Classical automata on promise problems

TL;DR

This paper investigates the state complexity of classical automata solving promise problems, revealing exponential gaps between different models and highlighting the unique capabilities of probabilistic automata.

Contribution

It provides new bounds and comparisons for classical automata on promise problems, including exponential gaps and the power of probabilistic automata.

Findings

Exponential gap between two-way nondeterministic and one-way alternating automata.

Promise problems can cause exponential gaps in automata state complexity.

One-way bounded-error probabilistic automata can solve problems unsolvable by other classical models.

Abstract

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.