In computing, a regular expression, also referred to as regex or regexp, provides a concise and flexible means for matching strings of text, such as particular characters, words, or patterns of characters. A regular expression is written in a formal language that can be interpreted by a regular expression processor, a program that either serves as a parser generator or examines text and identifies parts that match the provided specification.
The following examples illustrate a few specifications that could be expressed in a regular expression:
- The sequence of characters "car" appearing consecutively in any context, such as in "car", "cartoon", or "bicarbonate"
- The sequence of characters "car" occurring in that order with other characters between them, such as in "Icelander" or "chandler"
- The word "car" when it appears as an isolated word
- The word "car" when preceded by the word "blue" or "red"
- The word "car" when not preceded by the word "motor"
- A dollar sign immediately followed by one or more digits, and then optionally a period and exactly two more digits (for example, "$100" or "$245.99").
Regular expressions can be much more complex than these examples.
Regular expressions are used by many text editors, utilities, and programming languages to search and manipulate text based on patterns. Some of these languages, including Perl, Ruby, Awk, and Tcl, have fully integrated regular expressions into the syntax of the core language itself. Others like C, C++, .NET, Java, and Python instead provide access to regular expressions only through libraries. Utilities provided by Unix distributions—including the editor ed and the filter grep—were the first to popularize the concept of regular expressions.
As an example of the syntax, the regular expression
\bex can be used to search for all instances of the string "ex" that occur after "word boundaries" (signified by the \b). Thus \bex will find the matching string "ex" in two possible locations, (1) at the beginning of words, and (2) between two characters in a string, where one is a word character and the other is not a word character. For instance, in the string "Texts for experts", \bex matches the "ex" in "experts" but not in "Texts" (because the "ex" occurs inside a word and not immediately after a word boundary).Many modern computing systems provide wildcard characters in matching filenames from a file system. This is a core capability of many command-line shells and is also known as globbing. Wildcards differ from regular expressions in generally expressing only limited forms of patterns.
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). In most formalisms, if there is any regex that matches a particular set then there is an infinite number of such expressions. Most formalisms provide the following operations to construct regular expressions.- Boolean "or"
- Grouping
- Parentheses are used to define the scope and precedence of the operators (among other uses). For example,
gray|greyandgr(a|e)yare equivalent patterns which both describe the set of "gray" and "grey". - A quantifier after a token (such as a character) or group specifies how often that preceding element is allowed to occur. The most common quantifiers are the question mark
?, the asterisk*(derived from the Kleene star), and the plus sign+(Kleene cross).
?The question mark indicates there is zero or one of the preceding element. For example, colou?rmatches both "color" and "colour".*The asterisk indicates there are zero or more of the preceding element. For example, ab*cmatches "ac", "abc", "abbc", "abbbc", and so on.+The plus sign indicates that there is one or more of the preceding element. For example, ab+cmatches "abc", "abbc", "abbbc", and so on, but not "ac".
These constructions can be combined to form arbitrarily complex expressions, much like one can construct arithmetical expressions from numbers and the operations +, −, ×, and ÷. For example,
H(ae?|ä)ndel and H(a|ae|ä)ndel are both valid patterns which match the same strings as the earlier example, H(ä|ae?)ndel.The precise syntax for regular expressions varies among tools and with context; more detail is given in the Syntax section.
The origins of regular expressions lie in automata theory and formal language theory, both of which are part of theoretical computer science. These fields study models of computation (automata) and ways to describe and classify formal languages. In the 1950s, mathematician Stephen Cole Kleene described these models using his mathematical notation called regular sets.[1] The SNOBOL language was an early implementation of pattern matching, but not identical to regular expressions. Ken Thompson built Kleene's notation into the editor QED as a means to match patterns in text files. He later added this capability to the Unix editor ed, which eventually led to the popular search tool grep's use of regular expressions ("grep" is a word derived from the command for regular expression searching in the ed editor:
g/re/p where re stands for regular expression[2]). Since that time, many variations of Thompson's original adaptation of regular expressions have been widely used in Unix and Unix-like utilities including expr, AWK, Emacs, vi, and lex.Perl and Tcl regular expressions were derived from a regex library written by Henry Spencer, though Perl later expanded on Spencer's library to add many new features.[3] 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 including PHP and Apache HTTP Server. Part of the effort in the design of Perl 6 is to improve Perl's regular expression integration, and to increase their scope and capabilities to allow the definition of parsing expression grammars.[4] The result is a mini-language called Perl 6 rules, which are used to define Perl 6 grammar as well as provide a tool to programmers in the language. These rules maintain existing features of Perl 5.x regular expressions, but also allow BNF-style definition of a recursive descent parser via sub-rules.
