1. 1 FOL CS 531 Dr M M Awais Formal Logic The most widely used formal logic method is FIRST-ORDER PREDICATE LOGIC Reference: Chapter Two The predicate Calculus Luger’s Book Examples included from Norvig and Russel.

2. 2 FOL CS 531 Dr M M Awais First-order logic Whereas propositional logic assumes the world contains facts, first-order logic (like natural language) assumes the world contains –Objects: people, houses, numbers, colors, baseball games, wars, … –Relations: red, round, prime, brother of, bigger than, part of, comes between, … –Functions: father of, best friend, one more than, plus, …

3. 3 FOL CS 531 Dr M M Awais FOL PL only deals with facts PL cannot have variables Variables refer to things in the world You can deliberate over them How would represent the following fact in PL When you sterilize a jar all the bacteria are dead

4. 4 FOL CS 531 Dr M M Awais Syntax of FOL: Basic elements Constantsjohn, 2, lums,... Predicatesbrother, >,... Functionssqrt, leftsideOf,... VariablesX,Y,A,B... Connectives , , , ,  Equality= Quantifiers , 

5. 5 FOL CS 531 Dr M M Awais Syntax of FOL: Basic elements ConstantsJohn, Lums,... PredicatesBrother, >,... FunctionsSqrt, LeftsideOf,... Variablesx,y,a,b... Connectives , , , ,  Equality= Quantifiers ,  Alternate Convention also exists (Norvig uses the alternate)

6. 6 FOL CS 531 Dr M M Awais Truth in first-order logic Sentences are true with respect to a model and an interpretation Model contains objects (domain elements) and relations among them Interpretation specifies referents for constant symbols → objects predicate symbols → relations function symbols →functional relations An atomic sentence predicate(term 1,...,term n ) is true iff the objects referred to by term,,..., term, are in the relation referred to by predicate

7. 7 FOL CS 531 Dr M M Awais Alphabets-I Predicates, variables, functions,constants, connectives, quantifiers, and delimiters Constants: (first letter small) bLUEa color sanTROa car crowa bird Variables: (first letter capital) Dog: an element that is a dog, but unspecified Color:an unspecified color

8. 8 FOL CS 531 Dr M M AwaisAlphabets-II Function (evaluates to a constant / variable): Maps Sentences to Objects Ali is father of Babar father_of(ali) = baber father(babar) = ali Babar is son of Ali son_of(babar) = ali Interpretation has to be very clear. If you write father(baber), the answer should be ali For the above functions the arity is 1 (number of arguments to the function)

9. 9 FOL CS 531 Dr M M AwaisAlphabets-II Functions: 1) shahid likes zahidlikes(shahid) = zahid 2) atif likes abidlikes(atif) = abid 3)Constants to Variableslikes(X) = Y BINDINGS {X,Y} have two possible BINDINGS {X, Y} could be {shahid, zahid} Or {X,Y} could be {atif, abid} likes(X) =Y Substitutions: For 1 to be true: {shahid/X, zahid/Y} For 2 to be true: {atif/X, abid/Y}

10. 10 FOL CS 531 Dr M M AwaisAlphabets-II Predicate Maps Sentences to Truth Values (True/False) 1) Shahid is studentstudent(shahid) 2) Sana is a girlgirl(sana) 3) Father of baber is elder than Hamza elder(father(babar), hamza) For 1 and 2 arity is 1 and for 3 the arity is 2

11. 11 FOL CS 531 Dr M M AwaisAlphabets-II Predicate 1)Shahid is a good student student(shahid,good) or good_student(shahid) or is_good(shahid,student) 2) Sana is a friend of Saima, Sana and Saima both are girls friend_of(sana,saima)^girl(sana)^girl(saima) 3) Bill helps Fred helps(bill,fred)

12. 12 FOL CS 531 Dr M M Awais Atomic sentences Atomic sentence =predicate (term 1,...,term n ) or term 1 = term 2 term = function (term 1,...,term n ) or constant or variable brother(john,richard) greater(length(leftsideOf(squareA)), length(leftsideOf(squareB))) >(length(leftsideOf(squareA)), length(leftsideOf(squareB))) Functions cannot be atomic sentences

