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1 FOL CS 331 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.

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2 FOL CS 331 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, …

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3 FOL CS 331 Dr M M Awais Syntax of FOL: Basic elements Constantsjohn, 2, lums,... Predicatesbrother, >,... Functionssqrt, leftsideOf,... VariablesX,Y,A,B... Connectives , , , , Equality= Quantifiers ,

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4 FOL CS 331 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

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5 FOL CS 331 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

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6 FOL CS 331 Dr M M AwaisAlphabets-II Function: Maps Sentences to Objects Ali is father of Babarfather(babar) = ali father_of(baber) = 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)

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7 FOL CS 331 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}

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8 FOL CS 331 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

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9 FOL CS 331 Dr M M AwaisAlphabets-II Predicate 1)Shahid is a good student student(shahid,good) or good_student(shahid) 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)

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10 FOL CS 331 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

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11 FOL CS 331 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)

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12 FOL CS 331 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)

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13 FOL CS 331 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)

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14 FOL CS 331 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) ...

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15 FOL CS 331 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) )

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16 FOL CS 331 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)

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17 FOL CS 331 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)

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18 FOL CS 331 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

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19 FOL CS 331 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

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20 FOL CS 331 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

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21 FOL CS 331 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? 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)

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22 FOL CS 331 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) is in fact equivalence

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23 FOL CS 331 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

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24 FOL CS 331 Dr M M Awais Models for FOL: Example

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25 FOL CS 331 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

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26 FOL CS 331 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) ...

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27 FOL CS 331 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 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).

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28 FOL CS 331 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)

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29 FOL CS 331 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)]

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30 FOL CS 331 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)

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31 FOL CS 331 Dr M M Awais Rules: Wumpus world Perception – T,S,B percept([S,B,glitter],T) glitter(T) Reflex – T glitter(T) bestAction(grab,T)

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32 FOL CS 331 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)]

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33 FOL CS 331 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

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34 FOL CS 331 Dr M M Awais The electronic circuits domain One-bit full adder

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35 FOL CS 331 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 )

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36 FOL CS 331 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))

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37 FOL CS 331 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 ))

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38 FOL CS 331 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

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39 FOL CS 331 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

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40 FOL CS 331 Dr M M Awais Operations Unification: Algorithm for determining the subitutions needed to make two predicate calculus expressions match Skolemization: A method of removing or replacing existential quantifiers 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 S` and finally adding the results

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