Mining a class of decision problems for one-dimensional cellular automata

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dc.contributor.authorLobos F.
dc.contributor.authorGoles E.
dc.contributor.authorRuivo E.L.P.
dc.contributor.authorDe Oliveira P.P.B.
dc.contributor.authorMontealegre P.
dc.date.accessioned2024-03-12T23:59:30Z
dc.date.available2024-03-12T23:59:30Z
dc.date.issued2018
dc.description.abstract© 2018 Old City Publishing, Inc.Cellular automata are locally defined, homogeneous dynamical systems, discrete in space, time and state variables. Within the context of one-dimensional, binary, cellular automata operating on cyclic configurations of odd length, we consider the general decision problem: if the initial configuration satisfies a given property, the lattice should converge to the fixed-point of all 1s (1),orto 0, otherwise. Two problems in this category have been widely studied in the literature, the parity problem [1] and the density classification task [4]. We are interested in determining all cellular automata rules with neighborhood sizes of 2, 3, 4 and 5 cells (i.e., radius r of 0.5, 1, 1.5 and 2.5) that solve decision problems of the previous type. We have demonstrated a theorem that, for any given rule in those spaces, ensures the non existence of fixed points other than 0 and 1 for configurations of size larger than 22r, provided that the rule does not support different fixed points for any configuration with size smaller than or equal to 22r. In addition, we have a proposition that ensures the convergence to only 0 or 1 of any initial configuration, if the rule complies with given conditions. By means of theoretical and computational approaches, we determined that: for the rule spaces defined by radius 0.5 and r = 1, only 1 and 2 rules, respectively, converge to 1 or 0, to any initial configuration, and both recognize the same language, and for the rule space defined by radius r = 1.5, 40 rules satisfy this condition and recognize 4 different languages. Finally, for the radius 2 space, out of the 4,294,967,296 different rules, we were able to significantly filter it out, down to 40,941 candidate rules. We hope such an extensive mining should unveil new decision problems of the type widely studied in the literature.
dc.description.firstpage393
dc.description.issuenumber5-6
dc.description.lastpage405
dc.description.volume13
dc.identifier.issn1557-5977
dc.identifier.urihttps://dspace.mackenzie.br/handle/10899/35622
dc.relation.ispartofJournal of Cellular Automata
dc.rightsAcesso Restrito
dc.subject.otherlanguageDecision problems
dc.subject.otherlanguageDensity classification
dc.subject.otherlanguageOne-dimensional cellular automata
dc.subject.otherlanguageParity problem
dc.titleMining a class of decision problems for one-dimensional cellular automata
dc.typeArtigo
local.scopus.citations1
local.scopus.eid2-s2.0-85049358657
local.scopus.subjectComputational approach
local.scopus.subjectDecision problems
local.scopus.subjectDensity classification task
local.scopus.subjectDensity classifications
local.scopus.subjectInitial configuration
local.scopus.subjectNeighborhood size
local.scopus.subjectParity problems
local.scopus.subjectTime and state variables
local.scopus.updated2024-05-01
local.scopus.urlhttps://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85049358657&origin=inward
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