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

Abstract

Given a matrix A and vector b with polynomial entries in d real variables δ=(δ₁,…,δ subscript 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∈U there exists x satisfying A(δ)x≥b(δ) entry-wise. For d=1 we construct a polynomial time algorithm for deciding local feasibility. For d≥2 we show local feasibility is NP-hard.

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 ηₜ∈{0,1}ℤ the next state ηₜ₊₁(n) at each vertex n∈ superscript ℤ is obtained by ηₜ₊₁(n)=NAND(BSCδ(ηₜ(n−1)),BSCδ(ηt(n))). Here the binary symmetric channel BSCδ 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 𝛿₀ > 0 such that for all 0 < 𝛿 < 𝛿₀ the distribution of 𝜂ₜ converges to a unique stationary measure irrespective of the initial condition 𝜂₀.

We also consider the problem of broadcasting information on the 2D-grid of noisy binary-symmetric channels BSCδ, 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 in Makur, Mossel, Polyanskiy (ISIT 2021).