Complexity of Universality and Related Decision Problems for Unary Two-Dimensional Automata

2026-06-30

AI Summary

This study investigates the computational complexity of fundamental decision problems—universality, equivalence, inclusion, and disjointness—for unary two-dimensional automata. Focusing on deterministic and nondeterministic variants with restricted movement directions (specifically, three-way and two-way models), the work provides the first complete characterization of the complexity landscape across these automaton classes. By leveraging automata-theoretic techniques, complexity analysis, and careful reductions, the paper establishes tight upper and lower bounds for multiple problems. Notably, it proves that bounded universality is coNP-complete and precisely classifies several other problems within established complexity classes, including P, NL-hard, coNP-hard, and ELEMENTARY.

Abstract

A two-dimensional automaton is able to move its input head through its input word in four directions: upward, downward, leftward, and rightward. If we prevent the input head from moving upward, then we obtain a three-way two-dimensional automaton; preventing both upward and leftward movements results in a two-way two-dimensional automaton. While much is known about the decidability and complexity properties of the two-dimensional automaton model, the unary variant of this model is less studied. We show that the universality, equivalence, and inclusion problems for unary three-way deterministic two-dimensional automata are coNP-hard, while for the corresponding two-way model, the universality, equivalence, inclusion, and disjointness problems are in P. We further show that the universality, equivalence, and inclusion problems for unary two-way nondeterministic two-dimensional automata are coNP-hard and in ELEMENTARY; and the disjointness problem for the same model is NL-hard and in ELEMENTARY. Finally, we establish the decidability of a bounded variant of the universality problem for unary three-way nondeterministic two-dimensional automata, and show that this variant problem is coNP-complete.

Problem

• universality
• two-dimensional automata
• unary
• decision problems
• computational complexity

Innovation

• two-dimensional automata
• unary alphabet
• computational complexity
• universality problem
• coNP-hard