Page - 446 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 110 Ongoingworkof theauthor isconsideringasystematicanalysisof languagefamilies,basedon theSSWLdatabaseofsyntacticparameters,usingthiscodingtheorytechnique. Thiswill includean analysisofhowmuchtheconclusionsabout thespreadingofsyntacticparametersacross language familiesobtainedwith this techniquedependsondatapre-processing like the removalof spoiling featuresandwhatcanberetainedasanobjectivepropertyofasetof languages.Moreover,a further purposeofthisongoingstudyistocombinethecodingtheoryapproachandthemeasuresofcomplexity for groups of languages described in the present paperwith the spin glass dynamicalmodels of language change considered in [8], which was aimed at studying dynamically the spreading of syntacticparametersacrossgroupsof languages. Theaimis to introducecomplexitymeasuresbased oncoding theoryaspartof theenergy landscapeof thespinglassmodel, followingthesuggestion of [28], on analogies between the roles of complexity in the theory of computation and energy in physical theories. Theseresults, alongwithamoredetailedanalysisof thecodesandcodeparameters ofvarious languagefamilies,will appear in forthcomingwork. 2.6. ComparisonwithOtherBounds Anotherpossiblequestiononecanconsiderinthissettingishowthecodesobtainedfromsyntactic parametersofagivensetofnatural languagescomparewithotherknownfamiliesoferrorcorrecting codesandwithotherbounds in thespaceofcodeparameters. For instance, it isknownthatan important improvementover thebehaviorof typical random codescanbeobtainedbyconsideringcodesdeterminedbyalgebro-geometric curvesdefinedover a finite field Fq. Let Nq(X)=#X(Fq) be the number of points over Fq of the curve X, and let Nq(g)=maxNq(X), with themaximum taken over all genus g curves X overFq. As shown in Theorem2.3.22of [12], asymptotically theNq(g) satisfy theDrinfeld–Vladutbound A(q) := limsup q→∞ Nq(g) g ≤√q−1, andasshowninSection3.4.1of [12], thisdeterminesanalgebro-geometricbound αq(δ)≥RAG(δ)=1− 1A(q)−δ andtheasymptoticTsfasman–Vladut–Zinkbound αq(δ)≥RTVZ(δ)=1−(√q−1)−1−δ. TheTsfasman–Vladut–Zink lineRTVZ(δ) = 1−(√q−1)−1−δ lies entirelybelowtheGV line for q<49 (Theorem3.4.4of [12]). Aprobabilistic argument given in Section 3.4.2 of [12] shows that highly non-randomcodes comingfromalgebraiccurvescanbeasymptoticallybetter thanrandomcodes (forsufficiently largeq) as theyclusteraroundtheTVZline.However, forq=2orq=3,as in thecaseofcodes fromsyntactic parametersofgroupsof languages thatweconsiderhere, theTVZline liesbelowtheGVline,hence anyexample that liesabovetheGVboundalsobehavesbetter thanthe thealgebro-geometricbound. Suchexamples, like theonegivenabove, for the three languagesArabic,Wolof,Basque,areveryrare amongcodesobtainedfromsyntacticparametersof languages,as theyrequire thechoiceofagroup of languages that are all very far fromeachother syntactically,withvery large relativeHamming distancesbetweensyntacticparameters. Ontheotherhand,evenforcasesofgroupsof languages forwhich theresultingcodeparameters arebelowtheGVline, it is stillpossible toget someadditional informationbycomparingtheposition of the code parameters to other curves obtained fromother bounds, such as the Blokh–Zyablow 446