Page - 441 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 110 theGilbert–Varshamovcurve). Thisprovidesaprecisequantitativeboundtothepossiblespreadof syntacticparameterscomparedto thesizeof the family, in termsof thenumberofdifferent languages belongingto thesamehistorico-linguisticgroup. However,wealsoshowthat, ifoneconsiderssetsof languages thatdonotbelongto thesame historical-linguistic family, thenonecanobtaincodes that lieabovetheasymptoticbound,a fact that reflects, in code theoretic terms, themuchgreatervariabilityof syntacticparameters. The result is in itself not surprising, but thepointwewish tomake is that the theoryof error-correcting codes provides a natural settingwhere quantitative statements of this sort can bemadeusingmethods alreadydeveloped for thedifferent purposes of coding theory. We concludeby listing somenew linguisticquestions thatarisebyconsideringtheparametriccomparisonmethodunder thiscoding theoryperspective. 1.4. ComplexityofLanguagesandLanguageFamilies Thestudyofnatural languagesfromthepointofviewofcomplexitytheoryhasbeenofsignificant interest to linguists inrecentyears. Theapproaches typically followedfocusonassigningareasonable measureofcomplexity to individual languagesandcomparingcomplexitiesacrossdifferent languages. For example, anotionofmorphological complexitywas studied in [17]. Anapproach todefining Kolmogorov complexity of languages on the basis of syntactic parameterswasdeveloped in [18]. A notion of language complexity based on the production rules of a generative grammar was considered in [19], in the setting of (finite) formal languages. For amore general computational perspectiveon thecomplexityofnatural languages, see [20]. The ideaofdistinguishing languages bycomplexity isnotwithoutcontroversy inLinguistics.Avery interestinggeneraldiscussionof the problemanditsevolution in thefieldcanbefoundin[21]. In thepresentpaper,weargue in favorofa somewhatdifferentperspective,whereweassign anestimateofcomplexitynot to individual languagesbut togroupsof languages,andinparticular (historical) languagefamilies.Ourversionofcomplexity ismeasuringhow“spreadout” thesyntactic parameterscanbe,across the languages thatbelongto thesamefamily.Asweoutlinedintheprevious subsections, this ismeasuredbyassigningto the languagefamilyacode,whosecodewordsrecordthe syntacticparametersof the individual languages in thefamily, thencomputing itscodeparameters andevaluating thepositionof the resultingcodepointswith respect to curves like theasymptotic boundor theGilbert–Varshamovline. Thereasonwhythispositioncarriescomplexity information lies in thesubtle relationbetweentheasymptoticboundandKolmogorovcomplexity, recentlyderived byManinandtheauthor in [16],whichwewill reviewbriefly in thispaper. 2. LanguageFamiliesasCodes ThePrinciplesandParametersmodelofLinguisticsassigns toeverynatural languageLasetof binaryvaluesparameters thatdescribepropertiesof thesyntactic structureof the language. LetFbea language family, bywhichwemeanafinitecollectionF= {L1, . . . ,Lm}of languages. Thismaycoincidewitha family in thehistorical sense, suchas the Indo-Europeanfamily,orasmaller subsetof languagesrelatedbyhistoricoriginanddevelopment (e.g., the Indo-Iranian,orBalto–Svalic languages), or simplyanycollectionof languagesone is interested incomparingat theparametric level, even if theyarespreadacrossdifferenthistorical families. Wedenotebynbethenumberofparametersused in theparametriccomparisonmethod.Wedo notfix,apriori, avalue forn, andweconsider itavariableof themodel.Wewilldiscussbelowhow oneviews, inourperspective, the issueof the independenceofparameters. Afterfixinganenumerationof theparameters, that is, abijectionbetweenthesetofparameters andtheset{1,. . . ,n},weassociate toa language familyFacodeC=C(F) inFn2,withonecodeword foreachlanguageL∈F,withthecodewordw=w(L)givenbythelistofparametersw=(x1, . . . ,xn), xi∈F2 of the language. Forsimplicityofnotation,we justwriteL for thewordw(L) in the following. 441