A Truly Symbolic Linear-Time Algorithm for SCC Decomposition

Abstract

Decomposing a directed graph to its strongly connected components (SCCs) is a fundamental task in model checking. To deal with the state-space explosion problem, graphs are often represented symbolically using binary decision diagrams (BDDs), which have exponential compression capabilities. The theoretically-best symbolic algorithm for SCC decomposition is Gentilini et al’s $$\textsc {Skeleton}$$ S K E L E T O N algorithm, that uses O(n) symbolic steps on a graph of n nodes. However, $$\textsc {Skeleton}$$ S K E L E T O N uses $$\Theta (n)$$ Θ ( n ) symbolic objects, as opposed to (poly-)logarithmically many, which is the norm for symbolic algorithms, thereby relinquishing its symbolic nature. Here we present $$\textsc {Chain}$$ C H A I N , a new symbolic algorithm for SCC decomposition that also makes O(n) symbolic steps, but further uses logarithmic space, and is thus truly symbolic. We then extend $$\textsc {Chain}$$ C H A I N to $$\textsc {ColoredChain}$$ C O L O R E D C H A I N , an algorithm for SCC decomposition on edge-colored graphs, which arise naturally in model-checking a family of systems. Finally, we perform an experimental evaluation of $$\textsc {Chain}$$ C H A I N among other standard symbolic SCC algorithms in the literature. The results show that $$\textsc {Chain}$$ C H A I N is competitive on almost all benchmarks, and often faster, while it clearly outperforms all other algorithms on challenging inputs.