Question

Suppose we have some text and a regular expression that matches it. Question: if I apply the same expression to text backwards (starting from the last letter to the first one), will it still match?

regex —–> text

xereg –?–> txet

In practice that seems to work, the question is rather about what the theory says about the general case.

Answer 1

Not if you use the Kleene star – if you reverse the regex, you will end up with an invalid regex or one that matches a different pattern:

• ab* -> *ba (invalid syntax)
• a*b -> b*a (the first one matches aaab but not abbb, while the second one matches bbba but not baaa)

On the other hand, I’m quite sure that it would be possible to design an algorithm that, given a regex, produces a regex that matches the reverse strings. The following recursive algorithm should work (if r is a regex, rev(r) means the regex that matches the reversed strings):

• If r is a single symbol x, then rev(r) = x.
• If r is a union A|B, then rev(r) = rev(A)|rev(B).
• If r is a concatenation AB, then rev(r) = rev(B)rev(A).
• If r is a Kleene star A*, then rev(r) = rev(A)*.