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# Regular expression

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### Regular expression

The regular expression
(?<=\.) {2,}(?=[A-Z]) matches at least two spaces occurring after period (.) and before an upper case letter as highlighted in the text above.

In theoretical computer science and formal language theory, a regular expression (abbreviated regex or regexp and sometimes called a rational expression)[1][2] is a sequence of characters that define a search pattern, mainly for use in pattern matching with strings, or string matching, i.e. "find and replace"-like operations. The concept arose in the 1950s, when the American mathematician Stephen Kleene formalized the description of a regular language, and came into common use with the Unix text processing utilities ed, an editor, and grep (global regular expression print), a filter.

Regular expressions are so useful in computing that the various systems to specify regular expressions have evolved to provide both a basic and extended standard for the grammar and syntax; modern regular expressions heavily augment the standard. Regular expression processors are found in several search engines, search and replace dialogs of several word processors and text editors, and in the command lines of text processing utilities, such as sed and AWK.

Many programming languages provide regular expression capabilities, some built-in, for example Perl, JavaScript, Ruby, AWK, and Tcl, and others via a standard library, for example .NET languages, Java, Python and C++ (since C++11). Most other languages offer regular expressions via a library.

## Patterns

Each character in a regular expression is either understood to be a metacharacter with its special meaning, or a regular character with its literal meaning. Together, they can be used to identify textual material of a given pattern, or process a number of instances of it that can vary from a precise equality to a very general similarity of the pattern. The pattern sequence itself is an expression that is a statement in a language designed specifically to represent prescribed targets in the most concise and flexible way to direct the automation of text processing of general text files, specific textual forms, or of random input strings.

A very simple use of a regular expression would be to locate the same word spelled two different ways in a text editor, for example the regular expression
seriali[sz]e

matches both "serialise" and "serialize". A wildcard match can also achieve this, but wildcard matches differ from regular expressions in that wildcards are more limited in what they can pattern (having fewer metacharacters and a simple language-base). A usual context of wildcard characters is in globbing similar names in a list of files, whereas regular expressions are usually employed in applications that pattern-match text strings in general. For example, the regexp
^[ \t]+|[ \t]+$ matches excess whitespace at the beginning or end of a line. An advanced regexp used to match any numeral is ^[+-]?(\d+(\.\d+)?|\.\d+)([eE][+-]?\d+)?$

. See Examples for more examples.
Translating the Kleene star "s*": "zero or more of s".

A regular expression processor translates a regular expression into a nondeterministic finite automaton (NFA), which is then made deterministic and run on the target text string to recognize substrings that match the regular expression.

The picture shows the NFA scheme N(s
*

) obtained from the regex s
*

, where s denotes a simpler regex in turn, which has already been recursively translated to the NFA N(s).

## History

Regular expressions originated in 1956, when mathematician Stephen Cole Kleene described regular languages using his mathematical notation called regular sets.[3] These arose in theoretical computer science, in the subfields of automata theory (models of computation) and the description and classification of formal languages. Other early implementations of pattern matching include the SNOBOL language, which did not use regular expressions, but instead its own syntax.

Regular expressions entered popular use from 1968 in two uses: pattern matching in a text editor[4] and lexical analysis in a compiler.[5] Among the first appearances of regular expressions in program form was when Ken Thompson built Kleene's notation into the editor QED as a means to match patterns in text files.[4][6][7][8] For speed, Thompson implemented regular expression matching by just-in-time compilation (JIT) to IBM 7094 code on the Compatible Time-Sharing System, an important early example of JIT compilation.[9] 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 meaning "Global search for Regular Expression and Print matching lines"[10]). Around the same time when Thompson developed QED, a group of researchers including Douglas T. Ross implemented a tool based on regular expressions that is used for lexical analysis in compiler design.[5]

Many variations of these original forms of regular expressions were used in Unix[8] programs at Bell Labs in the 1970s, including vi, lex, sed, AWK, and expr, and in other programs such as Emacs. Regular expressions were subsequently adopted by a wide range of programs, with these early forms standardized in the POSIX.2 standard in 1992.

