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Recursive patterns

 

 

 

Consider the problem of matching a string in parentheses, allowing for unlimited nested parentheses. Without the use of recursion, the best that can be done is to use a pattern that matches up to some fixed depth of nesting. It is not possible to handle an arbitrary nesting depth.

 

For some time, Perl has provided a facility that allows regular expressions to recurse (amongst other things). It does this by interpolating Perl code in the expression at run time, and the code can refer to the expression itself. A Perl pattern using code interpolation to solve the parentheses problem can be created like this:

 

  $re = qr{\( (?: (?>[^()]+) | (?p{$re}) )* \)}x;


The (?p{...}) item interpolates Perl code at run time, and in this case refers recursively to the pattern in which it appears.

 

Obviously, PCRE cannot support the interpolation of Perl code. Instead, it supports special syntax for recursion of the entire pattern, and also for individual subpattern recursion. After its introduction in PCRE and Python, this kind of recursion was subsequently introduced into Perl at release 5.10.

 

A special item that consists of (? followed by a number greater than zero and a closing parenthesis is a recursive subroutine call of the subpattern of the given number, provided that it occurs inside that subpattern. (If not, it is a non-recursive subroutine call, which is described in the next section.) The special item (?R) or (?0) is a recursive call of the entire regular expression.

 

This PCRE pattern solves the nested parentheses problem (assume the PCRE_EXTENDED option is set so that white space is ignored):

 

  \( ( [^()]++ | (?R) )* \)

 

First it matches an opening parenthesis. Then it matches any number of substrings which can either be a sequence of non-parentheses, or a recursive match of the pattern itself (that is, a correctly parenthesized substring). Finally there is a closing parenthesis. Note the use of a possessive quantifier to avoid backtracking into sequences of non-parentheses.

 

If this were part of a larger pattern, you would not want to recurse the entire pattern, so instead you could use this:

 

  ( \( ( [^()]++ | (?1) )* \) )

 

We have put the pattern into parentheses, and caused the recursion to refer to them instead of the whole pattern.

 

In a larger pattern, keeping track of parenthesis numbers can be tricky. This is made easier by the use of relative references. Instead of (?1) in the pattern above you can write (?-2) to refer to the second most recently opened parentheses preceding the recursion. In other words, a negative number counts capturing parentheses leftwards from the point at which it is encountered.

 

It is also possible to refer to subsequently opened parentheses, by writing references such as (?+2). However, these cannot be recursive because the reference is not inside the parentheses that are referenced. They are always non-recursive subroutine calls, as described in the next section.

 

An alternative approach is to use named parentheses instead. The Perl syntax for this is (?&name); PCRE's earlier syntax (?P>name) is also supported. We could rewrite the above example as follows:

 

  (?<pn> \( ( [^()]++ | (?&pn) )* \) )

 

If there is more than one subpattern with the same name, the earliest one is used.

 

This particular example pattern that we have been looking at contains nested unlimited repeats, and so the use of a possessive quantifier for matching strings of non-parentheses is important when applying the pattern to strings that do not match. For example, when this pattern is applied to

 

  (aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa()

 

it yields "no match" quickly. However, if a possessive quantifier is not used, the match runs for a very long time indeed because there are so many different ways the + and * repeats can carve up the subject, and all have to be tested before failure can be reported.

 

At the end of a match, the values of capturing parentheses are those from the outermost level. If you want to obtain intermediate values, a callout function can be used (see below and the pcrecallout documentation). If the pattern above is matched against


  (ab(cd)ef)

 

the value for the inner capturing parentheses (numbered 2) is "ef", which is the last value taken on at the top level. If a capturing subpattern is not matched at the top level, its final captured value is unset, even if it was (temporarily) set at a deeper level during the matching process.

 

If there are more than 15 capturing parentheses in a pattern, PCRE has to obtain extra memory to store data during a recursion, which it does by using pcre_malloc, freeing it via pcre_free afterwards. If no memory can be obtained, the match fails with the PCRE_ERROR_NOMEMORY error.

 

Do not confuse the (?R) item with the condition (R), which tests for recursion. Consider this pattern, which matches text in angle brackets, allowing for arbitrary nesting. Only digits are allowed in nested brackets (that is, when recursing), whereas any characters are permitted at the outer level.

 

  < (?: (?(R) \d++  | [^<>]*+) | (?R)) * >


In this pattern, (?(R) is the start of a conditional subpattern, with two different alternatives for the recursive and non-recursive cases. The (?R) item is the actual recursive call.

 


Differences in recursion processing between PCRE and Perl

 

Recursion processing in PCRE differs from Perl in two important ways. In PCRE (like Python, but unlike Perl), a recursive subpattern call is always treated as an atomic group. That is, once it has matched some of the subject string, it is never re-entered, even if it contains untried alternatives and there is a subsequent matching failure. This can be illustrated by the following pattern, which purports to match a palindromic string that contains an odd number of characters (for example, "a", "aba", "abcba", "abcdcba"):

 

  ^(.|(.)(?1)\2)$


The idea is that it either matches a single character, or two identical characters surrounding a sub-palindrome. In Perl, this pattern works; in PCRE it does not if the pattern is longer than three characters. Consider the subject string "abcba":

 

At the top level, the first character is matched, but as it is not at the end of the string, the first alternative fails; the second alternative is taken and the recursion kicks in. The recursive call to subpattern 1 successfully matches the next character ("b"). (Note that the beginning and end of line tests are not part of the recursion).

 

Back at the top level, the next character ("c") is compared with what subpattern 2 matched, which was "a". This fails. Because the recursion is treated as an atomic group, there are now no backtracking points, and so the entire match fails. (Perl is able, at this point, to re-enter the recursion and try the second alternative.) However, if the pattern is written with the alternatives in the other order, things are different:

 

  ^((.)(?1)\2|.)$


This time, the recursing alternative is tried first, and continues to recurse until it runs out of characters, at which point the recursion fails. But this time we do have another alternative to try at the higher level. That is the big difference: in the previous case the remaining alternative is at a deeper recursion level, which PCRE cannot use.


To change the pattern so that it matches all palindromic strings, not just those with an odd number of characters, it is tempting to change the pattern to this:

 

  ^((.)(?1)\2|.?)$


Again, this works in Perl, but not in PCRE, and for the same reason. When a deeper recursion has matched a single character, it cannot be entered again in order to match an empty string. The solution is to separate the two cases, and write out the odd and even cases as alternatives at the higher level:

 

  ^(?:((.)(?1)\2|)|((.)(?3)\4|.))


If you want to match typical palindromic phrases, the pattern has to ignore all non-word characters, which can be done like this:

 

  ^\W*+(?:((.)\W*+(?1)\W*+\2|)|((.)\W*+(?3)\W*+\4|\W*+.\W*+))\W*+$

 

If run with the PCRE_CASELESS option, this pattern matches phrases such as "A man, a plan, a canal: Panama!" and it works well in both PCRE and Perl. Note the use of the possessive quantifier *+ to avoid backtracking into sequences of non-word characters. Without this, PCRE takes a great deal longer (ten times or more) to match typical phrases, and Perl takes so long that you think it has gone into a loop.

 


 


 

Philip Hazel

University Computing Service

Cambridge CB2 3QH, England.

Last updated: 12 November 2013

Copyright © 1997-2013 University of Cambridge.


 


 

 

 

 

 

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