https://hal.inria.fr/hal-01349053Geffert, ViliamViliamGeffertInstitute of Computer Science [Kosice] - P.J. Safarik UniversityYakaryilmaz, AbuzerAbuzerYakaryilmazLNCC / MCT - Laboratorio Nacional de Computação Cientifica [Rio de Janeiro]Classical Automata on Promise ProblemsHAL CCSD2015descriptional complexitypromise problemsnondeterministic automataprobabilistic automataalternating automata[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Episciences Iam, Coordination2016-07-26 17:21:592017-11-13 15:22:012016-07-26 17:56:37enJournal articleshttps://hal.inria.fr/hal-01349053/document10.46298/dmtcs.2138application/pdf1Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Second, despite of the existing quadratic gap between Las Vegas realtime probabilistic automata and one-way deterministic automata for language recognition, we show that, by turning to promise problems, the tight gap becomes exponential. Last, we show that the situation is different for one-way probabilistic automata with two-sided bounded-error. We present a family of unary promise problems that is very easy for these machines; solvable with only two states, but the number of states in two-way alternating or any simpler automata is not limited by a constant. Moreover, we show that one-way bounded-error probabilistic automata can solve promise problems not solvable at all by any other classical model.