The use of regular expressions in structured information standards for document and database modeling started in the 1960s and expanded in the 1980s when industry standards like ISO SGML (precursored by ANSI "GCA 101-1983") consolidated. The kernel of the structure specification language standards are regular expressions. Simple use is evident in the DTD element group syntax.
Regular expressions describe regular languages in formal language theory. They have thus the same expressive power as regular grammars. Regular expressions consist of constants and operators that denote sets of strings and operations over these sets, respectively. The following definition is standard, and found as such in most textbooks on formal language theory.[5][6] Given a finite alphabet Σ, the following constants are defined:
- (empty set) ∅ denoting the set ∅.
- (empty string) ε denoting the set containing only the "empty" string, which has no characters at all.
- (literal character)
ain Σ denoting the set containing only the character a.
The following operations are defined:
- (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. For example {"ab", "c"}|{"ab", "d", "ef"} = {"ab", "c", "d", "ef"}.
- (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 any finite number (including zero) of strings from R. For example, {"0","1"}* is the set of all finite binary strings (including the empty string), and {"ab", "c"}* = {ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "abcab", ... }.
To avoid parentheses it is assumed that the Kleene star has the highest priority, then concatenation and then set union. If there is no ambiguity then parentheses may be omitted. For example,
(ab)c can be written as abc, and a|(b(c*)) can be written as a|bc*. Many textbooks use the symbols ∪, +, or ∨ for alternation instead of the vertical bar.Examples:
a|b*denotes {ε, a, b, bb, bbb, ...}(a|b)*denotes the set of all strings with no symbols other than a and b, including the empty string: {ε, a, b, aa, ab, ba, bb, aaa, ...}ab*(c|ε)denotes the set of strings starting with a, then zero or more bs and finally optionally a c: {a, ac, ab, abc, abb, abbc, ...}
[edit] Expressive power and compactness
The formal definition of regular expressions is purposely parsimonious and avoids defining the redundant quantifiers
? and +, which can be expressed as follows: a+ = aa*, and a? = (a|ε). Sometimes the complement operator is added, to give a generalized regular expression; here Rc matches all strings over Σ* that do not match R. In principle, the complement operator is redundant, as it can always be circumscribed by using the other operators. However, the process for computing such a representation is complex, and the result may require expressions of a size that is double exponentially larger.[7][8]Regular expressions in this sense can express the regular languages, exactly the class of languages accepted by deterministic finite automata. There is, however, a significant difference in compactness. Some classes of regular languages can only be described by deterministic finite automata whose size grows exponentially in the size of the shortest equivalent regular expressions. The standard example are here the languages Lk consisting of all strings over the alphabet {a,b} whose kth-last letter equals a. On the one hand, a regular expression describing L4 is given by (a | b) * a(a | b)(a | b)(a | b). Generalizing this pattern to Lk gives the expression
On the other hand, it is known that every deterministic finite automaton accepting the language Lk must have at least 2k states. Luckily, there is a simple mapping from regular expressions to the more general nondeterministic finite automata (NFAs) that does not lead to such a blowup in size; for this reason NFAs are often used as alternative representations of regular languages. NFAs are a simple variation of the type-3 grammars of the Chomsky hierarchy.[5]
Finally, it is worth noting that many real-world "regular expression" engines implement features that cannot be described by the regular expressions in the sense of formal language theory; see below for more on this.
[edit] Deciding equivalence of regular expressions
As the examples show, 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, reduces each expression to a minimal deterministic finite state machine, and determines whether they are isomorphic (equivalent).
To what extent can this redundancy be eliminated? Kleene star and set union are required to find an interesting subset of regular expressions that is still fully expressive, but perhaps their use can be restricted. This is a surprisingly difficult problem. As simple as the regular expressions are, there is no method to systematically rewrite them to some normal form. The lack of axiom in the past led to the star height problem. Recently, Dexter Kozen axiomatized regular expressions with Kleene algebra.[9]A regular expression (regex or regexp for short) is a special text string for describing a search pattern. You can think of regular expressions as wildcards on steroids
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