9. Logica de ordinul intai. First-order logic. December 8, 2015
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1 9. Logica de ordinul intai. First-order logic December 8, 2015
2 Outline First order logic Sintaxa FOL Adevarul in FOL Propozitii atomice Propozitii complexe Cuantificatorul universal Cuantificatorul existential Egalitatea Baze de cunostinte in FOL
3 Pro si contra Logicii propozitionale (B 1,1 = (P 1,2 P 2,1 )) ((P 1,2 P 2,1 ) = B 1,1 ) Este declarativa: componentele sintactice (propozitiile) sunt legate de fapte, semantica este bazata pe o relatie de adevar intre propozitii si lumi posibile permite informatii partiale, disjunctive si negate: Citesc MaJoc,W 1,1 W 2,1 este compozitionala: intelesul pt B 1,1 P 1,2 este derivat din intelesul pt B 1,1 si pt P 1,2 intelesul este independent de context: (spre deosebire de limbajul natural: Look!) Cons: logica propozitionale este foarte limitata expresiv: nu se poate spune gropile cauzeaza racoare in patratele adiacente decat scriind o propozitie pt fiecare patrat
4 First-order logic In timp ce logica propozitionala presupune ca lumea contine fapte (facts), first-order logic presupune ca lumea contine: Obiecte : oameni, case, numere, teorii, culori, razboaie, secole, oferte comerciale,... substantive, fraze substantivale Relatii : unare - este rosu, rotund, prim,..., n-are este frate a, mai mare decat, inauntru, parte din, are culoarea, intamplat dupa,... verbe, fraze verbale] Functii : tata a, cel mai bun prieten, sfarsitul a,...o singura valoare pentru un anumit input Exemple: unu plus unu este egal cu doi, celulele vecine wumpus-ului sunt urat mirositoare, Maleficul rege John a condus Anglia in 1200
5 Logici Limbaj Ontological Commitment Epistemological Commitm Ce exista in lume Ce crede un agent despre fapte Logica propozitionala fapte true/false/unknown Logica predicatelor fapte, obiecte, relatii true/false/unknown de ordinul intai Logica temporala fapte, obiecte, relatii, momente true/false/unknown Logica probabilitatilor fapte degree of belief [0,1] Logica fuzzy fapte si grade de adevar [0, 1] known interval value
6 Modele pentru FOL: Exemplu crown person R brother brother left leg $ J on head person king left leg Obiecte: Richard, John, piciorul stang al lui Richard, piciorul stang al lui John, coroana Relatii: tuple - binare: brotherhood, onhead; unare: person, king, crown Functii: leftlegof
7 Sintaxa FOL: elemente de baza Constants KingJohn,2,Emag,... Predicates Brother,>,Person,King,Crown... Functions Sqrt, LeftLegOf,... Variables x,y,a,b,... Connectives,,,, Equality = Quantifiers,
8 Exemplu Considerand interpretarea (ce obiecte, relatii si functii sunt referite de simbolurile constante, predicate si functii): Richard Richard Regele Leu ( ) John regele John ( ) Brother relatia de frate Brother(Richard, John) este adevarata doar daca Regele Leu si regele John sunt in relatie de frate in model (frati) O alta interpretare: Richard coroana, John piciorul stang al regelui John. Observatie: in logica propozitionala poate exista un model in care ambele Inorat si Senin sunt false. Eliminarea modelelor care sunt inconsistente cu cunostintele noastre trebuie sa fie realizata de baza de cunostinte.
