Do CFLOBDDs Actually Make Use of Linear Structure?

Binary Decision Diagrams (BDDs) are a widely used data structure for efficient Boolean function representation. Context-Free-Language Ordered Binary Decision Diagrams (CFLOBDDs) are a recently introduced hierarchical data structure that can, in the best case, exhibit exponential compression over BDDs and double-exponential compression over decision trees. Roughly speaking, a CFLOBDD is a finite, acyclic, non-recursive hierarchical finite-state machine (HFSM) (with some additional restrictions). In this paper, we investigate the role of \emph{linear structure} in CFLOBDDs -- a property that connects them to Nested-Word Automata (NWAs) and Visibly Pushdown Automata (VPAs) -- and examine whether CFLOBDDs actively exploit this structure beyond their well-studied hierarchical properties. We demonstrate that linear structure, in conjunction with hierarchical structure, plays a crucial role in enabling CFLOBDDs to achieve efficient function compression. Furthermore, we show that removing linearity from CFLOBDDs leads to a significant blowup in representation size, resulting in degraded performance in the domain of quantum-circuit simulation.