2014/05/26 by Viliam Geffert, Geffert, Viliam, Abuzer Yakaryilmaz +1
Computer Science · #Computational Complexity (cs.CC) #FOS: Computer and information sciences #Formal Languages and Automata Theory (cs.FL) #cs.CC #cs.FL
paper · pdf · doi:10.48550/arxiv.1405.6671
21 pages, significantly improved with a correction (the single sentence statement given just before Corollary 4 in the previous version and (in the conference version) is not correct). A preliminary version appeared in DCFS2014 [vol. 8614 of LNCS, pp. 126--137, Springer-Verlag, 2014]
arxiv created 2014/10/16 · arxiv updated 2014/10/17
Promise 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.