13. 13 FOL CS 531 Dr M M Awais Alphabets-III Connectives: ^and vor ~not Implication Quantification All persons can see There is a person who cannot see Universal quantifiers  (ALL) Existential quantifiers  (There exists)

14. 14 FOL CS 531 Dr M M Awais Complex sentences Complex sentences are made from atomic sentences using connectives  S, S 1  S 2, S 1  S 2, S 1  S 2, S 1  S 2, sibling(ali,hamza)  sibling(hamza,ali) >(1,2)  ≤ (1,2) (1 is greater than 2 or less than equal to 2) >(1,2)   >(1,2) (1 is greater than 2 and is not greater than equal to 2)

15. 15 FOL CS 531 Dr M M Awais Examples My house is a blue, two-story, with red shutters, and is a corner house blue(my-house)^two-story(my-house)^red-shutters(my- house)^corner(my-house) Ali bought a scooter or a car bought(ali, car) v bought(ali, scooter) IF fuel, air and spark are present the fuel will combust present(spark)^present(fuel)^present(air) combustion(fuel)

16. 16 FOL CS 531 Dr M M Awais Universal quantification  Everyone at LUMS is smart:  X at(X, lums)  smart(X)  X P is true in a model m iff P is true with X being each possible object in the model Roughly speaking, equivalent to the conjunction of instantiations of P at(rabia,lums)  smart(rabia)  at(shahid,lums)  smart(shahid)  at(lums,lums)  smart(lums) ...

17. 17 FOL CS 531 Dr M M AwaisExamples All people need air  X[person(X) need_AIR(X)] The owner of the car also owns the boat [owner(X, car) ^ car(X, boat)] Formulate the following expression in the PC: “Ali is a computer science student but not a pilot or a football player” cs_STUDENT(ali)  (  pilot(ali)   ft_PLAYER(ali) )

18. 18 FOL CS 531 Dr M M Awais Examples Restate the sentence in the following way: 1.Ali is a computer science (CS) student 2.Ali is not a pilot 3.Ali is not a football player cs_student(ali)^ ~pilot(ali)^ ~football_player(ali)

19. 19 FOL CS 531 Dr M M AwaisExamples Studying fuzzy systems is exciting and applying logic is great fun if you are not going to spend all of your time slaving over the terminal  X(~slave_terminal(X) [fs_eciting(X)^logic_fun(X)]) Every voter either favors the amendment or despises it  X[voter(X) [favor(X, amendment) v despise(X,amendment)] ^ ~[favor(X, amendment) v despise(X, amendment)] (this part simply endorses the statement, may not be required)

20. 20 FOL CS 531 Dr M M Awais Undecidable Predicate For which exhaustive testing is required Example:  X likes(zahra, X) This sentence is computationally impossible to calculate Scope of problem domain is to be limited to remove this problem, –i.e., X is a variable representing final year female student in the AI class, compared to an X representing all the people in the city of Lahore

21. 21 FOL CS 531 Dr M M Awais Robotic Arm Example Represent the initial details of the system Generate sentences of descriptive and or implicative nature Modify the facts using new sentences a b c d

22. 22 FOL CS 531 Dr M M Awais Example: Robotic Arm Represent the initial details of the systems a b c d on(a,b) on( c,d) ontable(b) ontable(d) clear(a) clear(c) hand_empty

23. 23 FOL CS 531 Dr M M Awais Goal: To pick a block and place it over another block Define predicate: stack_on(X,Y) General sentence: Conditions Conclusions What could the conditions be? hand_empty clear (X) clear (Y) pick (X) put_on (X,Y) a b c d hand_empty ^ clear (X) ^ clear (Y) ^ pick (X) ^ put_on (X,Y) stack_on (X,Y)

