On the complexity of freezing automata networks of bounded pathwidth

TL;DR

This work investigates the computational complexity of freezing automata networks on graphs with bounded pathwidth, focusing on the specification-checking and model-checking problems. It proves that Regular Specification Checking ($REGSPEC$) is $NL$-complete under bounded pathwidth, using a logspace reduction from $STCON$ and a constructive encoding on a narrow grid; it also provides a detailed $DLOGSPACE$ reduction detailing how a sequence of blocks encodes reachability. It further shows that first-order logic with reachability predicates, $FO^+$, remains computationally hard to model-check even for one-dimensional freezing networks, by reducing the $HV$-domino CSP to a deterministic network and crafting a formula that captures vertical constraints. Together, the results delineate a sharp boundary between tractable and intractable dynamics under strong graph-structural restrictions, highlighting a complexity gap relative to prior results for graphs of bounded treewidth. The findings have implications for verification tasks in diffusion-like processes and self-assembly models where the underlying structure is nearly linear. $REGSPEC$ and $FO^+$ thus capture essential, distinct facets of the complexity landscape for freezing automata networks on narrow graphs.

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.

Figures (6)

• Figure 1: An example of a block for some graph $D$.
• Figure 2: Example of the verification dynamics for a periodic pattern. In this case, the pattern is given by the second row of a block representing the edge $(1,2)$ in some graph.
• Figure 3: Example of the verification dynamics for a marker. (Upper panel) A successful verification of a marker. If exactly one zone has only cells in state $1$ an acceptance state will be reached. (Middle panel) An error in the verification raised by a zone in which cells in state $0$ and $1$ were identified. In this case, an error state is propagated. (Lower panel) An error in the verification raised by the signal only reading cells in state $0$. In this case, an error state is propagated.
• Figure 5: Example of a verification dynamics which compares blocks and checks if the corresponding edges are both incident to the same node. If exactly one tail signal collide with a head it means that the previous property is verified and an error state is spread otherwise.
• Figure 6: Example of valid orbit with ${n=3}$ starting from a configuration testing the V-constraint $V_{2,2}$ and resulting in a positive output. The trajectory of the $Q_h$ head is reproduced on the $Q_t$-component to clarify the interactions. The vertical thick lines represent separations between consecutive blocks.

Theorems & Definitions (12)

• definition thmcounterdefinition
• definition thmcounterdefinition
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary
• proof
• theorem thmcountertheorem
• proof : Proof of Theorem \ref{['theo:nlhard']}
• lemma thmcounterlemma: HV-domino CSP
• proof