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Partial orders on words, minimal elements of regular languages, and state complexity. (English) Zbl 0786.68071

Having a paratial order (po) on strings, \(MIN(L)\) is the set of all minimal elements of the language \(L\) w.r.t. the po. The paper shows that if \(L\) is regular then so is \(MIN(L)\) for po’s like the generalized prefix (suffix, subsegment, subsequence) order; “generalized” supposes a po on the alphabet – the identity leads to the “usual” meaning. Note that a well-known result of Higman implies the finiteness of \(MIN(L)\) for the generalized subsequence order. Moreover, if \(L\) is recognized by a deterministic finite automaton (DFA) with \(n\) states then \(MIN(L)\) is recognizable by a DFA with roughly \(2^ n\) states. The author shows the upper bounds more precisely, thus extending and improving some results of Kundu.
Nevertheless, the main value of the paper lies in proving the lower bounds, by which the given upper bounds are shown to be tight, even if we consider nondeterministic FA’s for \(MIN(L)\). An exponential lower bound is also shown for the case with a 4-letter alphabet.
The ideas are also used for a new proof of the theorem by Sakoda and Sipser showing the exponential growth of the number of states going from an NFA to an NFA recognizing the complementary language.
For completeness, the author also shows that regularity of \(L\) implies regularity of \(MIN(L)\) w.r.t. the generalized dictionary and lexical orders; in that case the relevant growth of states is nor \(n^ 2\), respectively.

MSC:

68Q45 Formal languages and automata
68Q25 Analysis of algorithms and problem complexity
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References:

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