Fine-Grained Complexity of Safety Verification

Abstract

We study the fine-grained complexity of Leader Contributor Reachability ({textsf {LCR}} ) and Bounded-Stage Reachability ({textsf {BSR}} ), two variants of the safety verification problem for shared memory concurrent programs. For both problems, the memory is a single variable over a finite data domain. Our contributions are new verification algorithms and lower bounds. The latter are based on the Exponential Time Hypothesis ({textsf {ETH}} ), the problem {textsf {Set~Cover}} , and cross-compositions. {textsf {LCR}} is the question whether a designated leader thread can reach an unsafe state when interacting with a certain number of equal contributor threads. We suggest two parameterizations: (1) By the size of the data domain {texttt {D}} and the size of the leader {texttt {L}}, and (2) by the size of the contributors {texttt {C}}. We present algorithms for both cases. The key techniques are compact witnesses and dynamic programming. The algorithms run in {mathcal {O}}^*(({texttt {L}}cdot ({texttt {D}}+1))^{{texttt {L}}cdot {texttt {D}}} cdot {texttt {D}}^{{texttt {D}}}) and {mathcal {O}}^*(2^{{texttt {C}}}) time, showing that both parameterizations are fixed-parameter tractable. We complement the upper bounds by (matching) lower bounds based on {textsf {ETH}} and {textsf {Set~Cover}} . Moreover, we prove the absence of polynomial kernels. For {textsf {BSR}} , we consider programs involving {texttt {t}} different threads. We restrict the analysis to computations where the write permission changes {texttt {s}} times between the threads. {textsf {BSR}} asks whether a given configuration is reachable via such an {texttt {s}}-stage computation. When parameterized by {texttt {P}}, the maximum size of a thread, and {texttt {t}}, the interesting observation is that the problem has a large number of difficult instances. Formally, we show that there is no polynomial kernel, no compression algorithm that reduces the size of the data domain {texttt {D}} or the number of stages {texttt {s}} to a polynomial dependence on {texttt {P}} and {texttt {t}}. This indicates that symbolic methods may be harder to find for this problem.