies on a result in formal language theory that allows every Nondeterministic Finite State Machine (NFA) to be transformed into a deterministic finite state machine (DFA). The algorithm performs or simulates this transformation and then runs the resulting DFA on the input string, one symbol at a time. The latter process takes time linear in the length of the input string. More precisely, an input string of size n can be tested against a regular expression of size m in time O(n+2m) or O(nm), depending on the details of the implementation. This algorithm is often referred to as DFA. It is fast, but can be used only for matching and not for recalling grouped subexpressions. There is a variant that can recall grouped subexpressions, but its running time slows down to O(n2m).

The other algorithm is to match the pattern against the input string by backtracking. (This algorithm is sometimes called NFA, but this terminology is highly confusing.) Its running time can be exponential, which simple implementations exhibit when matching against expressions like "(a|aa)*b" that contain both alternation and unbounded quantification and force the algorithm to consider an exponential number of subcases. More complex implementations identify and speed up various common cases where they would otherwise run slowly.

Even though backtracking implementations only give an exponential guarantee in the worst case, they allow much greater flexibility and provide more expressive power. For instance any implementation that allows the use of backreferences, or implements the various improvements that Perl introduced, must use a backtracking implementation.

Some implementations try to provide the best of both algorithms by first running a fast DFA match to see if the string matches the regular expression at all, and only in that case perform a potentially slower backtracking match.

See also

References

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Articles

Tools

  • Crypter JavaScript-based RegExp tool
  • Expresso A free Regular Expression integrated development environment for developing, testing, and coding .NET regular expressions. Also includes a built-in tutorial and regular expression library.
  • JRegexpTester Free Java regexp testing tool.
  • JRX Web page using JavaScript to test regular expressions without a page reload
  • Kiki A front end to the Python re module for testing regular expressions against a sample text that provides extensive output about the results, including highlighting of groups within a match.
  • Kodos Free visual regular expression debugger.
  • KRegExpEditor open source KDE widget for visually creating regular expressions
  • LogBud Online regular expression testing for PHP
  • PCRE Workbench Simple regular expression tester for Windows, based on PCRE.
  • PowerGREP Feature-rich Windows application to search, search-and-replace and collect data using one or more regular expressions.
  • Redet A tool for creating and executing regular expressions in Tcl or any of 30+ other programs. Provides palettes constructed by executing a test suite, persistent history, ability to change locale while running, character entry tools and extensive help. Handles both matching and substitution. Fully supports Unicode. Runs on all major platforms.
  • RegexBuddy Complete tool for learning, creating, testing and saving regular expressions, with two-way conversion between regex syntax and plain English building blocks, and a debugger that lets you see inside the regex engine. Windows and Linux versions available.
  • RegexBuilder Simple .NET tool to test regular expressions.
  • The Regex Coach Powerful interactive learning system to hone your regexp skills. Windows and Linux versions available.
  • RegExSearchReplace Free tool to search-and-replace using a regular expression. Requires Java.
  • Regex.NET Tester Web page using .NET to test regular expressions online.
  • The Regulator Free regular expressions testing and learning tool. Build and verify a regular expression against any text input, file or web. Displays matching, splitting or replacement results within an easy to understand, hierarchical tree.
  • Rex V an AJAX PCRE, Posix and Javascript Regular Expression Evaluator, written in PHP.
  • Tcl Regular Expression Visualiser Tcl/Tk script for testing regular expressions.
  • TextTransformer Tool for building complete Parsers using the Boost.RegexC++ library with POSIX-Extended Syntax. A dialog to test regular expressions can be used freely. All sub-expressions and their matches are shown.
  • ^txt2regex$ Tool for building simple regular expressions from human language by answering the tool's questions. Generates regular expressions for 20 engines.
  • WebRegEx Free web-based tool to find regular expression matches in a given web page.
  • WhatsMyIP.org Regexp Tester Simple web page to test a regular expression in PHP.
  • Rad Software Regular Expression Designer is a free download that helps programmers learn, develop and test Regular Expressions. It is an interactive Windows application that is designed to be simple and easy to use. (Requires .NET Framework 1.1)

Programming libraries

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