{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/144875"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/144875","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"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).","abstract_html":"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 &gt; 0 such that for all 0 &lt; 𝛿 &lt; 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&gt;0 such that for all noise levels 0&lt;δ&lt;δ′₀ it is impossible to broadcast information for any processing function, as conjectured in Makur, Mossel, Polyanskiy (ISIT 2021).","abstract_has_math":false,"creators":["Guo, Chenghao"],"institution":"Massachusetts Institute of Technology","degree_name":"Master","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","school":null,"contributors":[],"advisors":["Bresler, Guy","Polyanskiy, Yury"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-05","date_published":"2022-05","updated_at":"2026-07-22T22:21:20Z","subjects":[],"languages":[],"rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"rights_urls":["http://rightsstatements.org/page/InC-EDU/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/144875","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Bresler, Guy","Polyanskiy, Yury"]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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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)."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["S.M."]},{"key":"dc:title","label":"Title","values":["Linear Programs with Polynomial Coefficients and Applications to 1D Cellular Automata"]}]}],"canonical_facts":{"dc:contributor.advisor":["Bresler, Guy","Polyanskiy, Yury"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science"],"dc:creator":["Guo, Chenghao"],"dc:date.accessioned":["2022-08-29T16:17:51Z"],"dc:date.available":["2022-08-29T16:17:51Z"],"dc:date.issued":["2022-05"],"dc:description.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)."],"dc:description.degree":["S.M."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/144875"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Linear Programs with Polynomial Coefficients and Applications to 1D Cellular Automata"],"dc:type":["Thesis"],"thesis:degree_name":["Master","Master of Science in Electrical Engineering and Computer Science"]},"updated_at":"2026-07-22T22:21:20Z"}