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entropy Article SyntacticParametersandaCodingTheory PerspectiveonEntropyandComplexityof LanguageFamilies MatildeMarcolli DepartmentofMathematics,California InstituteofTechnology,Pasadena,CA91125,USA;matilde@caltech.edu; Tel.: +1-626-395-4326 AcademicEditors: FrédéricBarbaresco,FrankNielsenandKevinH.Knuth Received: 14 January2016;Accepted: 18March2016;Published: 7April2016 Abstract: Wepresentasimplecomputationalapproachtoassigningameasureofcomplexityand information/entropyto familiesofnatural languages,basedonsyntacticparametersandthe theory oferrorcorrectingcodes.Weassociate toeach languageabinarystringofsyntacticparametersand toa languagefamilyabinarycode,withcodewords thebinarystringassociatedtoeach language. Wethenevaluate thecodeparameters (rateandrelativeminimumdistance)andthepositionof the parameterswithrespect totheasymptoticboundoferrorcorrectingcodesandtheGilbert–Varshamov bound. These bounds are, respectively, related to theKolmogorov complexity and the Shannon entropy of the code and this gives us a computationally simpleway to obtain estimates on the complexity and information, not of individual languages but of language families. This notion of complexity is related, fromthe linguisticpoint ofview to thedegreeofvariabilityof syntactic parameteracross languagesbelongingto thesame(historical) family. Keywords: syntax;principlesandparameters;error-correctingcodes;asymptoticbound;Kolmogorov complexity;Gilbert–Varshamovbound;Shannonentropy 1. Introduction Weproposeanapproach,basedonLongobardi’sparametriccomparisonmethod(PCM)andthe theoryoferror-correctingcodes, toaquantitativeevaluationof the“complexity”ofa languagefamily. Oneassociates toacollectionof languages tobeanalyzedwith thePCMabinary (or ternary) code withonecodewordforeach language in the familyandeachwordconsistingof thebinaryvaluesof thesyntacticparametersof that language. Theternarycaseallowsforanadditionalparameterstate that takes intoaccountcertainphenomenaofentailmentofparameters.Wethenconsideradifferent kindofparameters: thecodeparametersof the resultingcode,which incoding theoryaccount for theefficiencyof the codinganddecodingprocedures. These canbecomparedwith someclassical boundsofcodingtheory: theasymptoticbound, theGilbert–Varshamov(GV)bound, etc. Theposition of thecodeparameterswithrespect tosomeof theseboundsprovidesquantitative informationonthe variabilityof syntacticparameterswithinandacrosshistorical-linguistic families.Whilecomputations carriedout for languagesbelonging to the samehistorical familyyield codesbelowtheGVcurve, comparisonsacrossdifferenthistorical familiescangiveexamplesof isolatedcodes lyingabovethe asymptoticbound. 1.1. Principles andParameters Thegenerativeapproach to linguistics relieson thenotionof aUniversalGrammar (UG)and arelateduniversal listof syntacticparameters. In thePrinciplesandParametersmodel,developed since [1], these are thoughtof asbinaryvaluedparametersor “switches” that set thegrammatical Entropy2016,18, 110 439 www.mdpi.com/journal/entropy