Complexity of promise problems on classical and quantum automata

Abstract

We consider the promise problem AN,r1,r2 on a unary alphabet {σ} studied by Gruska et al. in [21]. This problem is formally defined as the pair AN,r1,r2 = (AN,r1yes,AN,r2 no), with 0 ≤ r1 _= r2 < N, AN,r1 yes = {σn | n ≡r1 mod N} and AN,r2 no = {σn | n ≡r2 mod N}. There, it is shown that a measure-once one-way quantum automaton can solve exactly AN,r1,r2 with only 3 basis states, while any one-way deterministic finite automaton requires d states, d being the smallest integer such that d | N and d _ (r2−r1) mod N. Here, we introduce the promise problem Diofa,N r1,r2 as an extension of AN,r1,r2 to general alphabets. Even for this problem, we show the same descriptional superiority of the quantum paradigm over one-way deterministic automata. Moreover, we prove that even by adding features to classical automata, namely nondeterminism, probabilism, two-way motion, we cannot obtain automata for AN,r1,r2 and Diofa,N r1,r2 smaller than one-way deterministic.