Chapter 3

1. Artificial Intelligence 605451 Chapter 3 Propositional calculus Predicate calculus Dr.Hassan Al-Tarawneh

2. Propositional Calculus Symbols • They symbols of propositional calculus are the propsositional symbols • P, Q, R, S ….. • Truth symbols • true, false • Connectives •     • An interpretation of a set of propositions is the assignment of truth value, either T or F • true is always assigned T • false is always assigned F

3. Propositional Calculus Semantics • Negation P,

4. Propositional Calculus Semantics • Conjunction, 

5. Propositional Calculus Semantics • Disjunction, 

6. Propositional Calculus Semantics • Implication,  • P is the premise or antecedent • Q is the conclusion or consequent

7. Propositional Calculus Semantics • Equivalence, 

8. Identities •  (P )  P • (P  Q)  P  Q • Contrapositive law (P  Q)  ( Q   P) • De Morgan’s Law  (P  Q)  ( P  Q)  (P  Q)  ( P  Q)

9. Identities • The commutative laws (P  Q)  (Q  P) (P  Q)  (Q  P) • The associative law ((P  Q)  R)  (P  (Q  R)) ((P  Q)  R)  (P  (Q  R))

10. Identities • The Distributive law(it is wrong ) P  (Q  R)  (P  Q)  (Q  R)) P  (Q  R)  (P  Q)  (Q  R)) • P(Q R) = (P  Q)  (P  R)

11. Predicate Calculus • In propositional calculus, • P would denote “it rained on Tuesday” • In predicate calculus • weather(tuesday, rain) • Using variables • weather(X, rain) • X is a day of the week

12. Predicate Calculus Symbols • The alphabet that makes up the symbols of the predicate calculus consists of • The set of letter, both upper- and lowercase • The set of digits, 0, 1, …, 9. • The underscore, _.

13. Predicate Calculus Symbols • Symbols in the predicate calculus begin with a letter and are followed by any sequence of these legal characters a R 6 9 p _ z • Example of characters not in the alphabet include : # % @ / & • Legitimate predicate calculus symbols include • George fire3 tom_and_jerry bill XXXX friend_of • Example of strings that are not legal symbols are • 3jack “no blacks allowed” ab%cd ***71 duck!!!

14. Predicate Calculus Symbols • likes(george, kate) • ( ) - parentheses • , - commas • . - period

15. Predicate Calculus Symbols • Truth symbols • true and false • Constants • Must be lower case • goerge, tree

16. Predicate Calculus Symbols • Variables • Represented by symbols • Begin with upper case • George, BILL, Kate (correct) • geORGE, bill (wrong) • Functions • Begin with lower case • Denoted mapping • Each function had an associated arity • father (jack) – one arity • plus(2,3) – two arity

17. Predicate Calculus Symbols • A predicate calculus term is either a constant, variable or function expression • Example cat constant times(2,3) function X variable blue constant mother(jane) function kate constant

18. Atomic sentences • An atomic sentence is a predicate constant of arity n, followed by n terms, t1, t2 … tn, enclosed in parentheses and separated by comas • The truth values, true and false are also atomic sentences • Example likes(george,kate) likes(X,george) likes(X,X) likes(george,sarah,tuesday) friends(bill,richard) friends(father_of(david),father_of(andrew))

19. Advantages of Predicate Calculus • Allows the access to individual components of a proposition • plays(ahmad,fooball) • Allows to determine the relationship between individuals or objects and their properties • Allows expressions to contain variable, enabling general assertions about classes of entities • plays(X, football) [X refer to humans that play football] • plays(ahmad, Y) [Y refer to class of games] • Through inference rules we can manipulate expression to infer new sentences • team_mates(X, Y) plays (X, football)  plays (Y, football) • team_mates(ahmad, ali) plays (ahmad, football)  plays (ali, football)

20. Predicate Calculus • We can combine atomic sentences using logical operators to from sentences in the predicate calculus. • Logical connectives are       is the universal quantifier, indicating that the sentence is true for all values of the  Is the existential quantifier indicating that the sentence is true for at least one value in the domain

21. Examples • Marcus was a man • man(marcus) • Marcus was a Pompeian • pompeian(marcus) • All pompeians were Romans •  X pompeian(X)  roman(X) • Caesar was a ruler • ruler(ceaser)

22. Examples • All romans were either loyal to Caesar or hated him •  X roman(X)  loyal_to(X, caeser)  hate(X, caeser) • Everyone it loyal to someone •  X  Y loyal_to(X,Y) • People try to assassinate rulers that they are not loyal to •  X  Y person(X)  ruler (Y)  tryassassinate (X,Y)   loyalto(X,Y) • Marcus tried to assassinate Caesar • tryassassinate(marcus,caesar)

