DecidetoattackstraightusingtheincompletePLorapplyaclosurepropertytosimplifytheproblemseemyhandwrittennotesofL15forexamplesalsomybookchapterThenproceedtoapplytheincompletePLtoshowthatLisnotregul ID: 201597
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^(prime(1))(1=2)_odd(1)^(prime(2))(2=2)_odd(2)^(prime(3))(3=2)_odd(3)^(prime(4))(4=2)_odd(4)^:::Trythis:Simplify:(8x2Nat:prime(x))(x=2)_odd(x))andexpressitintheaboveform.2(Incomplete)PumpingLemmas{whatwewillstudyUseourPumpingLemmaversionstoProvethatsomethingisNOTregular.Toseeacompleteversion,seethechapterexcerptfrommybookthatIhaveplacedonline.2.1BasicsofOUR(incomplete)PumpingLemmasLetusrstconsidersingletonalphabets.Oneimmediateobservationwecanmakeisthis:AllinniteregularlanguagesoverasingletonalphabethaveaDFAthatislassoshaped.Thelengthsofstringsformanultimatelyperiodicset(ifyoulookfarenoughbeyondsomepointp,eachelementafterpinthesetissomemultipleofkstepsawayfrompthroughp�k).ThisisnottrueofLwarpedSlinkyandsoweatleastgetonefactestablished:thelengthsofstringsinaregularlanguageisultimatelyperiodic.WehavetwoincompletePumpingLemmasthatwecanuse:theoneinSection12.1ofmybook,andtheonein12.1.1ofmybook.Hereistheproofstructure:SuspectthatLisnotregular.Lookforperiodicity,niteness,etc.Bevirtuallysureitisnotregular.Thengotothenextstep.Lengthsofstringsisnotacertaincriterion.Ifthelengthsarenotperiodic,thennotregular.Butnottheotherway.Example:f0n1njn0g2 DecidetoattackstraightusingtheincompletePLorapplyaclosurepropertytosimplifytheproblem(seemyhand-writtennotesofL15forexamples;alsomybookchapter).ThenproceedtoapplytheincompletePLtoshowthatLisnotregular.Asktheadversary(whoclaimsLisregular)foranumbermrepresentingthenumberofstatesintheminimalDFAofL.Pickastringwoflengthm.Splitwinaninterestingway.ShowthattherearepumpsthatgooutsideofL.ButifLwereregular,allpumpsstayinsideL.Hence,Lcan'tberegular!2.2J apPumpingLemmaTutorReadafulldescriptionathttp://www.cs.duke.edu/csed/jflap/tutorial/pumpinglemma/regular/index.html.Thisfollowsthe12.1versionofmyPumpinglemma.Here,Iprovidesomeextradetails.Iftheuser(U)goesrstandwins(a\YOUWIN"isshown),then,assumingthattheir\computer"movesaredesignedwell,thelanguageisregular.Thatmeans,wecan'pump'.Thatmeans,wecanpumpandstayinthelanguageforallpumps.Thus,ifUgoesrstandgetsa\tryagain,"aftermanytries,theuserassumeshe/shelost|thelanguageisthenlikelynotregular.Sitdownandproveitatthatpoint(alsohitthe\Explain"button).Ifthecomputer(C)goesrstandtheuserwins(a\YOUWIN"isshown),thenitmeansthecomputerloses.Thatis,theuserwassmartinpickingapartitionsuchthatthepumpwentoutsidethelanguage.Thatmeans,thecomputercouldnot,inallcases,pickapumpandstayinthelanguage(thatwouldbethecaseforaregularlanguage).Sowehaveevidence(intheformoftheuserchosenpartition)thatwecangooutsidethelanguageatleastinonecase.Thelanguageisnotregular.ButifCgoesrstandtheuserkeepsgetting\tryagain,"thelanguageislikelyregular.3YourAssignmentSubmityouranswersassimplewriteupsinEnglish.Bebrief,yetcomplete.Tomakegradingeasier,pleaseprovide(orsummarize)thequestionandthenyouranswer.SubmitasPDForTXTintoasg8viahandin.1.(10points)foreachlanguageLetthecomputergorst.Selecttheselanguages:L=fanbkc(n+k)jn0;k0gL=fanblakjn�5;l�3;klgShowhowyouwon(whichmeansyoucouldprovethatthelanguageisnotregular).Thenunderstandtheexplanationprovided,andcopyitdowninyouranswer.3