In the 1980s more complicated regular expressions arose in Perl, which originally derived from a regex library written by Henry Spencer (1986), who later wrote an implementation of Advanced Regular Expressions for Tcl.[11] The Tcl library is a hybrid NFA/DFA implementation with improved performance characteristics, earning praise from Jeffrey Friedl who said, "...it really seems quite wonderful."[12] Software projects that have adopted Spencer's Tcl regular expression implementation include PostgreSQL.[13] Perl later expanded on Spencer's original library to add many new features,[14] but has not yet caught up with Spencer's Advanced Regular Expressions implementation in terms of performance or Unicode handling.[15][16] 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.[17] 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 consists of regular expressions. Its use is evident in the DTD element group syntax.

Starting in 1997, 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.

Today regular expressions are widely supported in programming languages, text processing programs (particular lexers), advanced text editors, and some other programs. Regular expression support is part of the standard library of many programming languages, including Java and Python, and is built into the syntax of others, including Perl and ECMAScript. Implementations of regular expression functionality is often called a regular expression engine, and a number of libraries are available for reuse.

## Basic concepts

A regular expression, often called a pattern, is an expression used to specify a set of strings required for a particular purpose. A simple way to specify a finite set of strings is to list its elements or members. However, there are often more concise ways to specify the desired set of strings. For example, the set containing the three strings "Handel", "Händel", and "Haendel" can be specified by the pattern H(ä|ae?)ndel; we say that this pattern matches each of the three strings. In most formalisms, if there exists at least one regex that matches a particular set then there exists an infinite number of other regex that also match it—the specification is not unique. Most formalisms provide the following operations to construct regular expressions.

Boolean "or"
A vertical bar separates alternatives. For example, gray|grey can match "gray" or "grey".
Grouping
Parentheses are used to define the scope and precedence of the operators (among other uses). For example, gray|grey and gr(a|e)y are equivalent patterns which both describe the set of "gray" or "grey".
Quantification
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 plus).
 ? The question mark indicates zero or one occurrences of the preceding element. For example, colou?r matches both "color" and "colour". * The asterisk indicates zero or more occurrences of the preceding element. For example, ab*c matches "ac", "abc", "abbc", "abbbc", and so on. + The plus sign indicates one or more occurrences of the preceding element. For example, ab+c matches "abc", "abbc", "abbbc", and so on, but not "ac". {n}[18] The preceding item is matched exactly n times. {min,}[18] The preceding item is matched min or more times. {min,max}[18] The preceding item is matched at least min times, but not more than max times.

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.

## Formal language theory

Regular expressions describe regular languages in formal language theory. They have the same expressive power as regular grammars.

### Formal definition

Regular expressions consist of constants and operator symbols 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.[19][20] Given a finite alphabet Σ, the following constants are defined as regular expressions:

• (empty set) denoting the set .
• (empty string) ε denoting the set containing only the "empty" string, which has no characters at all.
• (literal character)
a

in Σ denoting the set containing only the character a.

Given regular expressions R and S, the following operations over them are defined to produce regular expressions:

• (concatenation) RS denotes the set of strings that can be obtained by concatenating a string in R and a string in S. For example {"ab", "c"}{"d", "ef"} = {"abd", "abef", "cd", "cef"}.
• (alternation) R | S denotes the set union of sets described by R and S. For example, if R describes {"ab", "c"} and S describes {"ab", "d", "ef"}, expression R | S describes {"ab", "c", "d", "ef"}.
• (Kleene star) R* denotes the smallest superset of set described by 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 set described by 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 alternation. 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 "b"s and finally optionally a "c": {"a", "ac", "ab", "abc", "abb", "abbc", ...}
• (0|(1(01*0)*1))*

denotes the set of binary numbers that are multiples of 3: { ε, "0", "00", "11", "000", "011", "110", "0000", "0011", "0110", "1001", "1100", "1111", "00000", ... }

### 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.[21][22]

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 here is the languages Lk consisting of all strings over the alphabet {a,b} whose kth-from-last letter equals a. On one hand, a regular expression describing L4 is given by (a\mid b)^*a(a\mid b)(a\mid b)(a\mid b). Generalizing this pattern to Lk gives the expression

(a\mid b)^*a\underbrace{(a\mid b)(a\mid b)\cdots(a\mid b)}_{k-1\text{ times}}. \,

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.[19]

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.

### Deciding equivalence of regular expressions

As seen in many of the examples above, there is more than one way to construct a regular expression to achieve the same results.

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

The redundancy can be eliminated by using Kleene star and set union 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. In 1991, Dexter Kozen axiomatized regular expressions with Kleene algebra;[23] see Kleene algebra#History for details.

## Syntax

A regexp pattern matches a target string. The pattern is composed of a sequence of atoms. An atom is a single point within the regexp pattern which it tries to match to the target string. The simplest atom is a literal, but grouping parts of the pattern to match an atom will require using ( ) as metacharacters. Metacharacters help form: atoms; quantifiers telling how many atoms (and whether it is a greedy quantifier or not); a logical OR character, which offers a set of alternatives, and a logical NOT character, which negates an atom's existence; and back references to refer to previous atoms of a completing pattern of atoms. A match is made, not when all the atoms of the string are matched, but rather when all the pattern atoms in the regular expression have matched. The idea is to make a small pattern of characters stand for a large number of possible strings, rather than compiling a large list of all the literal possibilities.

Depending on the regexp processor there are about fourteen metacharacters, characters that may or may not have their literal character meaning, depending on context, or whether they are "escaped", i.e. preceded by an escape sequence, in this case, the backslash \. Modern and POSIX extended regular expressions use metacharacters more often than their literal meaning, so to avoid "backslash-osis" it makes sense to have a metacharacter escape to a literal mode; but starting out, it makes more sense to have the four bracketing metacharacters ( ) and { } be primarily literal, and "escape" this usual meaning to become metacharacters. Common standards implement both. The usual metacharacters are {}[]()^$.|*+? and \. The usual characters that become metacharacters when escaped are dsw.DSW and N. ### Delimiters When entering a regular expression in a programming language, they may be represented as a usual string literal, hence usually quoted; this is common in C, Java, and Python for instance, where the regular expression re is entered as "re". However, they are often written with slashes as delimiters, as in /re/ for the regular expression re. This originates in ed, where / is the editor command for searching, and an expression /re/ can be used to specify a range of lines (matching the pattern), which can be combined with other commands on either side, most famously g/re/p as in grep ("global regex print"), which is included in most Unix-based operating systems, such as Linux distributions. A similar convention is used in sed, where search and replace is given by s/regexp/replacement/ and patterns can be joined with a comma to specify a range of lines as in /re1/,/re2/. This notation is particularly well-known due to its use in Perl, where it forms part of the syntax distinct from normal string literals. In some cases, such as sed and Perl, alternative delimiters can be used to avoid collision with contents, and to avoid having to escape occurrences of the delimiter character in the contents. For example, in sed the command s,/,X, will replace a / with an X, using commas as delimiters. ### Standards The IEEE POSIX standard has three sets of compliance: BRE,[24] ERE, and SRE for Basic, Extended, and Simple Regular Expressions. SRE is deprecated,[25] in favor of BRE, as both provide backward compatibility. The subsection below covering the character classes applies to both BRE and ERE. BRE and ERE work together. ERE adds ?, +, and |, and it removes the need to escape the metacharacters ( ) and { }, which are required in BRE. Furthermore, as long as the POSIX standard syntax for regular expressions is adhered to, there can be, and often is, additional syntax to serve specific (yet POSIX compliant) applications. Although POSIX.2 leaves some implementation specifics undefined, BRE and ERE provide a "standard" which has since been adopted as the default syntax of many tools, where the choice of BRE or ERE modes is usually a supported option. For example, GNU grep has the following options: "grep -E" for ERE, and "grep -G" for BRE (the default), and "grep -P" for Perl regular expressions. Perl regular expressions have become a de facto standard, having a rich and powerful set of atomic expressions. Perl has no "basic" or "extended" levels, where the ( ) and { } may or may not have literal meanings. They are always metacharacters, as they are in "extended" mode for POSIX. To get their literal meaning, you escape them. Other metacharacters are known to be literal or symbolic based on context alone. Perl offers much more functionality: "lazy" regular expressions, backtracking, named capture groups, and recursive patterns, all of which are powerful additions to POSIX BRE/ERE. (See lazy quantification below.) #### POSIX basic and extended In the POSIX standard, Basic Regular Syntax, BRE, requires that the metacharacters ( ) and { } be designated  and \{\}, whereas Extended Regular Syntax, ERE, does not. 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 - character is treated as a literal character if it is the last or the first (after the ^, if present) character within the brackets: [abc-], [-abc]. Note that backslash escapes are not allowed. The ] character can be included in a bracket expression if it is the first (after the ^) character: []abc]. [^ ] 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". Likewise, 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.
( ) 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. BRE mode requires .
\n Matches what the nth marked subexpression matched, where n is a digit from 1 to 9. This construct is vaguely defined in the POSIX.2 standard. 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.
{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. BRE mode requires \{m,n\}.

Examples:

• .at matches any three-character string ending with "at", including "hat", "cat", and "bat".
• [hc]at matches "hat" and "cat".
• [^b]at matches all strings matched by .at except "bat".
• [^hc]at matches all strings matched by .at other than "hat" and "cat".
• ^[hc]at matches "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. • $.$ matches any single character surrounded by "[" and "]" since the brackets are escaped, for example: "[a]" and "[b]". • s.* matches s followed by zero or more characters, for example: "s" and "saw" and "seed". #### POSIX extended 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, ab?c matches only "ac" or "abc". + Matches the preceding element one or more times. For example, ab+c matches "abc", "abbc", "abbbc", and so on, but not "ac". | The choice (also known as 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]+at matches "hat", "cat", "hhat", "chat", "hcat", "cchchat", and so on, but not "at". • [hc]?at matches "hat", "cat", and "at". • [hc]*at matches "hat", "cat", "hhat", "chat", "hcat", "cchchat", "at", and so on. • cat|dog matches "cat" or "dog". POSIX Extended Regular Expressions can often be used with modern Unix utilities by including the command line flag -E. #### Character classes The character class is the most basic regular expression concept after a literal match. It makes one small sequence of characters match a larger set of characters. For example, [A-Z] could stand for the upper case alphabet, and \d could mean any digit. Character classes apply to both POSIX levels. When specifying a range of characters, such as [a-Z] (i.e. lowercase a to upper-case z), the computer's locale settings determine the contents by the numeric ordering of the character encoding. They could store digits in that sequence, or the ordering could be abc...zABC...Z, or aAbBcC...zZ. So the POSIX standard defines a character class, which will be known by the regular expression processor installed. Those definitions are in the following table: POSIX Non-standard Perl/Tcl Vim ASCII Description [:alnum:] [A-Za-z0-9] Alphanumeric characters [:word:] \w \w [A-Za-z0-9_] Alphanumeric characters plus "_" \W \W [^A-Za-z0-9_] Non-word characters [:alpha:] \a [A-Za-z] Alphabetic characters [:blank:] \s [ \t] Space and tab \b \< \> (?<=\W)(?=\w)|(?<=\w)(?=\W) Word boundaries [:cntrl:] [\x00-\x1F\x7F] Control characters [:digit:] \d \d [0-9] Digits \D \D [^0-9] Non-digits [:graph:] [\x21-\x7E] Visible characters [:lower:] \l [a-z] Lowercase letters [:print:] \p [\x20-\x7E] Visible characters and the space character [:punct:] [][!"#$%&'()*+,./:;<=>?@\^_{|}~-] Punctuation characters
[:space:] \s \_s [ \t\r\n\v\f] Whitespace characters
\S \S [^ \t\r\n\v\f] Non-whitespace characters
[:upper:] \u [A-Z] Uppercase letters
[:xdigit:] \x [A-Fa-f0-9] Hexadecimal digits

POSIX character classes can only be used within bracket expressions. For example,  

 in Unicode,[32] where the Alphabetic property contains more than just Letters, and the Decimal_Number property contains more than [0-9]. 

 
$string1 = "Hello World\n"; if ($string1 =~ m/\w/) {
print "There is at least one alphanumeric ";
print "character in $string1 (A-Z, a-z, 0-9, _)\n"; }  \W Matches a non-alphanumeric character, excluding "_"; same as [^A-Za-z0-9_] in ASCII, and [^\p{Alphabetic}\p{GC=Mark}\p{GC=Decimal_Number}\p{GC=Connector_Punctuation}] in Unicode. $string1 = "Hello World\n";
if ($string1 =~ m/\W/) { print "The space between Hello and "; print "World is not alphanumeric\n"; }  \s Matches a whitespace character, which in ASCII are tab, line feed, form feed, carriage return, and space; in Unicode, also matches no-break spaces, next line, and the variable-width spaces (amongst others). $string1 = "Hello World\n";
if ($string1 =~ m/\s.*\s/) { print "There are TWO whitespace characters, which may"; print " be separated by other characters, in$string1";
}

