## The minimum consistent DFA problem cannot be approximated within any polynomial (1993)

Venue: | Journal of the Association for Computing Machinery |

Citations: | 83 - 4 self |

### BibTeX

@ARTICLE{Pitt93theminimum,

author = {Leonard Pitt and Manfred K. Warmuth},

title = {The minimum consistent DFA problem cannot be approximated within any polynomial},

journal = {Journal of the Association for Computing Machinery},

year = {1993},

volume = {40},

pages = {95--142}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. 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 with fewer than opt ~ states, where opt is the number of states in the minimum state 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 expression, or a regular grammar that is consistent with the sample. A similar nonapproximability result is presented for the problem of finding small consistent linear grammars. For the case of finding minimum consistent DFAs when the alphabet is not of constant size but instead is allowed to vay with the problem specification, the slightly

### Citations

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Citation Context ...smallest consistent DFA is NP-hard. D. Angluin (private communication) showed that it is NP-hard to determine whether there exists a two-state DFA consistent with given data. Trakhtenbrot and Barzdin =-=[24]-=- gave a polynomial-time algorithm for finding a smallest consistent DFA in the case where the sets POS and NEG together consist of all strings up to a given length. Angluin [5] extended Gold’s result,... |

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Citation Context ...is of size at most polynomially larger than the smallest DFA consistent with a sample over a two-letter alphabet, This significantly improves the lower bound on approximability due to Li and Vazirani =-=[18]-=-, which shows that a constant factor of ~ cannot be achieved. The same techniques are used to also show that the linear grammar consistency problem cannot be approximated within any polynomial factor ... |

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