Page - 447 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 110 boundor theKatsman–Tsfasman–Vladutbound, seeAppendixA.2.1of [12] forasummaryofall these differentbounds. For example, thefirst examplegivenabove,with the three languages Italian, Spanish, French andastringof six syntacticparameters, gives a codewith codeparameters that arebelowtheGV line,butaboveboth theBlokh–ZyablowandtheKatsman–Tsfasman–Vladut, accordingto the tableof asymptoticboundsgiven inAppendixA.2.4of [12]. 2.7. EntailmentandDependencyofParameters Inthediscussionabovewedidnotincorporateinourmodelthefactthatcertainsyntacticparameters canentailotherparameters insuchawaythatoneparticularvalueofoneof theparameters renders anotherparameter irrelevantornotdefined, see thediscussion inSection5.3.2of [3]. Onepossiblewaytoalter thepreviousconstructiontoaccount for thesephenomenais toconsider thecodesCassociatedto familiesof languagesascodes inFn3,wheren is thenumberofparameters,as before,andthesetofvalues isnowgivenby{−1,0,+1}=F3,with±1correspondingto thebinary valuesof theparameters thatareset foragivenlanguageandvalue0assignedtothoseparameters that aremade irrelevant for the given language, by entailment fromother parameters, or are not defined. Thisallowsus toconsider the full rangeofparametersused in [3,4].WerevisitExample2 consideredabove. Example3. LetC= {L1,L2,L3}be thecodeobtainedfromthe languagesL1=Arabic,L2=Wolof, and L3 =Basque, as a code inFn3 withn= 63, using the entire list ofparameters in [3]. The code parameters (R= 0.0252,δ= 0.4643)of this codeno longerviolate thePlotkinbound. In fact, the parameterssatisfyR<1−H3(δ) so thecodeCnowalso liesbelowtheGVbound. Thus, theeffectof includingtheentailedsyntacticparameters in thecomparisonspoils thecode parametersenoughthat theyfall in theareabelowtheGVbound. Notice thatwhatweproposehere isdifferent fromthecountingusedin[3],where therelative distances δH(L1,L2) are normalized with respect to the number of non-zero parameters (which therefore varieswith the choice of thepair (L1,L2)) rather than the total numbernofparameters. While thishas thedesiredeffectofgettingridof insignificantparameters that spoil thecode, ithas the undesirablepropertyofproducingcodeswithcodewordsofvarying lengths,whilecountingonly thoseparameters thathavenozero-valuesover theentire familyof languages,as inExample2avoids thisproblem.Adaptingthecodingtheoryresultsabout theasymptoticboundtocodeswithwordsof variable lengthmaybedesirable forother reasonsaswell,but itwill requirean investigationbeyond thescopeof thepresentpaper. Moregenerally, therearevariouskindsofdependenciesamongsyntacticparameters. Somesets ofhierarchical relationsarediscussed, for instance, in [29]. By thespoilingoperationsC i f ofcodesdescribedabove,weknowthat if someof thesyntactic parametersconsideredare functionsofotherparameters, theresultingcodeparametersofC i f are worse thantheparametersof thecodeCwhereonly independentparameterswereconsidered. Part of the reasonwhycodeparameters of groupsof languages in the family analyzed in [3] endupin theregionbelowtheasymptoticandtheGVboundmaybeanartifactof thepresenceof dependences among the chosen63 syntacticparameters. Fromthe coding theoryperspective, the parametriccomparisonmethodworksbestonasmallersetof independentparameters thanonalarger set that includesseveraldependencies. Entailmentrelationsbetweensyntacticparametersplayanimportantroleinthedynamicalmodels of languageevolutionsconstructed in [8],basedonspinglassmodels instatisticalphysics. Notice that the typeofentailment relationsweconsiderhereareonlyofa rather special form, whereaparameter ismadeundefinedbyeffectof thevalueofanotherparameter (hence theuseof thevalue0 for theundeterminedparameter). Therearemoregeneral formsofentailment thatwedo 447