\S Matches anything BUT a whitespace.
$string1 = "Hello World\n"; if ($string1 =~ m/\S.*\S/) {
print "There are TWO non-whitespace characters, which";
print " may be separated by other characters, in $string1"; }  \d Matches a digit; same as [0-9] in ASCII; in Unicode, same as the \p{Digit} or \p{GC=Decimal_Number} property, which itself the same as the \p{Numeric_Type=Decimal} property. $string1 = "99 bottles of beer on the wall.";
if ($string1 =~ m/(\d+)/) { print "$1 is the first number in '$string1'\n"; }  Output: 99 is the first number in '99 bottles of beer on the wall.'  \D Matches a non-digit; same as [^0-9] in ASCII or \P{Digit} in Unicode. $string1 = "Hello World\n";
if ($string1 =~ m/\D/) { print "There is at least one character in$string1";
print " that is not a digit.\n";
}

^ Matches the beginning of a line or string.
$string1 = "Hello World\n"; if ($string1 =~ m/^He/) {
print "$string1 starts with the characters 'He'\n"; }  $ Matches the end of a line or string.
$string1 = "Hello World\n"; if ($string1 =~ m/rld$/) { print "$string1 is a line or string ";
print "that ends with 'rld'\n";
}

\A Matches the beginning of a string (but not an internal line).
$string1 = "Hello\nWorld\n"; if ($string1 =~ m/\AH/) {
print "$string1 is a string "; print "that starts with 'H'\n"; }  \z Matches the end of a string (but not an internal line).[37] $string1 = "Hello\nWorld\n";
if ($string1 =~ m/d\n\z/) { print "$string1 is a string ";
print "that ends with 'd\\n'\n";
}

[^...] Matches every character except the ones inside brackets.
$string1 = "Hello World\n"; if ($string1 =~ m/[^abc]/) {
print "\$string1 contains a character other than ";
print "a, b, and c\n";
}


## Induction

Regular expressions can often be created ("induced" or "learned") based on a set of example strings. This is known as the induction of regular languages, and is part of the general problem of grammar induction in computational learning theory. Formally, given examples of strings in a regular language, and perhaps also given examples of strings not in that regular language, it is possible to induce a grammar for the language, i.e., a regular expression that generates that language. Not all regular languages can be induced in this way (see language identification in the limit), but many can. For example, the set of examples {1, 10, 100}, and negative set (of counterexamples) {11, 1001, 101, 0} can be used to induce the regular expression 1⋅0* (1 followed by zero or more 0s).

## Notes

1. ^
2. ^
3. ^
4. ^ a b
5. ^ a b
6. ^
7. ^
8. ^ a b Aho & Ullman 1992, 10.11 Bibliographic Notes for Chapter 10, p. 589.
9. ^ Aycock 2003, 2. JIT Compilation Techniques, 2.1 Genesis, p. 98.
10. ^
11. ^
12. ^
13. ^
14. ^
15. ^
16. ^
17. ^
18. ^ a b c grep(1) man page
19. ^ a b
20. ^
21. ^
22. ^
23. ^
24. ^ ISO/IEC 9945-2:1993 Information technology – Portable Operating System Interface (POSIX) – Part 2: Shell and Utilities, successively revised as ISO/IEC 9945-2:2002 Information technology – Portable Operating System Interface (POSIX) – Part 2: System Interfaces, ISO/IEC 9945-2:2003, and currently ISO/IEC/IEEE 9945:2009 Information technology – Portable Operating System Interface (POSIX®) Base Specifications, Issue 7
25. ^ The Single Unix Specification (Version 2)
26. ^
27. ^ a b
28. ^ Theorem 3 (p.9)
29. ^
30. ^
31. ^
32. ^ a b
34. ^ The character 'm' is not always required to specify a Perl match operation. For example, m/[^abc]/ could also be rendered as /[^abc]/`. The 'm' is only necessary if the user wishes to specify a match operation without using a forward-slash as the regex delimiter. Sometimes it is useful to specify an alternate regex delimiter in order to avoid "delimiter collision". See 'perldoc perlre' for more details.