Parallel symbolic state-space exploration is difficult,but what is the alternative?

State-space exploration is an essential first step in many modeling and analysis problems. Its goal is to find and store all the states reachable from the initial state(s) of a discrete-state high-level model described, for example, using pseudocode or Petri nets. The state space can then be used to answer important questions, such as "Is there a dead state?" and "Can variable n ever become negative?", or as the starting point for sophisticated investigations expressed in temporal logic.Unfortunately, the state space is often so large that ordinary explicit data structures and sequential algorithms simply cannot cope, prompting the exploration of parallel or symbolic approaches. The former uses multiple processors, from simple networks of workstations to expensive shared-memory supercomputers or, more recently, powerful multicore workstations, to satisfy the large memory and runtime requirements. The latter uses decision diagrams to compactly encode the large structured sets and relations manipulated during state-space generation.Both approaches have merits and limitations. Parallel explicit state-space generation is challenging, but close to linear speedup can be achieved, thus its scalability can be quite good; however, the analysis is ultimately and obviously limited by the amount of memory and number of processors available overall. Symbolic methods rely on the heuristic properties of decision diagrams, which can encode many, but by no means all, functions over a structured and exponentially large domain in polynomial space; here the pitfalls are subtler, as the performance of symbolic approaches can vary widely depending on the particular class of decision diagram chosen, on the order in which the variables describing the state are considered, and on many obscure algorithmic parameters.In this paper, we survey both approaches. Observing that symbolic approaches are often enormously more efficient than explicit ones for many practical models (although it is rarely obvious a priori whether this will be the case on a particular application), we argue for the need to parallelize symbolic state-space generation algorithms, so that we can realize the advantage of both approaches. Unfortunately, this is a very challenging endeavor, as the most efficient symbolic algorithm, Saturation, is inherently sequential. We conclude by discussing challenges, efforts, and promising directions toward this goal.