Reachability in parallel programs is polynomial in the number of threads

Abstract

Reachability in parallel finite-state programs equipped with interleaving semantics is an inherently difficult, important problem. Its complexity in the number of threads n, while keeping the thread-local–memory size and the shared-memory size bounded by constants, has been explored only poorly. We significantly narrow this gap by measuring: (i) the diameter, i.e., the longest finite distance realizable in the transition graph of the program, (ii) the local diameter, i.e., the maximum finite distance from any program state to any thread-local state, and (iii) the computational complexity of finding bugs. We prove that all these are majorized by polynomials in n and, in certain cases, by linear, logarithmic, or even constant functions in n; we make the bounds explicit whenever possible. Our results shed new light on the widely expressed claim that one of the major obstacles to analyzing parallel programs is the exponential state explosion in the number of threads.