Pros and Cons of Propositional Logic - Lecture Slides | CS 3600, Study notes of Computer Science

Download Pros and Cons of Propositional Logic - Lecture Slides | CS 3600 and more Study notes Computer Science in PDF only on Docsity! g woydeyg Q UELdVHD ODIDOT YUACUO-LSUI A g woydeyg 104 Ul pom sndunyy & saduaquas yim uny & 104 40 soljuewas pue xequds <> 2104 AUM © eullnO g woydeyg anjea jeAsaqUu! UMOU¥ Jaljaq Jo aesap umouyun /asjey /aniy umouyun /asjey /ansy umouyun /asjey/ansy yn} Jo asap + sje} syoe} soul} ‘suolze|ad ‘syalgo ‘sq0e4 suoljejad ‘syalgo ‘sq0e4 syoe} 2180] Azzn4 Asoayy Ayjigeqoid D180] jesodwa | 2130] Japs0-3S414 D180] Jeuolzisodosq qua} WW’) jea1Sojowaysidy quaWw}IWWO0) jea1d0|01UQ asensuey] [e1oues UI SOIsO'T g woydeyg EA SJalpiquent = Ayyenby S&S — +t A V SeAl}eUUO a ‘pf ‘a sajqeuen SL ObeqifaT “qubg — suorouny ct< ‘ayjoug — sayeoipad “On *% ‘uyopbursy — squeysuo> syUOUE[9 DISeG :JTOW jo xequAs Lg wydeyo (((uyor bur sz) fObeTIfaT)yuaT ‘((pavyory) fQbaT1f9T)YypbuaT) < (qupayuoryay, rpnyory ‘uyor bury ).ioyjorg “8'y IIQDLLDA JO YUDISUOI JO (“uwag ++ Tuta7)uorgoun f 2d] = 1.17 JO (“usp ++ Tusap)agnoipaLd Wi9 | 99U9}USS JIWOW S9DU9}UVS IIWOTV ModelsforFOL:Example RJ $ left legleft leg on head brother brother person person king crown Chapter810 IL g woydeyp JaPOW ayy Ul UO!JeJa1 pooysayioug ayy Ul ade uyor Buly |IAa ay} pue Yeayuo!y ay} pyeydiy ased ul ysnf ani} si (UYyor ‘punyoRy ).ayjo4g ‘uoneyasdsau! siy} Japuy) uoljejad pooysayjoug ay} — LaYyJO“g uyor Buly |IAa ayy — UYOLr yeayuoly ay} pueydiy <— pupyory YdIYM Ul UO!eJaJd4aqU! ayy JapIsuo7 ejdurexe Y4NAI, ModelsforFOL:Lots! Entailmentinpropositionallogiccanbecomputedbyenumeratingmodels WecanenumeratetheFOLmodelsforagivenKBvocabulary: Foreachnumberofdomainelementsnfrom1to∞ Foreachk-arypredicatePkinthevocabulary Foreachpossiblek-aryrelationonnobjects ForeachconstantsymbolCinthevocabulary ForeachchoiceofreferentforCfromnobjects... ComputingentailmentbyenumeratingFOLmodelsisnoteasy! Chapter812 Existentialquantification ∃〈variables〉〈sentence〉 SomeoneatStanfordissmart: ∃xAt(x,Stanford)∧Smart(x) ∃xPistrueinamodelmiffPistruewithxbeing somepossibleobjectinthemodel Roughlyspeaking,equivalenttothedisjunctionofinstantiationsofP (At(KingJohn,Stanford)∧Smart(KingJohn)) ∨(At(Richard,Stanford)∧Smart(Richard)) ∨(At(Stanford,Stanford)∧Smart(Stanford)) ∨... Chapter815 or g woydeyp jpsoyueis 3e 3OU SI OYM suosue SI a4ay} JI aNd SI (w)quvUg <= (psofunig 'x)Iy LE *E YIM dAlPoUUOD UleW 2yu} se <= Suisn -OeqSIWW UOLWWWO") E YUM aAlzaUUOD UleEW auy s! \V ‘AyjedidA] PIOAR 0} OYFeISIUL UOWIUIOD JoyJOUW Propertiesofquantifiers ∀x∀yisthesameas∀y∀x(why??) ∃x∃yisthesameas∃y∃x(why??) ∃x∀yisnotthesameas∀y∃x ∃x∀yLoves(x,y) “Thereisapersonwholoveseveryoneintheworld” ∀y∃xLoves(x,y) “Everyoneintheworldislovedbyatleastoneperson” Quantifierduality:eachcanbeexpressedusingtheother ∀xLikes(x,IceCream)¬∃x¬Likes(x,IceCream) ∃xLikes(x,Broccoli)¬∀x¬Likes(x,Broccoli) Chapter817 g woydeyp quased ajewiaj Ss auo SI JaYyJOW Saud (vh)burqug <> (fra)burjqus faa duyeWWAS SI BUI|qIS,, ‘(A‘x)burqug <= (f‘tx)uayjoug fh‘ax SBul|gis ae s¥aYyIOIG soouoyuEs YIM UNT  1% g woydeyp Buljgis Sjuased e Jo pjiyd e SI UIsnod YSU ‘((fi‘x)quaung \V (x)ajpuay) = (A‘x)sayjopy fit pn quased ajewiaj Ss auo SI JaYyJOW Saud (vh)burqug <> (fra)burjqus faa duyeWWAS SI BUI|qIS,, ‘(A‘x)burqug <= (f‘tx)uayjoug fh‘ax SBul|gis ae s¥aYyIOIG soouoyuEs YIM UNT Funwithsentences Brothersaresiblings ∀x,yBrother(x,y)⇒Sibling(x,y). “Sibling”issymmetric ∀x,ySibling(x,y)⇔Sibling(y,x). One’smotherisone’sfemaleparent ∀x,yMother(x,y)⇔(Female(x)∧Parent(x,y)). Afirstcousinisachildofaparent’ssibling ∀x,yFirstCousin(x,y)⇔∃p,psParent(p,x)∧Sibling(ps,p)∧ Parent(ps,y) Chapter822 Knowledgebaseforthewumpusworld “Perception” ∀b,g,tPercept([Smell,b,g],t)⇒Smelt(t) ∀s,b,tPercept([s,b,Glitter],t)⇒AtGold(t) Reflex:∀tAtGold(t)⇒Action(Grab,t) Reflexwithinternalstate:dowehavethegoldalready? ∀tAtGold(t)∧¬Holding(Gold,t)⇒Action(Grab,t) Holding(Gold,t)cannotbeobserved ⇒keepingtrackofchangeisessential Chapter825 Deducinghiddenproperties Propertiesoflocations: ∀x,tAt(Agent,x,t)∧Smelt(t)⇒Smelly(x) ∀x,tAt(Agent,x,t)∧Breeze(t)⇒Breezy(x) Squaresarebreezynearapit: Diagnosticrule—infercausefromeffect ∀yBreezy(y)⇒∃xPit(x)∧Adjacent(x,y) Causalrule—infereffectfromcause ∀x,yPit(x)∧Adjacent(x,y)⇒Breezy(y) Neitheroftheseiscomplete—e.g.,thecausalruledoesn’tsaywhether squaresfarawayfrompitscanbebreezy DefinitionfortheBreezypredicate: ∀yBreezy(y)⇔[∃xPit(x)∧Adjacent(x,y)] Chapter826 Keepingtrackofchange Factsholdinsituations,ratherthaneternally E.g.,Holding(Gold,Now)ratherthanjustHolding(Gold) SituationcalculusisonewaytorepresentchangeinFOL: Addsasituationargumenttoeachnon-eternalpredicate E.g.,NowinHolding(Gold,Now)denotesasituation SituationsareconnectedbytheResultfunction Result(a,s)isthesituationthatresultsfromdoingains PIT PIT PIT Gold PIT PIT PIT Gold S0 Forward S1 Chapter827 Makingplans InitialconditioninKB: At(Agent,[1,1],S0) At(Gold,[1,2],S0) Query:Ask(KB,∃sHolding(Gold,s)) i.e.,inwhatsituationwillIbeholdingthegold? Answer:{s/Result(Grab,Result(Forward,S0))} i.e.,goforwardandthengrabthegold ThisassumesthattheagentisinterestedinplansstartingatS0andthatS0 istheonlysituationdescribedintheKB Chapter830 Makingplans:Abetterway Representplansasactionsequences[a1,a2,...,an] PlanResult(p,s)istheresultofexecutingpins ThenthequeryAsk(KB,∃pHolding(Gold,PlanResult(p,S0))) hasthesolution{p/[Forward,Grab]} DefinitionofPlanResultintermsofResult: ∀sPlanResult([],s)=s ∀a,p,sPlanResult([a|p],s)=PlanResult(p,Result(a,s)) Planningsystemsarespecial-purposereasonersdesignedtodothistypeof inferencemoreefficientlythanageneral-purposereasoner Chapter831 ze g aoydey gy snjnzjed uoljenyis e uo aduaJajul Se Suluue|d aqejnwoj ued — JO4 ul aBuey> pue suolde Buiquosap 40} SuoIyUaAUOD — ‘snjnajed uolenys pom sndwnm auijap 0} Jua!diyyns :zamod anissaidxa paseasdu| suayiquenb ‘Ayjenba ‘sazedipaid ‘suoiouny ‘sjueysuod :xequAs — SaAlpWId DIZUeWAS de SUOI}e|a4 pue sjaf[qo — :2180] Japso-IS414 AreuIuIng