24. 24 FOL CS 531 Dr M M Awais Goal: To pick a block and place it over another block hand_empty ^ clear (X) ^ clear (Y) ^ pick (X) ^ put_on (X,Y)  stack_on (X,Y) hand_empty ^ clear (X)  pick (X) clear(Y) ^ pick (X)  put_on (X,Y) put_on (X,Y)  stack_on (X,Y) Semantically more correct hand_empty could be written as empty(hand), if hand_empty is in the knowledge base, then hand is empty otherwise false. put_on (X,Y)  stack_on (X,Y)

25. 25 FOL CS 531 Dr M M Awais Example: Robotic Arm Modify details of the systems a b c d on(b,a) on( c,d) ontable(b) ontable(d) clear(a) clear(c) hand_empty a b c d on(b,a) on( c,d) on(e,a) ontable(b) ontable(d) clear(c) clear(e) hand_empty e

26. 26 FOL CS 531 Dr M M Awais Models for FOL: Example

27. 27 FOL CS 531 Dr M M Awais A common mistake to avoid Represent: Everyone at LUMS is smart  X at(X,lums)  smart(X)  X at(X, lums)  smart(X) Common mistake: using  as the main connective with  : means “Everyone is at LUMS and everyone is smart” “Everyone at LUMS is smart” Typically,  is the main connective with 

28. 28 FOL CS 531 Dr M M Awais Existential quantification  Someone at LUMS is smart:  X at(X,lums)  smart(X)  X P is true in a model m iff P is true with X being some possible object in the model Roughly speaking, equivalent to the disjunction of instantiations of P at(sana,lums)  smart(sana)  at(bashir,lums)  smart(bashir)  at(lums,lums)  smart(lums) ...

29. 29 FOL CS 531 Dr M M Awais Another common mistake to avoid Typically,  is the main connective with  Common mistake: using  as the main connective with  :  X at(X,lums)  smart(X) is true even if there is anyone who is not at LUMS! Even if the antecedent is false the sentence can still be true (see the truth table of implication).

30. 30 FOL CS 531 Dr M M Awais Properties of quantifiers  X  Y is the same as  Y  X  X  Y is the same as  Y  X  X  Y is not the same as  Y  X  X  Y loves(X,Y) “There is a person who loves everyone in the world”  Y  X loves(X,Y) “Everyone in the world is loved by at least one person” Quantifier duality: each can be expressed using the other  X likes(X,car)  X  likes(X,car)  X likes(X,bread)  X  likes(X,bread)

31. 31 FOL CS 531 Dr M M Awais Equality term 1 = term 2 is true under a given interpretation if and only if term 1 and term 2 refer to the same object Sibling in terms of Parent:  X,Y sibling(X,Y)  [  (X = Y)   M,F  (M = F)  parent(M,X)  parent(F,X)  parent(M,Y)  parent(F,Y)]

32. 32 FOL CS 531 Dr M M Awais Using FOL The kinship domain: Brothers are siblings  X,Y brother(X,Y)  sibling(X,Y) One's mother is one's female parent  M,C mother(C) = M  (female(M)  parent(M,C)) “Sibling” is symmetric  X,Y sibling(X,Y)  sibling(Y,X)

33. 33 FOL CS 531 Dr M M Awais Rules: Wumpus world Norvig and Russel book for details(will provide you with handouts) Actions –Turn(Right), Turn(Left), Forward, Shoot, Grab, Release, Climb Reflex –  T glitter(T)  bestAction(grab,T)

34. 34 FOL CS 531 Dr M M Awais Rules: Wumpus world Norvig and Russel book for details(will provide you with handouts) Perception –  T,S,B percept([S,B,glitter],T)  glitter(T) Reflex –  T glitter(T)  bestAction(grab,T)

35. 35 FOL CS 531 Dr M M Awais Rules: Wumpus world Norvig and Russel book for details(will provide you with handouts) Perception –  T,S,B percept([S,B,glitter],T)  glitter(T) Reflex –  T glitter(T)  bestAction(grab,T)

