Checking Whether an Automaton Is Monotonic Is NP-complete

TL;DR

This paper proves that determining whether an automaton is monotonic or oriented is NP-complete, even with binary input alphabets, highlighting the computational difficulty of these problems.

Contribution

It establishes the NP-completeness of checking automaton monotonicity and oriented automaton properties, including for binary input alphabets.

Findings

Deciding automaton monotonicity is NP-complete.

Checking oriented automata is NP-complete.

Problems remain hard with binary input alphabets.

Abstract

An automaton is monotonic if its states can be arranged in a linear order that is preserved by the action of every letter. We prove that the problem of deciding whether a given automaton is monotonic is NP-complete. The same result is obtained for oriented automata, whose states can be arranged in a cyclic order. Moreover, both problems remain hard under the restriction to binary input alphabets.