9 Adevarul in logica predicatelor de ordinul intai Propozitiile sunt adevarate relativ la un model si o interpretare Un model contine unul sau mai multe obiecte (elementele domeniului si relatiile dintre elemente) Trebuie sa legam propozitiile la un model pentru a stabili adevarul lor interpretarea care specifica ce obiecte, relatii si functii sunt referite de simbolurile constante, predicate si functii simbolurile constante obiecte simbolurile predicate relatii simboilurile functii relatii functionale (functii) Entailmentul si validitatea sunt definite in termeni de toate modelele posibile si toate interpretarile posibile
10 FOL: Backus Naur Form (BNF )
11 Propozitii atomice Propozitii atomice = predicate(term 1,...,term n ) or term 1 = term 2 Term = function(term 1,...,term n ) or constant or variable Termenii sunt expresii logice care refera un obiect - cu nume sau fara E.g., Brother(John, Richard) > (Length(LeftLegOf(Richard)), Length(LeftLegOf (John)))
12 Propozitii complexe Propozitiile complexe sunt formate din propozitii atomice si conectori S, S 1 S 2, S 1 S 2, S 1 S 2, S 1 S 2 E.g. Brother(John, Richard) = Brother(Richard, John) >(1,2) (1,2) >(1,2) >(1,2)
13 Cuantificatorul universal variables sentence Everyone at Berkeley is smart: x At(x, Berkeley) Smart(x) x P este adevarat intr-un model m daca si numai daca P este adevarat cu x fiind toate obiectele posibile din model Poate fi considerat ca fiind echivalent cu conjunctia instantierilor lui P (At(John, Berkeley) Smart(John)) (At(Richard, Berkeley) Smart(Richard)) (At(Berkeley, Berkeley) Smart(Berkeley))...
14 Exemplu pt x King(x) Person(x) In model, x Richard Regele Leu, Regele John, piciorul stang al lui Richard, piciorul stang al lui John, coroana: (Richard Inima de Leu este rege Regele RIchard este o persoana Regele John este rege Regele John este o persoana Piciorul stang al lui Richard este rege Piciorul stang al lui Richard este o pe Piciorul drept al lui Richard este rege Piciorul drept al lui Richard este o pe Coroana este rege Coroana este o persoana Toate sunt adevarate, dar numai a doua spune intr-adevar ceva despre calitatea cuiva de a fi o persoana (daca consideram ca doar Regele John este rege). Observatie: implicatia este falsa doar cand premisele sunt adevarate si concluzia falsa.
15 Greseala tipica In general, este conectorul principal care apare impreuna cu Greseala tipica: utilizarea drept conector impreuna cu : x At(x, Berkeley) Smart(x)
16 Greseala tipica In general, este conectorul principal care apare impreuna cu Greseala tipica: utilizarea drept conector impreuna cu : x At(x, Berkeley) Smart(x) inseamna Everyone is at Berkeley and everyone is smart
17 Cuantificatorul existential variables sentence Cineva la Stanford este destept : x At(x, Stanford) Smart(x) x P este adevarata intr-un model m daca si numai daca P este adevarata cu x fiind un obiect posibil din model Este echivalent cu disjunctia dintre instantierile lui P (At(John, Stanford) Smart(KingJohn)) (At(Richard, Stanford) Smart(Richard)) (At(Stanford, Stanford) Smart(Stanford))...
18 O alta greseala comuna In general, este conectorul principal de utilizat cu Greseala: utilizarea cu : x At(x, Stanford) = Smart(x) este adevarata daca exista cineva care nu e la Stanford!
19 Quiz Cum se exprima propozitia: Regele John are o coroana pe cap? 1. x Crown(x) OnHead(x, John) 2. x Crown(x) OnHead(x, John) 3. x Crown(x) OnHead(x, John)
20 Quiz Cum se exprima propozitia: Regele John are o coroana pe cap? 1. x Crown(x) OnHead(x, John) CORECT 2. x Crown(x) OnHead(x, John) Regele John este o coroana Regele John este pe capul lui John este adevarata, premisele fiind false 3. x Crown(x) OnHead(x, John) Daca ceva este coroana, atunci se afla pe capul lui John
21 Proprietati ale cuantificatorilor x y este la fel cu y x Fratii sunt rude: x y Brother(x,y) Rude(x,y) Se poate scrie un singur cuantificator cu mai multe variabile: x,y Brother(x,y) Rude(x,y) x y este la fel cu y x Cineva citeste ceva x y nu e la fel cu y x 1. x y Loves(x, y) 2. y x Loves(x, y) 3. Everyone in the world is loved by at least one person?? 4. There is a person who loves everyone in the world??
