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

TL;DR

This paper develops algorithms for polynomial coefficient linear programs, proves complexity results, and applies these findings to establish ergodicity in 1D cellular automata and limits of information broadcasting in noisy grids.

Contribution

It introduces a polynomial-time algorithm for 1D polynomial coefficient linear programs and proves NP-hardness for higher dimensions, with applications to cellular automata and information theory.

Findings

Polynomial-time algorithm for 1D local feasibility

NP-hardness of local feasibility for dimensions ≥ 2

Computer-assisted proof of ergodicity in a 1D cellular automaton

Abstract

Given a matrix $A$ and vector $b$ with polynomial entries in $d$ real variables $\delta=(\delta_1,\ldots,\delta_d)$ 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 $\delta\in U$ there exists $x$ satisfying $A(\delta)x\ge b(\delta)$ 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 $\eta_t \in \{0,1\}^{\mathbb{Z}}$ the next state $\eta_{t+1}(n)$ at each vertex $n\in \mathbb{Z}$ is obtained by $\eta_{t+1}(n)= \text{NAND}\big(\text{BSC}_\delta(\eta_t(n-1)), \text{BSC}_\delta(\eta_t(n))\big)$. Here the binary symmetric channel $\text{BSC}_\delta$ takes a bit as input and flips it with probability $\delta$ (and leaves it unchanged with probability $1-\delta$). It is shown that there exists $\delta_0>0$ such that for all $0<\delta<\delta_0$ the distribution of $\eta_t$ converges to a unique stationary measure irrespective of the initial condition $\eta_0$. We also consider the problem of broadcasting information on the 2D-grid of noisy binary-symmetric channels $\text{BSC}_\delta$, where each node may apply an arbitrary processing function to its input bits. We prove that there exists $\delta_0'>0$ such that for all noise levels $0<\delta<\delta_0'$ it is impossible to broadcast information for any processing function, as conjectured by Makur, Mossel and Polyanskiy.