Seite - 445 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 110 TheGilbert–Varshamovcurvecanbecharacterizedintermsofthebehaviorofsufficientlyrandom codes, in thesenseof theShannonRandomCodeEnsemble, see [26,27],while theasymptoticbound canbecharacterized in termsofKolmogorovcomplexity, see [16]. 2.5. CodeParametersofLanguageFamilies Fromthecodingtheoryviewpoint, it isnatural toaskwhether therearecodesC, formedoutofa choiceofacollectionofnatural languagesandtheir syntacticparameters,whosecodeparameters lie abovetheasymptoticboundcurveR=α2(δ). For instance,acodeCwhosecodeparametersviolate thePlotkinbound(5)mustbean isolated codeabovetheasymptoticbound.ThismeansconstructingacodeCwithδ≥1/2, that is, suchthat anypairofcodewordsw =w′ ∈Cdifferbyat leasthalfof theparameters.Adirectexaminationof the listofparameters inTableAof [3]andFigure7of [4] showsthat it isverydifficult tofind,within thesamehistorical linguistic family (e.g., the Indo-Europeanfamily)pairsof languagesL1,L2with δH(L1,L2)≥ 1/2. For example, among the syntactic relativedistances listed inFigure7of [4] one findsonly thepair (Farsi,Romanian)witha relativedistanceof 0.5. Otherpairs comeclose to this value, forexampleFarsiandFrenchhavearelativedistanceof0.483,butFrenchandRomanianonly differby0.162. Onehasbetterchancesofobtainingcodesabovetheasymptoticboundifonecompares languages thatarenotsocloselyrelatedat thehistorical level. Example 2. Consider the set C = {L1,L2,L3}with languages L1 = Arabic, L2 = Wolof, and L3=Basque.Weexcludefromthe listofTableAof [3]all thoseparameters thatareentailedandmade irrelevantbysomeotherparameter inat leastoneof these threechosen languages. Thisgivesusa list of25remainingparameters,whichare thosenumberedas1–5,7, 10,20–21,25,27–29,31–32,34,37,42, 50–53,55–57 in [3], andthe followingthreecodewords: L1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 0 0 0 L2 1 1 1 0 0 1 1 0 1 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 1 L3 1 1 0 1 0 0 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 This example, although very simple and quite artificial in the choice of languages, already suffices toproduceacodeC that liesabovetheasymptoticbound. In fact,wehavedH(L1,L2)=16, dH(L2,L3)=13anddH(L1,L3)=13, so thatδ=0.52. SinceR>0, thecodepoint (δ,R)violates the Plotkinbound,hence it liesabovetheasymptoticbound. Itwouldbemore interesting tofinda codeC consisting of languages belonging to the same historical-linguistic family (outsideof the Indo-Europeangroup), that liesabovetheasymptoticbound. Suchexampleswouldcorrespond to linguistic families that exhibit avery strongvariabilityof the syntacticparameters, inawaythat isquantifiable throughthepropertiesofCasacode. Ifoneconsiders the22 Indo-European languages in [3]with theirparameters,oneobtainsacode C that isbelowtheGilbert–Varshamovline,hencebelowtheasymptoticboundbyEquation(8).Afew otherexamples, takenfromothernonIndo-Europeanhistorical-linguistic families, computedusing thoseparameters reported in theSSWLdatabase (forexample thesetofMalayo–Polynesian languages currentlyrecordedinSSWL)alsogivecodeswhosecodeparameters liebelowtheGilbert–Varshamov curve. One can conjecture that any codeC constructedout of natural languages belonging to the same historical-linguistic family will be below the asymptotic bound (or perhaps below the GV bound),whichwouldprovideaquantitativeboundonthepossible spreadofsyntacticparameters within a historical family, given the size of the family. Examples like the simple one constructed above,using languagesnotbelongingto thesamehistorical familyshowthat, to thecontrary,across different historical families one encounters a greater variability of syntactic parameters. To our knowledge,nosystematic studyofparametervariability fromthiscodingtheoryperspectivehasbeen implementedsofar. 445