Formal Logic. The most widely used formal logic method is FIRST-ORDER PREDICATE LOGIC

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1 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. CS 331 Dr M M Awais 1

2 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, CS 331 Dr M M Awais 2

3 Syntax of FOL: Basic elements Constants john, 2, lums,... Predicates brother, >,... Functions sqrt, leftsideof,... Variables X,Y,A,B... Connectives,,,, Equality = Quantifiers, CS 331 Dr M M Awais 3

4 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 CS 331 Dr M M Awais 4

5 Alphabets-I Predicates, variables, functions,constants, connectives, quantifiers, and delimiters Constants: (first letter small) blue a color santro a car crow a bird Variables: (first letter capital) Dog: an element that is a dog, but unspecified Color: an unspecified color CS 331 Dr M M Awais 5

6 Alphabets-II Function: Maps Sentences to Objects Ali is father of Babar father(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) CS 331 Dr M M Awais 6

7 Functions: Alphabets-II 1) shahid likes zahid likes(shahid) = zahid 2) atif likes abid likes(atif) = abid 3) Constants to Variables likes(x) = Y {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} CS 331 Dr M M Awais 7

8 Alphabets-II Predicate Maps Sentences to Truth Values (True/False) 1) Shahid is student student(shahid) 2) Sana is a girl girl(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 CS 331 Dr M M Awais 8

9 Predicate Alphabets-II 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) CS 331 Dr M M Awais 9

10 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 CS 331 Dr M M Awais 10

11 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) CS 331 Dr M M Awais 11

12 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) CS 331 Dr M M Awais 12

13 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(myhouse)^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) CS 331 Dr M M Awais 13

14 Universal quantification <variables> <sentence> 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) CS 331 Dr M M Awais 14

15 All people need air Examples 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) ) CS 331 Dr M M Awais 15

16 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) CS 331 Dr M M Awais 16

17 Examples 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) CS 331 Dr M M Awais 17

18 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 CS 331 Dr M M Awais 18

19 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 CS 331 Dr M M Awais 19

20 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 CS 331 Dr M M Awais 20

21 Goal: To pick a block and place it over another block Define predicate: General sentence: Conditions What could the conditions? a b c d stack_on(x,y) hand_empty clear (X) clear (Y) pick (X) put_on (X,Y) Conclusions hand_empty ^ clear (X) ^ clear (Y) ^ pick (X) ^ put_on (X,Y) stack_on (X,Y) CS 331 Dr M M Awais 21

22 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) Semantically more correct hand_empty ^ clear (X) pick (X) clear(y) ^ pick (X) put_on (X,Y) put_on (X,Y) stack_on (X,Y) 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 CS 331 Dr M M Awais 22

23 Example: Robotic Arm Modify details of the systems a c b d on(b,a) on( c,d) ontable(b) ontable(d) clear(a) clear(c) hand_empty e a b c d on(b,a) on( c,d) on(e,a) ontable(b) ontable(d) clear(c) clear(e) hand_empty CS 331 Dr M M Awais 23

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

25 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 CS 331 Dr M M Awais 25

26 Existential quantification <variables> <sentence> 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)... CS 331 Dr M M Awais 26

27 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). CS 331 Dr M M Awais 27

28 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) CS 331 Dr M M Awais 28

29 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)] CS 331 Dr M M Awais 29

30 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) CS 331 Dr M M Awais 30

31 Rules: Wumpus world Perception T,S,B percept([s,b,glitter],t) glitter(t) Reflex T glitter(t) bestaction(grab,t) CS 331 Dr M M Awais 31

32 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)] CS 331 Dr M M Awais 32

33 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 CS 331 Dr M M Awais 33

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

35 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 ) CS 331 Dr M M Awais 35

36 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) = 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)) CS 331 Dr M M Awais 36

37 The electronic circuits domain 5. Encode the specific problem instance type(x 1 ) = xor type(a 1 ) = and type(o 1 ) = or type(x 2 ) = xor type(a 2 ) = and 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 )) CS 331 Dr M M Awais 37

38 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 CS 331 Dr M M Awais 38

39 Summary First-order logic: objects and relations are semantic primitives syntax: constants, functions, predicates, equality, quantifiers Increased expressive power: sufficient to define wumpus world CS 331 Dr M M Awais 39

40 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 CS 331 Dr M M Awais 40

First-Order Logic. Chapter 8

First-Order Logic. Chapter 8 First-Order Logic Chapter 8 1 Outline Why FOL? Syntax and semantics of FOL Using FOL Wumpus world in FOL Knowledge engineering in FOL 2 Pros and cons of propositional logic Propositional logic is declarative