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• Количество слайдов: 40 FOL 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 FOL 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 FOL Syntax of FOL: Basic elements • • Constants Predicates Functions Variables Connectives Equality Quantifiers john, 2, lums, . . . brother, >, . . . sqrt, leftside. Of, . . . X, Y, A, B. . . , , = , CS 331 Dr M M Awais 3 FOL 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 predicate symbols function symbols → → → objects relations functional relations • An atomic sentence predicate(term 1, . . . , termn) 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 FOL Alphabets-I Predicates, variables, functions, constants, connectives, quantifiers, and delimiters Constants: (first letter small) b. LUEa color san. TRO 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 FOL Alphabets-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 thearity is 1 (number of arguments to the function) CS 331 Dr M M Awais 6 FOL Alphabets-II Functions: 1) shahid likes zahid 2) atif likes abid 3) Constants to Variables likes(shahid) = zahid likes(atif) = abid likes(X) = Y {X, Y} have two possible BINDINGS likes(X) =Y Substitutions: {X, Y} could be { shahid, zahid} Or {X, Y} could be { atif, abid} CS 331 Dr M M Awais For 1 to be true: {shahid/X, zahid/Y} For 2 to be true: {atif/X, abid/Y} 7 FOL Alphabets-II Predicate Maps Sentences to Truth Values (True/False) 1) Shahid is student(shahid) 2) Sana is a 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 FOL Alphabets-II Predicate 1) Shahid is a good 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 FOL Atomic sentences Atomic sentence = predicate (term 1, . . . , termn) or term 1 = term 2 term = function (term 1, . . . , termn) or constant or variable • brother(john, richard) • greater(length(leftside. Of (square. A)), length(leftside. Of (square. B))) • >(length(leftside. Of (square. A)), length(leftside. Of (square. B))) Functions cannot be atomic sentences CS 331 Dr M M Awais 10 FOL Alphabets-III Connectives: ^ and v or ~ not Implication Quantification All persons can see There is a person who cannot see Universal quantifiers Existential quantifiers (ALL) (There exists) CS 331 Dr M M Awais 11 FOL Complex sentences • Complex sentences are made from atomic sentences using connectives S, 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 is greater than 2 and is not greater than equal to 2) CS 331 Dr M M Awais 12 FOL 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 FOL 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) CS 331 Dr M M Awais at(lums, lums) smart(lums) 14 FOL 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 FOL 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 FOL 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 FOL 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 FOL 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 c b d CS 331 Dr M M Awais 19 FOL Example: Robotic Arm • Represent the initial details of the systems a c b d CS 331 Dr M M Awais on(a, b) on( c, d) ontable(b) ontable(d) clear(a) clear(c) hand_empty 20 FOL Goal: To pick a block and place it over another block Define predicate: General sentence: Conditions What could the conditions? a c b d stack_on(X, Y) Conclusions hand_empty clear (X) clear (Y) pick (X) put_on (X, Y) hand_empty ^ clear (X) ^ clear (Y) ^ pick (X) ^ put_on (X, Y) CS 331 Dr M M Awais stack_on (X, Y) 21 FOL 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) 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 FOL Example: Robotic Arm • Modify details of the systems a c b d e a c b d CS 331 Dr M M Awais on(b, a) on( c, d) ontable(b) ontable(d) clear(a) clear(c) hand_empty on(b, a) on( c, d) on(e, a) ontable(b) ontable(d) clear(c) clear(e) hand_empty 23 FOL Models for FOL: Example CS 331 Dr M M Awais 24 FOL A common mistake to avoid • Represent: Everyone at LUMS is smart 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 FOL 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) . . . CS 331 Dr M M Awais 26 FOL 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 FOL Properties of quantifiers • 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, bread) CS 331 Dr M M Awais 28 FOL 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 FOL 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 FOL Rules: Wumpus world • Perception – T, S, B percept([S, B, glitter], T) glitter(T) • Reflex – T glitter(T) best. Action(grab, T) CS 331 Dr M M Awais 31 FOL 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 FOL Knowledge engineering in FOL 1. Identify the task 1. Assemble the relevant knowledge 1. Decide on a vocabulary of predicates, functions, and constants 1. Encode general knowledge about the domain 1. Encode a description of the specific problem instance CS 331 Dr M M Awais 33 FOL The electronic circuits domain One-bit full adder CS 331 Dr M M Awais 34 FOL The electronic circuits domain 1. Identify the task – