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dc.contributor.authorStone, Bertrand
dc.contributor.authorStone, Bertrand A.
dc.date.accessioned2018-11-20T14:59:58Z
dc.date.issued2018-11
dc.identifier.issn1382-4090
dc.identifier.issn1572-9303
dc.identifier.urihttp://hdl.handle.net/1721.1/119219
dc.description.abstractCellular automata are discrete dynamical systems that consist of patterns of symbols on a grid, which change according to a locally determined transition rule. In this paper, we will consider cellular automata that arise from polynomial transition rules, where the symbols are integers modulo some prime p. We consider the asymptotic behavior of the line complexity sequence a[subscript T](k), which counts, for each k, the number of coefficient strings of length k that occur in the automaton. We begin with the modulo 2 case. For a polynomial T(x)=c[subscript0] + c[subscript 1]x+...+c[subscript n]x[superscript n] with c[subscript 0],c[subscript n] ≠ 0, we construct odd and even parts of the polynomial from the strings0[subscript c[subscript 1]c[subscript 3]c[subscript 5] ... c[subscript 1+2 |_(n-1)/2_|] and c[subscript 0]c[subscript 2]c[susscript 4] .... c[subscript 2[_ n/2_|] respectively. We prove that a[subscript T][superscript (k)] satisfies recursions of a specific form if the odd and even parts of T are relatively prime. We also define the order of such a recursion and show that the property of “having a recursion of some order” is preserved when the transition rule is raised to a positive integer power. Extending to a more general setting, we consider an abstract generating function ϕ(z = ∑[subscript k=1][superscript ∞] α(k)z[superscript k] which satisfies a functional equation relating ϕ(z) and (z[superscript p]) . We show that there is a continuous, piecewise quadratic function f on [1 / p, 1] for which lim[subscript k→ ∞] (α(k)/k[superscript 2] - f(p[superscript -⟨log[subscript p]k))) = 0 (here ⟨y⟩ = y - ⌊y⌋). We use this result to show that for certain positive integer sequences s[subscript k] (x)→ ∞ with a parameter x ∈ [1/p,1], the ratio α(s[subscript k](x))/s[subscript k](x)[superscript 2] tends to f(x), and that the limit superior and inferior of α(k)/k[superscript 2] are given by the extremal values of f. Keywords: Cellular automata, Line complexity, Additive transition rules, Asymptotic estimatesen_US
dc.publisherSpringer USen_US
dc.relation.isversionofhttps://doi.org/10.1007/s11139-017-9957-7en_US
dc.rightsArticle is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.en_US
dc.sourceSpringer USen_US
dc.titleLine complexity asymptotics of polynomial cellular automataen_US
dc.typeArticleen_US
dc.identifier.citationStone, Bertrand. “Line Complexity Asymptotics of Polynomial Cellular Automata.” The Ramanujan Journal, vol. 47, no. 2, Nov. 2018, pp. 383–416.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Humanities. Music and Theater Arts Sectionen_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.contributor.mitauthorStone, Bertrand A.
dc.relation.journalThe Ramanujan Journalen_US
dc.eprint.versionAuthor's final manuscripten_US
dc.type.urihttp://purl.org/eprint/type/JournalArticleen_US
eprint.statushttp://purl.org/eprint/status/PeerRevieweden_US
dc.date.updated2018-10-12T03:38:15Z
dc.language.rfc3066en
dc.rights.holderSpringer Science+Business Media, LLC
dspace.orderedauthorsStone, Bertranden_US
dspace.embargo.termsYen_US
dspace.embargo2019-09-01T04:00:00Z
mit.licensePUBLISHER_POLICYen_US


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