22 Proprietati ale cuantificatorilor x y este la fel cu y x Fratii sunt rude: x y Brother(x,y) Rude(x,y) Se poate scrie un singur cuantificator cu mai multe variabile: x,y Brother(x,y) Rude(x,y) x y este la fel cu y x Cineva citeste ceva x y nu e la fel cu y x 1. x y Loves(x, y) 2. y x Loves(x, y) 3. Everyone in the world is loved by at least one person 2 4. There is a person who loves everyone in the world 1 Dualitatea cuantificatorilor: fiecare poate fi exprimat prin intermediul celuilalt x Likes(x, IceCream) x Likes(x, IceCream) x Likes(x, Broccoli) x Likes(x, Broccoli)
23 Exemple Brothers are siblings x,y Brother(x,y) Sibling(x,y).?
24 Exemple Brothers are siblings x,y Brother(x,y) Sibling(x,y). Sibling is symmetric x,y Sibling(x,y) Sibling(y,x).?
25 Exemple Brothers are siblings x,y Brother(x,y) Sibling(x,y). Sibling is symmetric x,y Sibling(x,y) Sibling(y,x). One s mother is one s female parent x,y Mother(x,y) (Female(x) Parent(x,y)).?
26 Exemple Brothers are siblings x,y Brother(x,y) Sibling(x,y). Sibling is symmetric x,y Sibling(x,y) Sibling(y,x). One s mother is one s female parent x,y Mother(x,y) (Female(x) Parent(x,y)). A first cousin is a child of a parent s sibling x, y FirstCousin(x, y) p,ps Parent(p,x) Sibling(ps,p) Parent(ps,y)
27 Egalitatea term 1 = term 2 este adevarat intr-o anumita interpretare daca si numai daca term 1 si term 2 se refera la acelasi obiect E.g., 2 = 2 father(john) = Henry x,y Brother(x,Richard) Brother(y,Richard) (x = y) - Richard are cel putin doi frati E.g., Sibling (frati si surori) x,y Sibling(x,y) [ (x = y) m,f (m = f) Parent(m,x) Parent(f,x) Parent(m,y) Parent(f,y)]
28 Interactiunea cu baze de cunostinte FOL: Tell, Ask Sa presupunem ca in lumea wumpus, agentul foloseste logica predicatelor de ordinul intai si percepe un miros si racoare, dar nu si stralucire, lovitura de perete sau urlet la momentul t = 5: Tell(KB, Percept([Smell, Breeze, None, None, None], 5)) Ask(KB, a BestAction(a, 5)) Se poate extrage din KB o anumita actiune t = 5? Actiuni disponibile: Turn(Left), Turn(Right), Forward, Shoot, Grab, Climb
29 Interactiunea cu baze de cunostinte FOL: Tell, Ask Sa presupunem ca in lumea wumpus, agentul foloseste logica predicatelor de ordinul intai si percepe un miros si racoare, dar nu si stralucire, lovitura de perete sau urlet la momentul t = 5: Tell(KB, Percept([Smell, Breeze, None, None, None], 5)) Ask(KB, a BestAction(a, 5)) Se poate extrage din KB o anumita actiune t = 5? Actiuni disponibile: Turn(Left), Turn(Right), Forward, Shoot, Grab, Climb Raspuns: Yes,{a/Shoot} substitutie (binding list) Ask(KB, S) returneaza toate substitutiile σ astfel incat KB = Sσ
30 Wumpus world in FOL - perceptii si actiuni Perception t,s,g,m,c Percept([s,Breeze,g,m,c],t) = Breeze(t) t,s,b,m,c Percept([s,b,Glitter,m,c],t) = Glitter(t) Reflex: t AtGold(t) = BestAction(Grab, t) Reflex cu stare interna: are aurul deja? t AtGold(t) Holding(Gold, t) = Action(Grab, t)
