Formal language theory
Regular expressions can be expressed in terms of 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 a string with no characters.
- (literal character) a in Σ denoting a character in the language.
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. Many textbooks use the symbols ∪, +, or ∨ for alternation instead of the vertical bar. For example {"ab", "c"}∪{"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 zero or more strings in R. For example, {"ab", "c"}* = {ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "abcab", ... }.
The above constants and operators form a Kleene algebra.
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 can be 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 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, ...}
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; ~R denotes the set of all strings over Σ* that are not in R. The complement operator is redundant, as it can always be expressed by using the other operators (although the process for computing such a representation is complex, and the result may be exponentially larger).
Regular expressions in this sense can express the regular languages, exactly the class of languages accepted by finite state automata. 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. On the other hand, there is a simple mapping from regular expressions to 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 expressions.
We can also study expressive power within the formalism. 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? 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. The lack of axiomatization in the past led to the star height problem. Recently, Cornell University professor Dexter Kozen axiomatized regular expressions with Kleene algebra.
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.
POSIX
POSIX Basic Regular Expressions
Traditional Unix regular expression syntax followed common conventions but often differed from tool to tool. The IEEE POSIX Basic Regular Expressions (BRE) standard (released alongside an alternative flavor called Extended Regular Expressions or ERE) was designed mostly for backward compatibility with the traditional (Simple Regular Expression) syntax but provided a common standard which has since been adopted as the default syntax of many Unix regular expression tools, though there is often some variation or additional features. Many such tools also provide support for ERE syntax with command line arguments.
In the BRE syntax, most characters are treated as literals — they match only themselves (i.e., a matches "a"). The exceptions, listed below, are called metacharacters or metasequences.
| Metacharacter | Description |
|---|---|
. | Matches any single character (many applications exclude newlines, and exactly which characters are considered newlines is flavor, character encoding, and platform specific, but it is safe to assume that the line feed character is included). Within POSIX bracket expressions, the dot character matches a literal dot. For example, a.c matches "abc", etc., but [a.c] matches only "a", ".", or "c". |
[ ] | A bracket expression. Matches a single character that is contained within the brackets. For example, [abc] matches "a", "b", or "c". [a-z] specifies a range which matches any lowercase letter from "a" to "z". These forms can be mixed: [abcx-z] matches "a", "b", "c", "x", "y", or "z", as does [a-cx-z]. The |
[^ ] | 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 from "a" to "z". As above, literal characters and ranges can be mixed. |
^ | Matches the starting position within the string. In line-based tools, it matches the starting position of any line. |
$ | Matches the ending position of the string or the position just before a string-ending newline. In line-based tools, it matches the ending position of any line. |
BRE: \( \)ERE: ( ) | Defines a marked subexpression. The string matched within the parentheses can be recalled later (see the next entry, \n). A marked subexpression is also called a block or capturing group. |
\n | Matches what the nth marked subexpression matched, where n is a digit from 1 to 9. This construct is theoretically irregular and was not adopted in the POSIX ERE syntax. Some tools allow referencing more than nine capturing groups. |
* | Matches the preceding element zero or more times. For example, ab*c matches "ac", "abc", "abbbc", etc. [xyz]* matches "", "x", "y", "z", "zx", "zyx", "xyzzy", and so on. \(ab\)* matches "", "ab", "abab", "ababab", and so on. |
BRE: \{m,n\}ERE: {m,n} | Matches the preceding element at least m and not more than n times. For example, a\{3,5\} matches only "aaa", "aaaa", and "aaaaa". This is not found in a few, older instances of regular expressions. |
Examples:
.atmatches any three-character string ending with "at", including "hat", "cat", and "bat".[hc]atmatches "hat" and "cat".[^b]atmatches all strings matched by.atexcept "bat".^[hc]atmatches "hat" and "cat", but only at the beginning of the string or line.[hc]at$matches "hat" and "cat", but only at the end of the string or line.
POSIX Extended Regular Expressions
The meaning of metacharacters escaped with a backslash is reversed for some characters in the POSIX Extended Regular Expression (ERE) syntax. With this syntax, a backslash causes the metacharacter to be treated as a literal character. So, for example, \( \) is now ( ) and \{ \} is now { }. Additionally, support is removed for \n backreferences and the following metacharacters are added:
| Metacharacter | Description |
|---|---|
? | Matches the preceding element zero or one time. For example, ba? matches "b" or "ba". |
+ | Matches the preceding element one or more times. For example, ba+ matches "ba", "baa", "baaa", and so on. |
| | The choice (aka alternation or set union) operator matches either the expression before or the expression after the operator. For example, abc|def matches "abc" or "def". |
Examples:
[hc]+atmatches "hat", "cat", "hhat", "chat", "hcat", "ccchat", and so on, but not "at".[hc]?atmatches "hat", "cat", and "at".cat|dogmatches "cat" or "dog".