36. 36 FOL CS 531 Dr M M Awais Rules: Wumpus world Norvig and Russel book for details(will provide you with handouts) Perception –  T,S,B percept([S,B,glitter],T)  glitter(T) Reflex –  T glitter(T)  bestAction(grab,T)

37. 37 FOL CS 531 Dr M M Awais Deducing Squares/Properties What are Adjacent Squares  X,Y,A,B adjacent([X,Y],[A,B])  [A,B]  {[X+1,Y], [X-1,Y],[X,Y+1],[X,Y-1]} Properties of squares:  S,T at(agent,S,T)  breeze(T)  breezy(S) Squares are breezy near a pit: Diagnostic rule---infer cause from effect  S breezy(S)  adjacent(R,S)  pit(R) Causal rule---infer effect from cause  R pit(R)  [  S adjacent(R,S)  breezy(S)]

38. 38 FOL CS 531 Dr M M Awais Deducing Squares/Properties Diagnostic rule---infer cause from effect  S breezy(S)  adjacent(R,S)  pit(R) Causal rule---infer effect from cause  R pit(R)  [  S adjacent(R,S)  breezy(S)]

39. 39 FOL CS 531 Dr M M Awais Knowledge engineering in FOL 1.Identify the task 2.Assemble the relevant knowledge 3.Decide on a vocabulary of predicates, functions, and constants 4.Encode general knowledge about the domain 5.Encode a description of the specific problem instance 6.Pose queries to the inference procedure and get answers 7.Debug the knowledge base

40. 40 FOL CS 531 Dr M M Awais Inferencing in FOL Interpretation of the objects/variables etc… Denotation of terms Find whether an interpretation holds for a particular sentence

41. 41 FOL CS 531 Dr M M Awais Interpretation I/Denotation ‘U’ set of Objects ( U universe of discourse) Map constants symbols to objects in U I(zahid), U={zahid, shahid, raza, sana, …} ‘T’ I(X)what happens now? Map predicates symbols to the relations between the objects on U Map function symbols to functions on U I(predicate(t1, t2))I(predicate)(I(t1), I(t2))

42. 42 FOL CS 531 Dr M M Awais Holds (meta rule) holds (predicate(t1,t2), I) iff  I(predicate)

43. 43 FOL CS 531 Dr M M Awais Example: Robotic Arm Find if the following sentence holds or not a b c d on(a,b) on( c,d) ontable(b) ontable(d) clear(a) clear(c) hand_empty stack_on (X,Y) I(X) I(Y) I(stack_on) holds (stack_on(X,Y),I) iff  I(stack_on)) ??? No stack_on is not in ‘U’ ‘on’ Holds but under the following interpretations {a/X, b/Y} or {c/X, d/Y} clear(a) holds but clear(b) does not hold

44. 44 FOL CS 531 Dr M M Awais Quantifiers

45. 45 FOL CS 531 Dr M M Awais FOL: Example What is U? Define the following: Predicates: above2 circle1 oval1 square1 triangle1 Function; hat2

46. 46 FOL CS 531 Dr M M Awais FOL: Example What is U? I(object) = I(above) = Find whether I(square) =????

47. 47 FOL CS 531 Dr M M Awais Reason it Out holds (sqaure(object),I) ? YES holds (above(object,hat(object)),I) ? I(hat(object))= sqaure holds(above(triangle,sqaure),I)? NO Hence I(square) = FALSE

48. 48 FOL CS 531 Dr M M Awais Find the truth of holds (  X, oval(X)), I) ? I(sqaure)={ } Note for the students Should have solved it when you come next time.

