On equivalence and emptiness problems of multi-letter (measure many) quantum finite automata

On equivalence and emptiness problems of multi-letter (measure many) quantum finite automata Abstract In this paper, we study some decision problems both for multi-letter quantum finite au-tomata and measure many multi-letter quantum finite automata. We first show that given a k1-letter quantum finite automaton A1 and a k2-letter quantum finite automaton A2 over the same input alphabet Σ, they are equivalent if and only if they are (n 2 1 + n 2 2 − 1)|Σ| k−1 + k-equivalent where ni, i = 1, 2, are the number of states in Ai respectively, and k = max{k1, k2}. By applying a method, due to the author, used to deal with the equivalence problem of measure many one-way quantum finite automata, we also show that a k1-letter measure many quantum finite automaton A1 and a k2-letter measure many quantum finite automaton A2 are equivalent if and only if they are (n 2 1 + n 2 2 − 1)|Σ| k−1 + k-equivalent where ni, i = 1, 2, are the number of states in Ai respectively, and k = max{k1, k2}. Next, we study the emptiness problem of those two kinds of quantum finite automata. We show that whether the language recognized by a k-letter quantum finite automaton with non-strict cut-point is empty is undecidable, but we leave open the emptiness of language reorganized by a k-letter quantum finite automaton with strict cutpoint. We also show that whether the languages recognized by a k-letter measure many quantum finite automaton with both nonstrict and strict cutpoints are undecidable. And the direct consequences of the above outcomes are summarized in the paper.