The minimum consistent DFA problem cannot be approximated within and polynomial

The minimum consistent DFA problem is that of finding a DFA with as few states as possible that is consistent with a given sample (a finite collection of words, each labeled as to whether the DFA found should accept or reject). Assuming that P ≠ NP, it is shown that for any constant k, no polynomial time algorithm can be guaranteed to find a consistent DFA of size optk, where opt is the size of a smallest DFA consistent with the sample. This result holds even if the alphabet is of constant size two, and if the algorithm is allowed to produce an NFA, a regular grammar, or a regular expression that is consistent with the sample. Similar hardness results are described for the problem of funding small consistent linear grammars.