A Neighborhood of Infinity: Fast incremental regular expression matching with monoids

The Problem Consider this problem: Fix a regular expression R. Suppose you have a string of length N. There's not much you can do about it, you'll likely have to scan all N characters to test so see if the string matches R. But once you've performed the test, how fast can you test the string again if a small edit is made to it? It seems that in general you'd have to rescan the entire string, or at