We extend the scope of context-free-language (CFL) reachability to a new class of infinite-state systems. Parikh's Theorem is a useful tool for solving CFL-reachability problems for transition systems that consist of commuting transition relations. It implies that the image of a context-free language under a homomorphism into a commutative monoid is semi-linear, and that there is a linear-time algorithm for constructing a Presburger arithmetic formula that represents it. However, for many transition systems of interest, transitions do not commute.

In this paper, we introduce almost-commuting transition systems, which pair finite-state control with commutative components, but which are in general not commutative. We extend Parikh's theorem to show that the image of a context-free language under a homomorphism into an \textit{almost-commuting monoid is semi-linear and that there is a polynomial-time algorithm for constructing a Presburger arithmetic formula that represents it. This result yields a general framework for solving CFL-reachability problems over \textit{almost-commuting transition systems}. We describe several examples of systems within this class. Finally, we examine closure properties of almost-commuting monoids that can be used to modularly compose almost-commuting transition systems while remaining in the class.