Operations on Automata with All States Final

A language L is prefix-closed if w ∈ L implies that every prefix of w is in L. It is known that a regular language is prefix-closed if and only if it is accepted by a nondeterministic finite automaton (NFA) with all states final [18]. In the minimal incomplete deterministic finite automaton (DFA) for a prefix-closed language, all the states are final as well. The authors of [18] examined several questions concerning NFAs with all states final. They proved that the inequivalence problem for NFAs with all states final is PSPACE-complete in the binary case, but polynomially solvable in the unary case. Next, they showed that minimizing a binary NFA with all states final is PSPACE-hard, and that deciding whether a given NFA accepts a language that is not prefixclosed is PSPACE-complete, while the same problem for DFAs can be solved in polynomial time. The NFA-to-DFA conversion and complementation of NFAs with all states final have been also considered in [18], and the tight bound 2n for the first problem, and the lower bound 2n−1 for the second one have been obtained. The quotient complexity of prefix-closed languages has been studied in [5]. The quotient of a language L by the string w is the set Lw = {x | wx ∈ L}. The quotient complexity of a language L, κ(L), is the number of distinct quotients of L. Quotient complexity is defined for any language, and it is finite if and only if the language is regular. The quotient automaton of a regular language L is the DFA ({Lw | w ∈ Σ∗},Σ, ·,Lε ,F), where Lw ·a = Lwa, and a quotient Lw is final if it contains the empty string. The quotient automaton of L is a minimal complete DFA for L, so quotient complexity is the same as the state complexity of L which is defined as the number of states in the minimal DFA for L. In [5], the tight bounds on the quotient complexity of basic regular operation have been obtained, and to prove upper bounds, the properties of quotients have been used rather than automata constructions.