31 Deducerea proprietatilor ascunse Pozitia agentului la momentul t: At(Agent, s, t) x,s 1,s 2,t At(Agent,s 1,t) At(Agent,s 2,t) s 1 = s 2
32 Deducerea proprietatilor ascunse Pozitia agentului la momentul t: At(Agent, s, t) x,s 1,s 2,t At(Agent,s 1,t) At(Agent,s 2,t) s 1 = s 2 Proprietati ale pozitiilor x,t At(Agent,x,t) Breeze(t) = Breezy(x)
33 Deducerea proprietatilor ascunse Pozitia agentului la momentul t: At(Agent, s, t) x,s 1,s 2,t At(Agent,s 1,t) At(Agent,s 2,t) s 1 = s 2 Proprietati ale pozitiilor x,t At(Agent,x,t) Breeze(t) = Breezy(x) Adiacenta a doua celule x,y,a,b Adjacent([x,y],[a,b]) (x = a (y = b 1 y = b +1)) (y = b (x = a 1 x = a+1))
34 Deducerea proprietatilor ascunse Pozitia agentului la momentul t: At(Agent, s, t) x,s 1,s 2,t At(Agent,s 1,t) At(Agent,s 2,t) s 1 = s 2 Proprietati ale pozitiilor x,t At(Agent,x,t) Breeze(t) = Breezy(x) Adiacenta a doua celule x,y,a,b Adjacent([x,y],[a,b]) (x = a (y = b 1 y = b +1)) (y = b (x = a 1 x = a+1)) Celulele sunt cu vant in vecinatatea gropilor:
35 Deducerea proprietatilor ascunse Pozitia agentului la momentul t: At(Agent, s, t) x,s 1,s 2,t At(Agent,s 1,t) At(Agent,s 2,t) s 1 = s 2 Proprietati ale pozitiilor x,t At(Agent,x,t) Breeze(t) = Breezy(x) Adiacenta a doua celule x,y,a,b Adjacent([x,y],[a,b]) (x = a (y = b 1 y = b +1)) (y = b (x = a 1 x = a+1)) Celulele sunt cu vant in vecinatatea gropilor: Diagnostic rule infer cause from effect y Breezy(y) = x Pit(x) Adjacent(x,y) y Breezy(y) = x Pit(x) Adjacent(x,y)
36 Deducerea proprietatilor ascunse Pozitia agentului la momentul t: At(Agent, s, t) x,s 1,s 2,t At(Agent,s 1,t) At(Agent,s 2,t) s 1 = s 2 Proprietati ale pozitiilor x,t At(Agent,x,t) Breeze(t) = Breezy(x) Adiacenta a doua celule x,y,a,b Adjacent([x,y],[a,b]) (x = a (y = b 1 y = b +1)) (y = b (x = a 1 x = a+1)) Celulele sunt cu vant in vecinatatea gropilor: Diagnostic rule infer cause from effect y Breezy(y) = x Pit(x) Adjacent(x,y) y Breezy(y) = x Pit(x) Adjacent(x,y) Definitie pentru predicatul Breezy y Breezy(y) [ x Pit(x) Adjacent(x,y)]
37 Deducerea proprietatilor ascunse Pozitia agentului la momentul t: At(Agent, s, t) x,s 1,s 2,t At(Agent,s 1,t) At(Agent,s 2,t) s 1 = s 2 Proprietati ale pozitiilor x,t At(Agent,x,t) Breeze(t) = Breezy(x) Adiacenta a doua celule x,y,a,b Adjacent([x,y],[a,b]) (x = a (y = b 1 y = b +1)) (y = b (x = a 1 x = a+1)) Celulele sunt cu vant in vecinatatea gropilor: Diagnostic rule infer cause from effect y Breezy(y) = x Pit(x) Adjacent(x,y) y Breezy(y) = x Pit(x) Adjacent(x,y) Definitie pentru predicatul Breezy y Breezy(y) [ x Pit(x) Adjacent(x,y)] Causal rule infer effect from cause x Pit(x) ( y Adjacent(x,y) Breezy(y))
