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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.

##### MSC:
 68Q45 Formal languages and automata 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68Q25 Analysis of algorithms and problem complexity
ALGOL 60
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