Complexity of Unary Exclusive Nondeterministic Finite Automata

TL;DR

This paper explores the descriptional complexity and decision problem complexities of unary exclusive nondeterministic finite automata (XNFA), revealing exponential state costs for simulations and coNP-completeness for key decision problems.

Contribution

It provides tight bounds on state costs for simulating XNFA with DFA/NFA in the unary case and analyzes the computational complexity of fundamental decision problems.

Findings

State costs for simulation are exponential in the unary case.

Emptiness, universality, inclusion, and equivalence are coNP-complete.

Membership problem is NL-complete.

Abstract

Exclusive nondeterministic finite automata (XNFA) are nondeterministic finite automata with a special acceptance condition. An input is accepted if there is exactly one accepting path in its computation tree. If there are none or more than one accepting paths, the input is rejected. We study the descriptional complexity of XNFA accepting unary languages. While the state costs for mutual simulations with DFA and NFA over general alphabets differ significantly from the known types of finite automata, it turns out that the state costs for the simulations in the unary case are in the order of magnitude of the general case. In particular, the state costs for the simulation of an XNFA by a DFA or an NFA are $e^{\theta(\sqrt{n \cdot ln{n}})}$. Conversely, converting an NFA to an equivalent XNFA may cost $e^{\theta(\sqrt{n \cdot ln{n}})}$ states as well. All bounds obtained are also tight in the order of magnitude. Finally, we investigate the computational complexity of different decision problems for unary XNFA and it is shown that the problems of emptiness, universality, inclusion, and equivalence are coNP-complete, whereas the general membership problem is NL-complete.