Muddy Children other logic puzzles and temporal dynamic epistemic logic

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Muddy Children other logic puzzles and temporal dynamic epistemic logic




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Presentations text content in Muddy Children other logic puzzles and temporal dynamic epistemic logic

MuddyChildrenThreebrilliantchildrengototheparktoplay.Whentheirfathercomesto ndthem,heseesthattwoofthemhavemudontheirforeheads.Hethensays,\ Atleastoneofyouhasmudonyourforehead ",andthenasks,\ Doyouknowifyouhavemudonyourforehead? "Thechildrensimultaneouslyrespond,\ No ".Thefatherrepeatshisquestion,\ Doyouknowifyouhavemudonyourforehead? "andthistimethetwochildrenwithmuddyforeheadssimultaneouslyanswer,\ Yes,Ihave! "whilethere-mainingchildanswers,\ No ". 2 InductionBasecase: Suppose1childwasmuddy. Afterthefatheran-nounces,\ Atleastoneofyouhasmudonyourforehead ":  Whatdoesthemuddychildknow?Thatheis

themuddyone  Whatdothecleanchildrenconsiderpossible?Either1or2muddychildrentotal.Whenthefatherasks,\ Doyouknowifyouhavemudonyourforehead? ",themuddychildwillrespond,\ Yes,Iam! ". 4 Inductivehypothesis: Assumethatitiscommonknowledgethatatleastkchildrenaremuddyifthefatheraskedk1timesifthechildrenknewiftheyweremuddy,andeachtimetheyallanswered,\No" Ifmorethank+1childrenaremuddy,thentheyallwouldhaveknownallalongthatatleastk+1aremuddy.Considertwocases:kchildrenaremuddy:Eachmuddychildseesonlyk1muddychildren,butknowsthatatleastkaremuddy.Hethenanswer,\ Yes,Iam! "totheirfatherskthquestion.k+1childr

enaremuddy:Theirfatherasksiftheknowiftheyaremuddy,andtheyallanswer,\ No ",whichisnottheexpectedresponseiftherehadbeenexactlykmuddychildren.Thustheyallknowthereareatleastk+1muddychildren. 6 Rightafterthefatherannounces,\ Atleastoneofyouhasmudonyourforehead ":  mmm ooC// 99ByytttttttttOOA  mmc ccB##HHHHHHHHHOOA  mcm ooC// OOA  m cc  cc m GG eeB%%JJJJJJJJJ  cmmGG ooC//  c m cGG Notethatatmcc,Aiscertainmccisthecorrectstate,atcmc,Biscertaincmcisthecorrectstate,atccm,Ciscertainccmisthecorrectstate. 8 StateModel(AKAEpistemicModel)Fixaset  of\prop

erties"or\atomicpropositions"andaset A ofagents.S=(S; A !;kk),where 1. Sisaset 2. A!isan\epistemicrelation"overS,thatissA!tmeansAconsiderstpossibleatstates. 3. kkisfunctionfrom  toP(S).OftenA!isanequivalencerelationandiscalledan\indistin-guishabilityrelation". 10 LanguageforMuddyChildrenLet A =fA;B;Cg.Let  =f(Am);(Ac);(Bm);(Bc);(Cm);(Cc)g,whereweread(Am)as\Aismuddy",and(Ac)as\Aisclean",etc.ThenforexampleC(Am)^B:(Am)^:A(Am)canbereadas\CbelievesthatAismuddy,BbelievesthatAisnotmuddy,andAdoesnotbelieveheismuddy(Amaybeuncertain)". 12 DualityWecouldde neA'tobe:A:'read\Adoesnotbelievetha

t'isnotthecase".Thereissimilardualitybetween9and8:9x'(x)isequivalentto:8x:'(x).andcanbetreatedaslocal8and9respectively. 14 DynamicsAddtothelanguageformulasoftheform:  [ !] ' ,readas\ ' resultsfromanytruthfulannouncementof "or\If cantruthfullybeannounced,then ' willbecaseaftertheannouncement".Example:[ :A(Am)^:B(Bm)^:C(Cm) !] B(Bm) ,readas\Afteritisannouncedthat eachagentdoesnotknowheismuddy , Bknowsthatheismuddy ".SuchaslanguageiscalledPublicAnnouncementLogic,whichisakindofDynamicEpistemicLogic. 16 DualityforActionsDe neh !i' : [ !] : '\Itis not thecasethat if

cantruthfullybeannounced, then announcingitwillresultin : '."or\ canbetruthfullyannounced,andannouncing

willresultin'" 18 AddingPrevious-TimeOperatorIncludeformulasoftheform:  Y'readas\Previously'".SuchalanguageiscalledTemporalPublicAnnouncementLogic. 20 ModelforSumandProductConstructamodel(S;S!;P!;kk)suchthatS=f(n;m):1nm100g(n;m)S!(n0;m0)i n+m=n0+m0.(n;m)P!(n0;m0)i nm=n0m0.k (x=n) k=f(a;b):a=ngk (y=m) k=f(a;b):b=mg. 22 ModelCheckerProgramsThereisauniquesolutiontoSumandProduct.Itcanbesolvedbyacomputerprogram.DEMO(DynamicEpistemicMOdelling)isanexistingprogram,writteninHaskell,thatcancheckwhetheraformulainDynamicEpistemicLogicistrueatacertainstate.DEMOhasbeenusedtosolveanequivalentversionof

SumandProductthatdoesnotusethepasttense. 24 SurpriseExamPuzzleCanateachertruthfullyannouncethefollowing:\Youwillhaveanexamduringclassnextweek,buttheexamwillbeasurpriseinthatyouwillnotknowtheexamwillbegiventhatdayuntilitisactuallygivenout."?AssumethelastdayofclassisFriday.CantheexambegivenoutonFriday?WhataboutThursday?Wednesday? 26 4. D.Gerbrandy.BisimulationsonPlanetKripke.Dissertation,ILLC,1998. 5. B.Kooi:ProbabilisticDynamicEpistemicLogic.JournalofLogicLanguageandInformation,12(4):381-408,2003. 6. J.Sack.TemporalLanguagesforEpistemicPrograms.ToappearintheJournalofLogicLanguageandInformation.

2007 7. A.Yap.ProductUpdateandLookingBackward.ms.2005,StanfordUniversity.Alsoatwww.illc.uva.nl/lgc/papers/bms-temporal.pdf


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