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Deciding determinism of unary languages. (English) Zbl 1332.68122
Summary: In this paper, we investigate the complexity of deciding determinism of unary languages. First, we give a method to derive a set of arithmetic progressions from a regular expression \(E\) over a unary alphabet, and establish relations between numbers represented by these arithmetic progressions and words in \(\mathcal{L}(E)\). Next, we define a problem relating to arithmetic progressions and investigate the complexity of this problem. Then by a reduction from this problem we show that deciding determinism of unary languages is coNP-complete. Finally, we extend our derivation method to expressions with counting, and prove that deciding whether an expression over a unary alphabet with counting defines a deterministic language is in \(\Pi_2^{\mathrm p}\). We also establish a tight upper bound for the size of the minimal DFA for expressions with counting.

68Q45 Formal languages and automata
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
68Q25 Analysis of algorithms and problem complexity
Full Text: DOI
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