The Hardness of Local Certification of Finite-State Dynamics
CoRR(2023)
Abstract
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.
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