POSIX Extended Regular Expressions can often be used with modern Unix utilities by including the command line flag -E.
POSIX character classes
Since many ranges of characters depend on the chosen locale setting (i.e., in some settings letters are organized as abc...zABC...Z, while in some others as aAbBcC...zZ), the POSIX standard defines some classes or categories of characters as shown in the following table:
| POSIX | Perl | ASCII | Description |
|---|---|---|---|
[:alnum:] | [A-Za-z0-9] | Alphanumeric characters | |
[:word:] | \w | [A-Za-z0-9_] | Alphanumeric characters plus "_" |
\W | [^\w] | non-word character | |
[:alpha:] | [A-Za-z] | Alphabetic characters | |
[:blank:] | [ \t] | Space and tab | |
[:cntrl:] | [\x00-\x1F\x7F] | Control characters | |
[:digit:] | \d | [0-9] | Digits |
\D | [^\d] | non-digit | |
[:graph:] | [\x21-\x7E] | Visible characters | |
[:lower:] | [a-z] | Lowercase letters | |
[:print:] | [\x20-\x7E] | Visible characters and spaces | |
[:punct:] | [-!"#$%&'()*+,./:;<=>?@[\\\]_`{|}~] | Punctuation characters | |
[:space:] | \s | [ \t\r\n\v\f] | Whitespace characters |
\S | [^\s] | non-whitespace character | |
[:upper:] | [A-Z] | Uppercase letters | |
[:xdigit:] | [A-Fa-f0-9] | Hexadecimal digits |
POSIX character classes can only be used within bracket expressions. For example, [[:upper:]ab] matches the uppercase letters and lowercase "a" and "b".
In Perl regular expressions, [:print:] matches [:graph:] union [:space:]. An additional non-POSIX class understood by some tools is [:word:], which is usually defined as [:alnum:] plus underscore. This reflects the fact that in many programming languages these are the characters that may be used in identifiers. The editor Vim further distinguishes word and word-head classes (using the notation \w and \h) since in many programming languages the characters that can begin an identifier are not the same as those that can occur in other positions.
Note that what the POSIX regular expression standards call character classes are commonly referred to as POSIX character classes in other regular expression flavors which support them. With most other regular expression flavors, the term character class is used to describe what POSIX calls bracket expressions.
Perl-derivative regular expressions
Perl has a more consistent and richer syntax than the POSIX basic (BRE) and extended (ERE) regular expression standards. An example of its consistency is that \ always escapes a non-alphanumeric character. Another example of functionality possible with Perl but not POSIX-compliant regular expressions is the concept of lazy quantification (see the next section).
Due largely to its expressive power, many other utilities and programming languages have adopted syntax similar to Perl's — for example, Java, JavaScript, PCRE, Python, Ruby, Microsoft's .NET Framework, and the W3C's XML Schema all use regular expression syntax similar to Perl's. Some languages and tools such as PHP support multiple regular expression flavors. Perl-derivative regular expression implementations are not identical, and many implement only a subset of Perl's features. With Perl 5.10, this process has come full circle with Perl incorporating syntax extensions originally from Python, PCRE, the .NET Framework, and Java.
Simple Regular Expressions
Simple Regular Expressions is a syntax that may be used by historical versions of application programs, and may be supported within some applications for the purpose of providing backward compatibility, These forms of regular expression syntax are considered to be deprecated[4] and should not be used.
Lazy quantification
The standard quantifiers in regular expressions are greedy, meaning they match as much as they can, only giving back as necessary to match the remainder of the regex. For example, someone new to regexes wishing to find the first instance of an item between < and > symbols in this example:
Another whale explosion occurred on <January 26>, <2004>.
...would likely come up with the pattern <.*>, or similar. However, this pattern will actually return "<January 26>, <2004>" instead of the "<January 26>" which might be expected, because the * quantifier is greedy — it will consume as many characters as possible from the input, and "January 26>, <2004" has more characters than "January 26".
Though this problem can be avoided in a number of ways (e.g., by specifying the text that is not to be matched: <[^>]*>), modern regular expression tools allow a quantifier to be specified as lazy (also known as non-greedy, reluctant, minimal, or ungreedy) by putting a question mark after the quantifier (e.g., <.*?>), or by using a modifier which reverses the greediness of quantifiers (though changing the meaning of the standard quantifiers can be confusing). By using a lazy quantifier, the expression tries the minimal match first. Though in the previous example lazy matching is used to select one of many matching results, in some cases it can also be used to improve performance when greedy matching would require more backtracking.
Patterns for non-regular languages
Many features found in modern regular expression libraries provide an expressive power that far exceeds the regular languages. For example, the ability to group subexpressions with parentheses and recall the value they match 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 (.*)\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 numerous modern tools, is NP-hard.
However, many tools, libraries, and engines that provide such constructions still use the term regular expression for their patterns. This has led to a nomenclature where the term regular expression has different meanings in formal language theory and pattern matching. For this reason, some people have taken to using the term regex or simply pattern to describe the latter. Larry Wall (author of Perl) writes in Apocalypse 5:
| " | 'Regular 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).[3] |
Implementations and running times
There are at least three essentially different algorithms that decide if and how a given regular expression matches a string.
The oldest and fastest two rely 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 DFA can be constructed explicitly and then run on the resulting input string one symbol at a time. Constructing the DFA for a regular expression of size m has the time and memory cost of O(2m), but it can be run on a string of size n in time O(n). An alternative approach is to simulate the NFA directly, essentially building each DFA state on demand and then discarding it at the next step, possibly with caching. This keeps the DFA implicit and avoids the exponential construction cost, but running cost rises to O(nm). The explicit approach is called the DFA algorithm and the implicit approach the NFA algorithm. As both can be seen as different ways of executing the same DFA, they are also often called the DFA algorithm without making a distinction. These algorithms are fast, but can only be used for matching and not for recalling grouped subexpressions, lazy quantification, and several other features commonly found in modern regular expression libraries.[5]
The third algorithm is to match the pattern against the input string by backtracking. This algorithm is commonly called NFA, but this terminology can be 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 exponentially increasing number of sub-cases. More complex implementations will often identify and speed up or abort common cases where they would otherwise run slowly.
Although backtracking implementations only give an exponential guarantee in the worst case, they provide much greater flexibility and expressive power. For example, any implementation which allows the use of backreferences, or implements the various extensions introduced by Perl, 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.
Regular expressions and Unicode
Regular expressions were originally used with ASCII characters. Many regular expression engines can now handle Unicode. In most respects it makes no difference what the character set is, but some issues do arise when extending regular expressions to support Unicode.
- Supported encoding. Some regular expression libraries expect the UTF-8 encoding, while others might expect UTF-16, or UTF-32.
- Supported Unicode range. Many regular expression engines support only the Basic Multilingual Plane, that is, the characters which can be encoded with only 16 bits. Currently, only a few regular expression engines can handle the full 21-bit Unicode range.
- Extending ASCII-oriented constructs to Unicode. For example, in ASCII-based implementations, character ranges of the form
[x-y]are valid wherever x and y are codepoints in the range [0x00,0x7F] and codepoint(x) ≤ codepoint(y). The natural extension of such character ranges to Unicode would simply change the requirement that the endpoints lie in [0x00,0x7F] to the requirement that they lie in [0,0x10FFFF]. However, in practice this is often not the case. Some implementations, such as that of gawk, do not allow character ranges to cross Unicode blocks. A range like [0x61,0x7F] is valid since both endpoints fall within the Basic Latin block, as is [0x0530,0x0560] since both endpoints fall within the Armenian block, but a range like [0x0061,0x0532] is invalid since it includes multiple Unicode blocks. Other engines, such as that of the Vim editor, allow block-crossing but limit the number of characters in a range to 128.
- Case insensitivity. Some case-insensitivity flags affect only the ASCII characters. Other flags affect all characters. Some engines have two different flags, one for ASCII, the other for Unicode. Exactly which characters belong to the POSIX classes also varies.
- Cousins of case insensitivity. As the English alphabet has case distinction, case insensitivity became a logical feature in text searching. Unicode introduced alphabetic scripts without case like Devanagari. For these, case sensitivity is not applicable. For scripts like Chinese, another distinction seems logical: between traditional and simplified. In Arabic scripts, insensitivity to initial, medial, final and isolated position may be desired.
- Normalization. Unicode introduced combining characters. Like old typewriters, plain letters can be followed by non-spacing accent symbols to form a single accented letter. As a consequence, two different code sequences can result in identical character display.
- New control codes. Unicode introduced amongst others, byte order marks and text direction markers. These codes might have to be dealt with in a special way.
- Introduction of character classes for Unicode blocks and Unicode general character properties. In Perl and the
java.util.regexlibrary, classes of the form\p{InX}match characters in block X and\P{InX}match the opposite. Similarly,\p{Armenian}matches any character in the Armenian block, and\p{X}matches any character with the general character property X. For example,\p{Lu}matches any upper-case letter.
Uses of regular expressions
Regular expressions are useful in the production of syntax highlighting systems, data validation, and many other tasks.
While regular expressions would be useful on search engines such as Google or Live Search, processing them across the entire database could consume excessive computer resources depending on the complexity and design of the regex. Although in many cases system administrators can run regex-based queries internally, most search engines do not offer regex support to the public. A notable exception is Google Code Search.
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