Page - 453 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 110 has the advantage that it is very rich in linguistic information. However, it is at the same time computationallyverydifficult torealize. Whatweareproposinghere is amuchsimplerway toobtainanestimateof complexity for a languagefamily{L1, . . . ,Lk},which isnotbasedonestimatingcomplexityof the individual languages in the family,butwhich isaimedatdetectinghowspreadoutanddiversifiedthesyntacticparameters areacross the family,byestimatingthepositionof thecodepoint (R(C),δ(C))of theassociatedcode Cwithrespect to theasymptoticboundR=αq(δ). Thiscanbeestimatedintermsof thedistance to othercurves in thespaceofcodeparameters (R,δ) thatconstrain theasymptoticboundfromabove andbelow, suchas thePlotkinbound,Hammingbound, andGilbert–Varshamovbound, as in the examplesdiscussed in theprevioussections. 4.Conclusions Weproposedanapproachtoestimatingentropyandcomplexityofgroupsofnatural languages (language families), basedon the linguisticparametric comparisonmethod(PCM)of [2,22]via the mathematical theoryof error-correcting codes, by assigning a code to a family of languages to be analyzedwith thePCM,andinvestigating itsposition in thespaceofcodeparameters,withrespect to theasymptoticboundandtheGVbound.Wehaveshownthat thereareexamplesof languagesnot belongingtothesamehistorical-linguistic familythatyield isolatedcodesabovetheasymptoticbound, while languagesbelongingto thesamehistorical-linguistic familyappear togiverise tocodesbelow thebound, thoughamoresystematicanalysiswouldbeneededtomapcodeparametersofdifferent languagegroups.Wehavealsoshownthat, fromthesecodingtheoryperspective, it ispreferable to excludefromthePCMall thoseparameters thatareentailedandmade irrelevantbyotherparameters, as thosespoil thepropertiesof theresultingcodeandproducecodeparameters thatareartificially low withrespect to theasymptoticboundandtheGVbound. Theapproachproposedhere,basedonthePCMandthetheoryoferror-correctingcodes, suggests a fewnewlinguisticquestions thatmaybesuitable for treatmentwithcodingtheorymethods: 1. Do languagesbelonging to thesamehistorical-linguistic familyalwaysyieldcodesbelowthe asymptoticboundor theGVbound?Howoftendoes thesamehappenacrossdifferent linguistic families?Howmuchcancodeparametersbe improvedbyeliminatingspoilingeffectscausedby dependenciesandentailmentofsyntacticparameters? 2. Codesnear theGVcurveare typicallycomingfromtheShannonRandomCodeEnsemble,where codewordsand lettersof codewordsbehave like independent randomvariables, see [26,27]. Are there familiesof languageswhoseassociatedcodesare locatednear theGVbound?Dotheir syntacticparametersmimic theuniformPoissondistributionof randomcodes? 3. Theasymptoticboundforerror-correctingcodeswasrelated in [16] toKolmogorovcomplexity, andthemeasureofcomplexity for language families thatweproposedhere isestimated in terms of thepositionof thecodepointwithrespect to theasymptoticbound. Thereareothernotionsof complexity,mostnotably the typeoforganizedcomplexitiesdiscussedin[33–35].Canthesebe relatedto loci in thespaceofcodeparameters?Whatdotheserepresentwhenappliedtocodes obtainedfromsyntacticparametersofasetofnatural languages? 4. Is thereamoredirect linguisticcomplexitymeasureassociatedtoa familyofnatural languages thatwouldrelate to thepositionof theresultingcodeaboveorbelowtheasymptoticbound? 5. Codesandtheasymptoticboundinthespaceofcodeparameterswererecentlystudiedusing methods fromquantumstatisticalmechanics, operator algebra and fractal geometry, [24,36]. Can some of these mathematical methods be employed in the linguistic parametric comparisonmethod? The observational results reported here are still preliminary. The following topics should beconsolidated: 453