# Liveness in Broadcast Networks

## TL;DR

This paper presents polynomial-time algorithms for liveness and fair liveness verification in broadcast networks, resolving open complexity questions and enabling model checking against linear temporal logic specifications.

## Contribution

It introduces efficient algorithms for liveness verification in broadcast networks, closing a decade-long complexity gap and extending to model checking with temporal logic.

## Key findings

- Polynomial-time algorithm for Liveness Verification.
- Polynomial-time algorithm for Fair Liveness Verification.
- Reduction techniques for model checking problems.

## Abstract

We study liveness and model checking problems for broadcast networks, a system model of identical clients communicating via message passing. The first problem that we consider is Liveness Verification. It asks whether there is a computation such that one of the clients visits a final state infinitely often. The complexity of the problem has been open since 2010 when it was shown to be P-hard and solvable in EXPSPACE. We close the gap by a polynomial-time algorithm. The algorithm relies on a characterization of live computations in terms of paths in a suitable graph, combined with a fixed-point iteration to efficiently check the existence of such paths. The second problem is Fair Liveness Verification. It asks for a computation where all participating clients visit a final state infinitely often. We adjust the algorithm to also solve fair liveness in polynomial time. Both problems can be instrumented to answer model checking questions for broadcast networks against linear time temporal logic specifications. The first problem in this context is Fair Model Checking. It demands that for all computations of a broadcast network, all participating clients satisfy the specification. We solve the problem via the Vardi-Wolper construction and a reduction to Liveness Verification. The second problem is Sparse Model Checking. It asks whether each computation has a participating client that satisfies the specification. We reduce the problem to Fair Liveness Verification.