On the complexity of freezing automata networks of bounded pathwidth

TL;DR

This paper investigates the computational complexity of freezing automata networks with bounded pathwidth, revealing hardness results for trace checking and first-order property decision problems.

Contribution

It demonstrates that even with the restrictive graph structure of bounded pathwidth, certain decision problems remain computationally hard.

Findings

Trace specification checking is NL-complete.

Deciding first-order properties with reachability is NP-hard.

Abstract

An automata network is a graph of entities, each holding a state from a finite set and evolving according to a local update rule which depends only on its neighbors in the network's graph. It is freezing if there is an order on the states such that the state evolution of any node is non-decreasing in any orbit. They are commonly used to model epidemic propagation, diffusion phenomena like bootstrap percolation or cristal growth. Previous works have established that, under the hypothesis that the network graph is of bounded treewidth, many problems that can be captured by trace specifications at individual nodes admit efficient algorithms. In this paper we study the even more restricted case of a network of bounded pathwidth and show two hardness results that somehow illustrate the complexity of freezing dynamics under such a strong graph constraint. First, we show that the trace specification checking problem is NL-complete. Second, we show that deciding first order properties of the orbits augmented with a reachability predicate is NP-hard.