23. Predicate Calculus Sentences • Every atomic sentence is a sentence • If s is a sentence, then also its negation, s. • If s1 and s2 is a sentence, then also their conjunctions, s1 s2. • If s1 and s2 is a sentence, then also their disjunctions, s1 s2. • If s1 and s2 is a sentence, then also their implication, s1  s2. • If s1 and s2 is a sentence, then also their equivalence, s1  s2. • If X is a variable and s is a sentence, then  X is a sentence • If X is a variable and s is a sentence, then  X is a sentence

24. Predicate Calculus Semantic • friends(george,susie) = T if true • friends(george,kate) = F if false • friends(george,X) • X is a place holder • Substitute X with katie or susie • X is for all constants • Can have any other name, Y or PEOPLE

25. Predicate Calculus Semantic • In predicate calculus variable must be quantified • universally • existentially • Free variables • If it is not within the scope of the universal or existential quantifier • Closed expression • If its variables are quantified • Ground expression • Has no variables at all • X (p(X)  q (Y)  r(X)) • X is universally quantified to p(X) and r(X)

26. Universal Quantification • Problem computing the truth table of a sentence • All possible values of a variable must be tested to see whether the expression remains true • X likes(george,X) • X is all humans

27. Universal Quantification • Propositional calculus does not use variables • Has finite no of possible truth assignment • Can test all assignments • Done with truth table

28. Existential Quantifier • Evaluating the truth expression is also not easy. • Determine the truth by trying substitution until one is found that makes the expression true. • If the domain is infinite, algorithm will not halt

29. Universal & Existential Quantification • Relationships   X p(X)  X  p(X)   X p(X)   X  p(X)  X p(X)   Y p(Y)  X q(X)   Y q(Y)  X (p(X)  q(X))   X p(X)   Y p(Y)  X (p(X)  q(X))   X p(X)   Y p(Y)

30. Clauses to Predicate Calculus • If it doesn’t rain on Monday, Tom will go to the mountains weather(rain,Monday)  go(tom,mountains) • Emma is a Doberman and a good dog gooddog(emma)  isa(emma,doberman) • All basketball players are tall  X(basketball_player(X)  tall(X)) • Some people like anchovies X (person(X)  likes(X,anchovies)) • If wished were horses, beggars would ride equal(wishes,horses)  ride(beggars) • Nobody likes taxes   X likes(X, takes)

31. Inference Rules • Logical inference • Ability to infer correct expression from a set of true assertions • New expression consistent with all previous interpretation of the original expression

32. Inference Rules • A mechanical means of producing new predicate calculus sentences from other sentences • Example : modus ponens

33. Inference Rules • If the sentence P and P  Q are know to be true, then modus ponens lets us infer Q • Under the inference rule modus tollens, if P  Q is know to be true and Q is known to be false, we can infer P.

34. Inference Rules • Elimination allows us to infer the truth of either of the conjuncts from the truth of a conjunctive sentence. For instance, P  Q lets us conclude P and Q are true. • Introduction allows us to infer the truth of conjunction from the truth of its conjuncts. For instance, if P and Q are true, then P  Q is true

35. Inference Rules • Universal instantiation states that if any universally quantified variable in a true sentence is replaced by an appropriate term from the domain, the results is a true sentence • Thus, if a is from the domain of X, X p(X) lets us infer p(a)

36. Modus ponen • X (man(X)  mortal (X)) • man(socrates) • (man(socrates)  mortal (socrates))

37. Unification • Unification is an algorithm for determining the substitutions needed to make two predicate calculus expression match • foo(X,a,goo(Y)) • legal substitutes • foo(fred,a,goo(Z)) {fred/X, Z/Y} • foo(W,a,goo(jack)) {W/X, jack/Y} • foo(Z,a,goo(moo(Z))) {Z/X, moo(Z)/Y} • X/Y indicates X is substituted for the variable Y in the original expression

38. Unification Issues • Variable can be substituted by constant, but constant cannot be replaced with a constant • Not allowed p(p(p(…X))) • Must maintain consistency across all occurrence of the variable in the expression matched • X = socrates in man(X) and mortal(X)

39. Unification Issues • Once a variable is substituted, future unification and inference must take the value of this substitution into account • {X/Y, W/Z} {V/X} {a/V, f(b)/W} => {a/Y,f(b)/Z} • Use most general unification • p(X) p(Y) => {fred/X, fred/Y} • most general => {Z/X, Z/Y}