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A regular expression (abbreviated as regexp, regex or regxp) 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 has a powerful regular expression engine built directly into its 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.
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A regular expression, often called a pattern, is an expression that describes a set of strings without actually listing its elements. For example, the three strings Handel, Händel, and Haendel are described by the pattern "H(a|ä|ae)ndel". Most formalisms provide the following constructions, which are introduced here by example.
The constructions can be freely combined within the same expression, so that "H(ae?|ä)ndel" is the same as "H(a|ae|ä)ndel".
The precise syntax for regular expressions varies among tools and application areas, and is described in more detail below.
The origin 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, then into the Unix editor ed and eventually into grep. 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 regular expressions were derived from regex written by Henry Spencer. This in turn evolved into pcre (http://www.pcre.org/) (Perl Compatible Regular Expressions), a library built by Philip Hazel used by many modern tools. The integration of regular expressions in computer languages is still very poor and part of the effort in the design of the future Perl6 (http://dev.perl.org/perl6) is this very integration. This is the subject of Apocalypse 5 (http://dev.perl.org/perl6/apocalypse/A05.html).
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:
and the following operations:
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:
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. 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.
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.
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. |
| * |
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| \{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. |
Old versions of grep did not support the alternation operator "|".
Examples:
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:
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:
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", modern regexes give an expressive power that far exceeds the regular languages.
For example, the ability to group supexpressions with brackets and recall them in the same expression means that a regex can match strings of repeated words like "papa" or "WikiWiki". The regex for these strings is just "(.*)\1". However, the set of these strings viewed as a formal language is not regular, nor is it context-free. Regex matching with an unbounded number of back references, as supported by a number of modern tools, is NP-complete.
This leads to a confusing nomenclature where the term "regular expression" in formal language theory no longer agrees with the term "regular expression" in pattern matching. It has been suggested to use the term "regex" or just "pattern" for the latter.
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(nm). This algorithm is often referred to as DFA. It is fast, but can be used only for matching and not for recalling grouped subexpressions.
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 many 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. Even though backtracking implementations give no running time guarantee in the worst case, they allow much greater flexibility and provide more expressive power.
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.
Regular expressions with Perl examples
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