Alternating, private alternating, and quantum alternating realtime automata

TL;DR

This paper explores the computational limits of various automata models, demonstrating undecidability of the emptiness problem in several cases and identifying languages recognized by quantum automata with multiple alternations.

Contribution

It introduces new undecidability results for automata models and characterizes the languages recognized by quantum automata with multiple alternations.

Findings

Emptiness problem undecidable for alternating one-counter automata on unary alphabets.

Emptiness problem undecidable for private alternating finite automata.

Unary squares language recognized by alternating quantum automata with two alternations.

Abstract

We present new results on realtime alternating, private alternating, and quantum alternating automaton models. Firstly, we show that the emptiness problem for alternating one-counter automata on unary alphabets is undecidable. Then, we present two equivalent definitions of realtime private alternating finite automata (PAFAs). We show that the emptiness problem is undecidable for PAFAs. Furthermore, PAFAs can recognize some nonregular unary languages, including the unary squares language, which seems to be difficult even for some classical counter automata with two-way input. Regarding quantum finite automata (QFAs), we show that the emptiness problem is undecidable both for universal QFAs on general alphabets, and for alternating QFAs with two alternations on unary alphabets. On the other hand, the same problem is decidable for nondeterministic QFAs on general alphabets. We also show that the unary squares language is recognized by alternating QFAs with two alternations.