New results on classical and quantum counter automata

TL;DR

This paper demonstrates that quantum and probabilistic one-counter automata can outperform deterministic ones on certain promise problems and language recognition tasks, revealing new separations and refuting previous conjectures.

Contribution

It establishes new separation results between quantum, probabilistic, and deterministic counter automata, and disproves a conjecture about probabilistic blind-counter automata.

Findings

Quantum one-counter automata are more powerful than probabilistic ones on promise problems.

Probabilistic one-blind-counter automata can recognize the Kleene closure of the equality language.

Several separation results between blind and non-blind counter automata.

Abstract

We show that one-way quantum one-counter automaton with zero-error is more powerful than its probabilistic counterpart on promise problems. Then, we obtain a similar separation result between Las Vegas one-way probabilistic one-counter automaton and one-way deterministic one-counter automaton. We also obtain new results on classical counter automata regarding language recognition. It was conjectured that one-way probabilistic one blind-counter automata cannot recognize Kleene closure of equality language [A. Yakaryilmaz: Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. and Applic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and also show several separation results for blind/non-blind counter automata.