Linear Programs with Polynomial Coefficients and Applications to 1D Cellular Automata

Given a matrix $A$ and vector $b$ with polynomial entries in $d$ real variables $δ=(δ_1,\ldots,δ_d)$ we consider the following notion of feasibility: the pair $(A,b)$ is locally feasible if there exists an open neighborhood $U$ of $0$ such that for every $δ\in U$ there exists $x$ satisfying $A(δ)x\ge b(δ)$ entry-wise. For $d=1$ we construct a polynomial time algorithm for deciding local feasibility. For $d \ge 2$ we show local feasibility is NP-hard. This also gives the first polynomial-time algorithm for the asymptotic linear program problem introduced by Jeroslow in 1973. As an application (which was the primary motivation for this work) we give a computer-assisted proof of ergodicity of the following elementary 1D cellular automaton: given the current state $η_t \in \{0,1\}^{\mathbb{Z}}$ the next state $η_{t+1}(n)$ at each vertex $n\in \mathbb{Z}$ is obtained by $η_{t+1}(n)= \text{NAND}\big(\text{BSC}_δ(η_t(n-1)), \text{BSC}_δ(η_t(n))\big)$. Here the binary symmetric channel $\text{BSC}_δ$ takes a bit as input and flips it with probability $δ$ (and leaves it unchanged with probability $1-δ$). It is shown that there exists $δ_0>0$ such that for all $0<δ<δ_0$ the distribution of $η_t$ converges to a unique stationary measure irrespective of the initial condition $η_0$. We also consider the problem of broadcasting information on the 2D-grid of noisy binary-symmetric channels $\text{BSC}_δ$, where each node may apply an arbitrary processing function to its input bits. We prove that there exists $δ_0'>0$ such that for all noise levels $0<δ<δ_0'$ it is impossible to broadcast information for any processing function, as conjectured by Makur, Mossel and Polyanskiy.