Do CFLOBDDs Actually Make Use of Linear Structure?

TL;DR

This paper investigates whether CFLOBDDs leverage linear structure for efficient Boolean function representation, revealing that linearity is crucial for their compression capabilities and performance in quantum-circuit simulation.

Contribution

It demonstrates the importance of linear structure in CFLOBDDs and shows that removing linearity causes significant size blowup and performance degradation.

Findings

Linear structure is essential for CFLOBDD compression.

Removing linearity increases representation size significantly.

Degradation observed in quantum-circuit simulation performance.

Abstract

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.