49. 49 FOL CS 531 Dr M M Awais The electronic circuits domain One-bit full adder

50. 50 FOL CS 531 Dr M M Awais The electronic circuits domain 1.Identify the task –Does the circuit actually add properly? (circuit verification) 2.Assemble the relevant knowledge –Composed of wires and gates; Types of gates (AND, OR, XOR, NOT) –Irrelevant: size, shape, color, cost of gates 3.Decide on a vocabulary –Alternatives: type(x 1 ) = xor type(x 1, xor) xor(x 1 )

51. 51 FOL CS 531 Dr M M Awais The electronic circuits domain 4.Encode general knowledge of the domain –  T 1,T 2 connected(T 1, T 2 )  signal(T 1 ) = signal(T 2 ) –  T signal(T) = 1  signal(T) = 0 –1 ≠ 0 –  T 1,T 2 connected(T 1, T 2 )  connected(T 2, T 1 ) –  G type(G) = OR  signal(out(1,G)) = 1   N signal(in(N,G)) = 1 –  G type(G) = AND  signal(out(1,G)) = 0   N signal(in(N,G)) = 0 –  G type(G) = XOR  signal(out(1,G)) = 1  signal(in(1,G)) ≠ signal(in(2,G)) –  G type(G) = NOT  signal(out(1,G)) ≠ signal(in(1,G))

52. 52 FOL CS 531 Dr M M Awais The electronic circuits domain 5.Encode the specific problem instance type(x 1 ) = xortype(x 2 ) = xor type(a 1 ) = and type(a 2 ) = and type(o 1 ) = or connected(out(1,x 1 ),in(1,x 2 ))connected(in(1,c 1 ),in(1,x 1 )) connected(out(1,x 1 ),in(2,a 2 ))connected(in(1,c 1 ),in(1,a 1 )) connected(out(1,o 2 ),in(1,o 1 )) connected(in(2,c 1 ),in(2,x 1 )) connected(out(1,a 1 ),in(2,o 1 )) connected(in(2,c 1 ),in(2,a 1 )) connected(out(1,x 2 ),out(1,c 1 )) connected(in(3,c 1 ),in(2,x 2 )) connected(out(1,o 1 ),out(2,c 1 )) connected(in(3,c 1 ),in(1,a 2 ))

53. 53 FOL CS 531 Dr M M Awais The electronic circuits domain 6.Pose queries to the inference procedure What are the possible sets of values of all the terminals for the adder circuit?  I 1,I 2,I 3,O 1,O 2 signal(in(1,c_1)) = I 1  signal(in(2,c 1 )) = I 2  signal(in(3,c 1 )) = I 3  signal(out(1,c 1 )) = O 1  signal(out(2,o 1 )) = O 2 7.Debug the knowledge base May have omitted assertions like 1 ≠ 0

54. 54 FOL CS 531 Dr M M Awais Summary First-order logic: –objects and relations are semantic primitives –syntax: constants, functions, predicates, equality, quantifiers Increased expressive power: sufficient to define wumpus world

55. 55 FOL CS 531 Dr M M Awais Operations/Processes Entailment: Unification: Algorithm for determining the substitutions needed to make two predicate calculus expressions match Skolemization: A method of removing or replacing existential quantifiers using Skolem function Composition: If S and S` are two substitutions sets, then the composition of S and S` (SS`) is obtained by applying the elements of S’ to the elements of ` and adding the S’ to S

56. 56 FOL CS 531 Dr M M Awais Examples (Alternate Convention) Superscript represents arity of predicates/functions

57. 57 FOL CS 531 Dr M M Awais Entailment A knowledge base entails a sentence IFF the sentence holds in every interpretation in which the knowledge base holds In Propositional Logic: A knowledge base entails a sentence IFF the sentence is TRUE in every interpretation that makes the knowledge base TRUE

58. 58 FOL CS 531 Dr M M Awais Entailment (follows from) KB entails S: for every Interpretation I, if KB holds in I, then S holds in I

59. 59 FOL CS 531 Dr M M Awais Entailment Computing entailment is impossible in general, because there are Infinitely many possible interpretations Even computing holds is impossible for interpretations with infinite universes

60. 60 FOL CS 531 Dr M M Awais Example Alternate Convention

61. 61 FOL CS 531 Dr M M Awais Example Alternate Convention

62. 62 FOL CS 531 Dr M M Awais Example Alternate Convention