Broadcasting on Two-Dimensional Regular Grids

We study an important specialization of the general problem of broadcasting on directed acyclic graphs, namely, that of broadcasting on two-dimensional (2D) regular grids. Consider an infinite directed acyclic graph with the form of a 2D regular grid, which has a single source vertex <inline-form...

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Vydané v:IEEE transactions on information theory Ročník 68; číslo 10; s. 6297 - 6334
Hlavní autori: Makur, Anuran, Mossel, Elchanan, Polyanskiy, Yury
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York IEEE 01.10.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:
Apexes
Automata
Automata theory
Boolean
Boolean algebra
Boundary conditions
Broadcasting
Cellular automata
Cellular communication
Communication networks
Discrete time systems
Graph theory
linear code
Linear programming
Martingales
Noise level
Noise levels
Noise measurement
oriented bond percolation
Percolation
potential function
Principal component analysis
probabilistic cellular automata
Probabilistic logic
Statistical analysis
supermartingale
Two dimensional displays
ISSN:0018-9448, 1557-9654
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Shrnutí:We study an important specialization of the general problem of broadcasting on directed acyclic graphs, namely, that of broadcasting on two-dimensional (2D) regular grids. Consider an infinite directed acyclic graph with the form of a 2D regular grid, which has a single source vertex <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> at layer 0, and <inline-formula> <tex-math notation="LaTeX">k + 1 </tex-math></inline-formula> vertices at layer <inline-formula> <tex-math notation="LaTeX">k \geq 1 </tex-math></inline-formula>, which are at a distance of <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula>. Every vertex of the 2D regular grid has outdegree 2, the vertices at the boundary have indegree 1, and all other non-source vertices have indegree 2. At time 0, <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> is given a uniform random bit. At time <inline-formula> <tex-math notation="LaTeX">k \geq 1 </tex-math></inline-formula>, each vertex in layer <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> receives transmitted bits from its parents in layer <inline-formula> <tex-math notation="LaTeX">k-1 </tex-math></inline-formula>, where the bits pass through independent binary symmetric channels with common crossover probability <inline-formula> <tex-math notation="LaTeX">\delta \in \left({0,\frac {1}{2}}\right) </tex-math></inline-formula> during the process of transmission. Then, each vertex at layer <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> with indegree 2 combines its two input bits using a common deterministic Boolean processing function to produce a single output bit at the vertex. The objective is to recover <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> with probability of error better than <inline-formula> <tex-math notation="LaTeX">\frac {1}{2} </tex-math></inline-formula> from all vertices at layer <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> as <inline-formula> <tex-math notation="LaTeX">k \rightarrow \infty </tex-math></inline-formula>. Besides their natural interpretation in the context of communication networks, such broadcasting processes can be construed as one-dimensional (1D) probabilistic cellular automata, or discrete-time statistical mechanical spin-flip systems on 1D lattices, with boundary conditions that limit the number of sites at each time <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">k+1 </tex-math></inline-formula>. Inspired by the literature surrounding the "positive rates conjecture" for 1D probabilistic cellular automata, we conjecture that it is impossible to propagate information in a 2D regular grid regardless of the noise level <inline-formula> <tex-math notation="LaTeX">\delta </tex-math></inline-formula> and the choice of common Boolean processing function. In this paper, we make considerable progress towards establishing this conjecture, and prove using ideas from percolation and coding theory that recovery of <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> is impossible for any <inline-formula> <tex-math notation="LaTeX">\delta \in \left({0,\frac {1}{2}}\right) </tex-math></inline-formula> provided that all vertices with indegree 2 use either AND or XOR for their processing functions. Furthermore, we propose a detailed and general martingale-based approach that establishes the impossibility of recovering <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> for any <inline-formula> <tex-math notation="LaTeX">\delta \in \left({0,\frac {1}{2}}\right) </tex-math></inline-formula> when all NAND processing functions are used if certain structured supermartingales can be rigorously constructed. We also provide strong numerical evidence for the existence of these supermartingales by computing several explicit examples for different values of <inline-formula> <tex-math notation="LaTeX">\delta </tex-math></inline-formula> via linear programming.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3177667