Complexity of Liveness in Parameterized Systems

TL;DR

This paper analyzes the computational complexity of liveness verification in leader contributor systems, providing improved algorithms and matching lower bounds, thus advancing understanding of the problem's inherent difficulty.

Contribution

It introduces refined algorithms for liveness verification, connecting reachability and cycle detection, and settles an open problem on complexity bounds.

Findings

Polynomial-time cycle detection algorithm for liveness verification.

Improved (L + D)^O(L + D)-time algorithm for leader state and data domain parameterization.

Matching lower bounds confirm the problem's inherent NP-completeness.

Abstract

We investigate the fine-grained complexity of liveness verification for leader contributor systems. These consist of a designated leader thread and an arbitrary number of identical contributor threads communicating via a shared memory. The liveness verification problem asks whether there is an infinite computation of the system in which the leader reaches a final state infinitely often. Like its reachability counterpart, the problem is known to be NP-complete. Our results show that, even from a fine-grained point of view, the complexities differ only by a polynomial factor. Liveness verification decomposes into reachability and cycle detection. We present a fixed point iteration solving the latter in polynomial time. For reachability, we reconsider the two standard parameterizations. When parameterized by the number of states of the leader L and the size of the data domain D, we show an (L + D)^O(L + D)-time algorithm. It improves on a previous algorithm, thereby settling an open problem. When parameterized by the number of states of the contributor C, we reuse an O*(2^C)-time algorithm. We show how to connect both algorithms with the cycle detection to obtain algorithms for liveness verification. The running times of the composed algorithms match those of reachability, proving that the fine-grained lower bounds for liveness verification are met.