Page - 440 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 110 structure of a given language. Their universalitymakes it possible to obtain comparisons, at the syntactic level,betweenarbitrarypairsofnatural languages. APCMwas introduced in [2]asaquantitativemethodinhistorical linguistics, forcomparisonof languageswithinandacrosshistorical familiesat thesyntactic insteadof the lexical level. Evidence wasgiven in [3,4] that thePCMgives reliable informationonthephylogenetic treeof the familyof Indo-Europeanlanguages. ThePCMreliesessentiallyonconstructingametriconafamilyof languagesbasedontherelative Hammingdistancebetweenthesetsofparametersasameasureof relatedness. Thephylogenetic tree is thenconstructedonthebasisof thisdatumofrelativedistances, see [3]. Moreworkonsyntacticphylogenetic reconstructions, involvinga larger setof languagesand parameters isongoing, [5]. Syntacticparametersofworld languageshavealsobeenusedrecently for investigationsonthe topologyandgeometryofsyntactic structuresandforstatisticalphysicsmodels of languageevolution, [6–8]. Publiclyavailabledataofsyntacticparametersofworldlanguagescanbeobtainedfromdatabases such as Syntactic Structures ofWorld Languages (SSWL) [9] or TerraLing [10] orWorldAtlas of LanguageStructures (WALS)[11]. Thedataofsyntacticparametersusedinthepresentpaperaretaken fromTableAof [3]. 1.2. SyntacticParameters,CodesandCodeParameters Our purpose in this paper is to connect the PCM approach to the mathematical theory of error-correcting codes. We associate a code to anygroupof languages onewishes to analyze via thePCM,whichhasonecodewordforeach language. Ifoneusesanumbernofsyntacticparameters, thenthecodeCsits inthespaceFn2,wheretheelementsofF2={0,1}correspondtothetwo∓possible valuesofeachparameter,andthecodewordofa language is thestringofvaluesof itsnparameters. WealsoconsideraversionwithcodesonanalphabetF3 of three letterswhichallowsfor thepossibility that someof theparametersmaybemade irrelevantbyentailment fromotherparameters. In thiscase weuse the letter0∈F3 for the irrelevantparametersandthenonzerovalues±1for theparameters thatareset in the language. Inthetheoryoferror-correctingcodes,see[12],oneassignstoacodeC⊂Fnq twocodeparameters: R= logq(#C)/n, the transmissionrateof thecode,andδ= d/n therelativeminimumdistanceof the code,where d is themiminumHammingdistancebetweenpairsofdistinct codewords. It iswell knownincodingtheory that“goodcodes”are those thatmaximizebothparameters, compatiblywith several constraints relatingRandδ. Consider the function f :Cq→ [0,1]2 fromthespaceCqofq-ary codes to theunit square, thatassigns toacodeC its codeparameters, f(C)= (δ(C),R(C)). Apoint (δ,R) in therangeof f hasfinite (respectively, infinite)multiplicity if thepreimage f−1(δ,R) isafinite set (respectively,an infiniteset). Itwasprovedin[13] that there isacurveR=αq(δ) in thespaceof codeparameters, theasymptoticbound, that separatescodepoints thatfilladenseregionandthat have infinitemultiplicity fromisolatedcodepoints thatonlyhavefinitemultiplicity. Thesebetterbut moreelusivecodesare typicallyobtainedthroughalgebro-geometricconstructions, see [13–15]. The asymptoticboundwasrelatedtoKolmogorovcomplexity in [16]. 1.3. PositionwithRespect to theAsymptoticBound Givenacollectionof languagesonewants tocompare throughtheir syntacticparameters,onecan asknaturalquestionsabout thepositionof theresultingcode in thespaceofcodeparametersandwith respect to theasymptoticbound. The theoryoferror correctingcodes tellsus that codesabove the asymptoticboundareveryrare. Indeed,weconsideredvarioussetsof languages,andforeachchoice ofasetof languagesweconsideredanassociatedcode,withacodewordforeach language in theset, givenbyits listof syntacticparameters.Whencomputingthecodeparametersof theresultingcode, onefinds that, inarangeofcaseswelookedat,whenthe languages in thechosensetbelongto the samehistorical-linguistic family theresultingcode liesbelowtheasymptoticbound(andin factbelow 440