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An actual regular expression is a grammar for a regular language:
brilliant.org/wiki/regular-l
Most implementations (Perl-style, etc.) are cursed and support a lot more via state machine hacks (backreferences, etc.).
re2, Go regexp and Rust regex are actually always regular.
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re2, Go regexp and Rust regex also make sure to provide an O(n) time complexity guarantee.
It's also possible to use an index for the approach they use to make it really fast for a search engine:
swtch.com/~rsc/regexp/re
That's how this works:
codesearch.debian.net