Classical automata on promise problems

In automata theory, promise problems have been mainly examined for quantum automata. In this paper, we focus on classical automata and obtain some new results regarding the succinctness of models and their computational powers. We start with a negative result. Recently, Ambainis and YakaryIlmaz (2012) introduced a quantumly very cheap family of unary promise problems, i.e. solvable exactly by using only a single qubit. We show that two-way nondeterminism does not have any advantage over realtime determinism for this family of promise problems. Secondly, we present some basic facts for classical models: The computational powers of deterministic, nondeterministic, alternating, and Las Vegas probabilistic automata are the same. Then, we show that any gap of succinctness between any two of deterministic, nondeterministic, and alternating automata models for language recognition cannot be violated on promise problems. On the other hand, we show that the tight quadratic gap between Las Vegas realtime probabilistic automata and realtime deterministic automata given for language recognition can be replaced with a tight exponential gap on promise problems. Lastly, we show how the situation can be different when considering two-sided bounded-error. Similar to quantum case, we present a probabilistically very cheap family of unary promise problems, i.e. solvable by a 2-state automaton with bounded-error. Then, we show that this family is not cheap for any of the aforementioned classical models. Moreover, we show that bounded-error probabilistic automata are more powerful than any other classical model on promise problems.