Languages given by Finite Automata over the Unary Alphabet

This paper studies the complexity of operations with finite automata and the complexity of their decision problems when the alphabet is unary. (1) This paper improves the upper bound of the equality problem of unary nondeterministic automata from an exponential in the second root to an exponential in the third root of the number of states. This almost matches a known lower bound based on the exponential time hypothesis by Fernau and Krebs. (2) It is established that the standard regular operations of union, intersection, complementation and Kleene star cause either only a polynomial or a quasipolynomial blow-up. Concatenation of two $n$-state ufas, in worst case, causes a blow-up from $n$ to a function with an exponent of sixth root of $n$ or worse. Decision problems of finite formulas using regular operations and comparing languages given by $n$-state unambiguous automata, in worst case, require an exponential-type of time under the Exponential Time hypothesis and this complexity goes down to quasipolynomial time in the case that the concatenation of languages is not used in the formula. Merely comparing two languages given by $n$-state ufas in Chrobak Normal Form is in LOGSPACE. (3) Starting from this research, membership of the infinite word given by a unary alphabet language in a fixed regular language of infinite words is shown to be as difficult as constructing the dfa of that language from the given automaton.