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 (e.g., "aaabbbccc")

While regular expressions are traditionally associated with parsing regular grammars, modern implementations like PCRE offer extended capabilities. This raises the question: can we use PCRE to parse more complex, context-sensitive grammars?

Specifically, let's explore the challenge of matching strings following the context-sensitive grammar {a^n b^n c^n; n > 0} (e.g., "aaabbbccc").

Regex Solution:

Inspired by NullUserException's initial attempt, we have devised the following regex:

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

Explanation:

• The outer section ~^...$~ ensures the string starts and ends with the desired pattern.
• Inside the parentheses, ~?(a(?-1)?b)c~ is a positive lookahead assertion that checks for an equal number of as and bs.
• ~a ~ matches any number of as.
• ~b(?-1)?c~ matches an equal number of bs followed by cs.
• The (?-) in (?-1) allows us to match arbitrary sequences of as and bs that satisfy the lookahead assertion.

Test Cases:

We tested the regex against various inputs, including:

• "aabbcc" - Matches successfully
• "aaabbbccc" - Matches successfully
• "aaabbbcc" - Fails to match (correct)
• "aaaccc" - Fails to match (correct)
• "aabcc" - Fails to match (correct)
• "abbcc" - Fails to match (correct)

Conclusion:

This regex demonstrates the remarkable power of modern regular expressions, extending beyond regular grammars to match even context-sensitive grammars like {a^n b^n c^n}. This ability to parse more complex patterns opens up new possibilities for regex applications.

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