Approximate satisfiability and equivalence

Inspired by Property Testing, for every ε> 0 we relax the classical satisfiability U | = F between a finite structure U of a class K and a formula F, to a notion of ε-satisfiability U |=ε F, and relax the classical equivalence F1 ≡ F2 between two formulas F1 and F2 to ε-equivalence F1 ≡ε F2. We consider strings and trees with the norm of the edit distance with moves, and show that, unlike their exact counterparts, these approximate notions can be efficiently decided. We use a statistical embedding of words (resp. trees) into `1, which generalizes the original Parikh mapping, obtained by sampling O(f(ε)) finite samples of the words (resp. trees). We give a tester for equality and membership in any regular language, in time independent of the size of the structure. Using our geometrical embedding, we can also test the equivalence between two regular properties over words, defined by regular expressions or Monadic Second Order formulas. Our equivalence tester has polynomial time complexity in the size of the automaton (or regular expression), for any fixed ε, whereas the exact version of the equivalence problem is PSPACE-complete. We also prove versions of some of these results for trees, but with a worse time complexity