Can PCRE Match Context-Sensitive Grammars like {a^n b^n c^n}?

Matching Context-Sensitive Grammars with PCRE: {a^n b^n c^n}

While regular expressions have historically been associated with regular grammars, modern implementations like PCRE exhibit advanced capabilities that extend beyond this theoretical framework. One such ability is the expression of context-sensitive grammars.

In this vein, the question arises: can PCRE parse {a^n b^n c^n}, a grammar that cannot be described by regular expressions? The answer, as demonstrated by several users, is an astounding "yes."

One particularly compelling solution presented by the user involves the use of a positive lookahead assertion:

~^
    (?=(a(?-1)?b)c)
     a+(b(?-1)?c)
$~x

Explanation:

Without the lookahead assertion, the regex checks for an arbitrary number of as, followed by an equal number of bs and cs. However, this does not ensure that the number of as equals the number of bs, as required by the grammar.

The lookahead assertion, specifically (?=(a(?-1)?b)c), addresses this problem. It matches sequences of as, followed by bs, each time ensuring that the number of as matches the number of bs. The inclusion of c ensures that the match consumes all of the bs.

Conclusion:

This example showcases the remarkable power of modern regex implementations. PCRE's ability to parse non-context-free grammars highlights its versatility and capacity for expressing complex patterns. This challenges the long-held belief that regex is solely capable of capturing regular languages.

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