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Minimal consistent DFA from sample strings

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Abstract

We review the grammatical inference problem for regular languages which aims to generate a deterministic finite automaton from a representative set of training sample strings known to be in or not in the language. Although the general problem of producing a minimal DFA consistent with a given sample is known to be NP-hard, it is possible to generate minimal consistent DFA in polynomial time if certain constraints are satisfied by the given samples. In this work we propose a new algorithm which generates minimal DFA if the given training samples satisfy a certain sufficient condition. On the negative side, we also show that this problem is indeed hard, such that even for a more restricted class of training sets, the problem of generating minimal consistent DFA is already intractable.

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Notes

  1. Although the general results seem negative, Gold has also proved that from a given regular language, it is possible to generate a characteristic sample set S, such that learning from that training set yields a minimal DFA in polynomial time.

  2. Recall that Myhill–Nerode theorem defines equivalence classes on members of a language \(L\subseteq {\varSigma }^*\). If \(w_1,w_2\in {\varSigma }^*\) are defined as equivalent, then for all \(w\in {\varSigma }^*\), \(w_1\cdot w\) and \(w_2\cdot w\) are either both accepted by the language or both rejected by the language. The number of such equivalence classes for members of a regular language is always finite, which is equivalent to the number of states in a minimal DFA.

  3. This is similar to [5], in which it is shown that there may be multiple minimal cover-automata that are consistent with a given finite input sample. Also in the original TB algorithm, the construction of \(\delta _M(w,a)\) is nondeterministic as if \(w\cdot a\in S^+\setminus {\texttt {red}}\), the choice of \(w'\) is any compatible state in \({\texttt {red}}\).

  4. Philosophically, one may argue that when studying an automaton it is more reasonable to assume that the acceptability of all prefixes of a word should be observed.

  5. Being maximal means when compared with others there needs to have semantic evidences, either by showing it is larger or smaller by containment, or neither larger nor smaller (i.e., incomparable by an evidence trace). The term semantic-completeness enforces that such a measure always exists at least for the maximal elements.

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  1. Chenyi Zhang

    Present address: Jinan University, Guangzhou, China

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    Zhang, C. Minimal consistent DFA from sample strings. Acta Informatica 57, 657–670 (2020). https://doi.org/10.1007/s00236-020-00365-8

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