38 Deducerea proprietatilor ascunse Pozitia agentului la momentul t: At(Agent, s, t) x,s 1,s 2,t At(Agent,s 1,t) At(Agent,s 2,t) s 1 = s 2 Proprietati ale pozitiilor x,t At(Agent,x,t) Breeze(t) = Breezy(x) Adiacenta a doua celule x,y,a,b Adjacent([x,y],[a,b]) (x = a (y = b 1 y = b +1)) (y = b (x = a 1 x = a+1)) Celulele sunt cu vant in vecinatatea gropilor: Diagnostic rule infer cause from effect y Breezy(y) = x Pit(x) Adjacent(x,y) y Breezy(y) = x Pit(x) Adjacent(x,y) Definitie pentru predicatul Breezy y Breezy(y) [ x Pit(x) Adjacent(x,y)] Causal rule infer effect from cause x Pit(x) ( y Adjacent(x,y) Breezy(y)) Obs: regula cauzala nu spune daca celulele mai indepartate sunt cu vant sau nu
39 Axioma starii urmatoare: cuantificare peste timp t HaveArrow(t +1) HaveArrow(t) Action(Shoot,t))
40 Quiz 1. Care dinte urmatoarele translatari in FOL sunt corecte sintactic si semantic pentru Niciun caine nu musca niciun copil al stapanului? (No dog bites a child of its owner 1.1 x Dox(x) Bites(x, Child(Owner(x))) 1.2 x,y Dog(x) Child(y,Owner(x)) Bites(x,y) 1.3 x Dog(x) ( y Child(y,Owner(x)) Bites(x,y)) 1.4 x Dog(x) ( y Child(y,Owner(x)) Bites(x,y)) 2. Translatati in FOL: Everyones DNA is unique and is derived from their parents DNA. (DNA oricui este unic si este derivat din cel al parintilor). Hint: puteti folosi functia DNA si predicatul DerivedFrom(u,v,w) care inseamna ca u este derivat din v si w.
41 Quiz 1. Care dinte urmatoarele translatari in FOL sunt corecte sintactic si semantic pentru Niciun caine nu musca niciun copil al stapanului? (No dog bites a child of its owner 1.1 x Dox(x) Bites(x, Child(Owner(x))) child nu e o functie 1.2 x,y Dog(x) Child(y,Owner(x)) Bites(x,y) CORECT 1.3 x Dog(x) ( y Child(y,Owner(x)) Bites(x,y)) CORECT 1.4 x Dog(x) ( y Child(y,Owner(x)) Bites(x,y)) implicatie impreuna cu cuantificator existential 2. Translatati in FOL: Everyones DNA is unique and is derived from their parents DNA. (DNA oricui este unic si este derivat din cel al parintilor). Hint: puteti folosi functia DNA si predicatul DerivedFrom(u,v,w) care inseamna ca u este derivat din v si w. x,y( (x = y) (DNA(x) = DNA(y))) DerivedFrom(DNA(x), DNA(Mother(x)), DNA(Father(x)))
42 Leul si unicornul Raymond Smullyan: Leul minte Luni Marti si Miercuri si spune adevarul in restul zilelor. Unicornul minte Joi, Vineri si SAmbata si spune adevarul in rest. Alice ii intalneste pe cei doi si acestia ii spun urmatoarele: Leul: Ieri a fost una dintre zilele mele de minciuna Unicornul: Ieri a fost una dintre zilele mele de minciuna Ce zi a fost ieri?
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44 Rezumat First order logic Sintaxa FOL Adevarul in FOL Propozitii atomice Propozitii complexe Cuantificatorul universal Cuantificatorul existential Egalitatea Baze de cunostinte in FOL
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