We study context-bounded verification of liveness properties of multi-threaded, shared-memory programs, where each thread can spawn additional threads. Our main result shows that context-bounded fair termination is decidable for the model; context-bounded implies that each spawned thread can be context switched a fixed constant number of times. Our proof is technical, since fair termination requires reasoning about the composition of unboundedly many threads each with unboundedly large stacks. In fact, techniques for related problems, which depend crucially on replacing the pushdown threads with finite-state threads, are not applicable. Instead, we introduce an extension of vector addition systems with states (VASS), called VASS with balloons (VASSB), as an intermediate model; it is an infinite-state model of independent interest. A VASSB allows tokens that are themselves markings (balloons). We show that context bounded fair termination reduces to fair termination for VASSB. We show the latter problem is decidable by showing a series of reductions: from fair termination to configuration reachability for VASSB and thence to the reachability problem for VASS. For a lower bound, fair termination is known to be non-elementary already in the special case where threads run to completion (no context switches). We also show that the simpler problem of context-bounded termination is 2EXPSPACE-complete, matching the complexity bound---and indeed the techniques---for safety verification. Additionally, we show the related problem of fair starvation, which checks if some thread can be starved along a fair run, is also decidable in the context-bounded case. The decidability employs an intricate reduction from fair starvation to fair termination. Like fair termination, this problem is also non-elementary.
A fundamental problem in refinement verification is to check that the language of behaviors of an implementation is included in the language of the specification. We consider the refinement verification problem where the implementation is a multithreaded shared memory system modeled as a multistack pushdown automaton and the specification is an input-deterministic multistack pushdown language. Our main result shows that the context-bounded refinement problem, where we ask that all behaviors generated in runs of bounded number of context switches belong to a specification given by a Dyck language, is decidable and coNP-complete. The more general case of input-deterministic languages follows, with the same complexity. Context-bounding is essential since emptiness for multipushdown automata is already undecidable, and so is the refinement verification problem for the subclass of regular specifications. Input-deterministic languages capture many non-regular specifications of practical interest and our result opens the way for algorithmic analysis of these properties. The context-bounded refinement problem is coNP-hard already with deterministic regular specifications; our result demonstrates that the problem is not harder despite the stronger class of specifications. Our proof introduces several general techniques for formal languages and counter programs and shows that the search for counterexamples can be reduced in non-deterministic polynomial time to the satisfiability problem for existential Presburger arithmetic. These techniques are essential to ensure the coNP upper bound: existing techniques for regular specifications are not powerful enough for decidability, while simple reductions lead to problems that are either undecidable or have high complexities. As a special case, our decidability result gives an algorithmic verification technique to reason about reference counting and re-entrant locking in multithreaded programs.
Thread pooling is a common programming idiom in which a fixed set of worker threads are maintained to execute tasks concurrently. The workers repeatedly pick tasks and execute them to completion. Each task is sequential, with possibly recursive code, and tasks communicate over shared memory. Executing a task can lead to more new tasks being spawned. We consider the safety verification problem for thread-pooled programs. We parameterize the problem with two parameters: the size of the thread pool as well as the number of context switches for each task. The size of the thread pool determines the number of workers running concurrently. The number of context switches determines how many times a worker can be swapped out while executing a single task---like many verification problems for multithreaded recursive programs, the context bounding is important for decidability. We show that the safety verification problem for thread-pooled, context-bounded, Boolean programs is EXPSPACE-complete, even if the size of the thread pool and the context bound are given in binary. Our main result, the EXPSPACE upper bound, is derived using a sequence of new succinct encoding techniques of independent language-theoretic interest. In particular, we show a polynomial-time construction of downward closures of languages accepted by succinct pushdown automata as doubly succinct nondeterministic finite automata. While there are explicit doubly exponential lower bounds on the size of nondeterministic finite automata accepting the downward closure, our result shows these automata can be compressed. We show that thread pooling significantly reduces computational power: in contrast, if only the context bound is provided in binary, but there is no thread pooling, the safety verification problem becomes 3EXPSPACE-complete. Given the high complexity lower bounds of related problems involving binary parameters, the relatively low complexity of safety verification with thread-pooling comes as a surprise.
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