Deciding determinism of unary languages

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 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 Π2p. We also establish a tight upper bound for the size of the minimal DFA for expressions with counting.