Complexity classification of monomial equations AX = B over FDDS

Determine the computational complexity of solving monomial equations of the form AX = B over the semiring of finite discrete-time dynamical systems (FDDS), specifically deciding whether the problem is NP-hard, solvable in polynomial time, or exhibits intermediate complexity.

Background

The paper studies polynomial equations over the semiring of finite discrete-time dynamical systems (FDDS), where addition corresponds to alternative execution and multiplication to synchronous parallel execution. While general polynomial equations with variables on both sides are undecidable, the authors focus on the decidable and NP-contained variant with a constant right-hand side.

They provide polynomial-time algorithms for classes such as pseudo-injective polynomials and certain cases of AX = B when A is pseudo-cancelable. However, for unrestricted monomial equations AX = B, the global complexity classification remains unknown, motivating the explicit open problem below.