The Hardness of Local Certification of Finite-State Dynamics

Finite-State Dynamics (FSD) is one of the simplest and constrained distributed systems. An FSD is defined by an $n$-node network, with each node maintaining an internal state selected from a finite set. At each time-step, these nodes synchronously update their internal states based solely on the states of their neighboring nodes. Rather than focusing on specific types of local functions, in this article, our primary focus is on the problem of determining the maximum time required for an FSD to reach a stable global state. This global state can be seen as the acceptance state or as the output of a distributed computation. For fixed $k$ and $q$, we define the problem $\text{convergence}(k,q)$, which consists of deciding if a $q$-state FSD converges in at most $k$ time-steps. Our main focus is to study the problem $\text{convergence}$ from the perspective of distributed certification, with a focus on the model of proof-labeling schemes (PLS). First, we study the problem $\text{convergence}$ on arbitrary graphs and show that every PLS has certificates of size $\Theta(n^2)$ (up to logarithmic factors). Then, we turn to the restriction of the problem on graphs of maximum degree $\Delta$. Roughly, we show that the problem admits a PLS with certificates of size $\Delta^{k+1}$, while every PLS requires certificates of size at least $2^{k/6} \cdot 6/k$ on graphs of maximum degree 3.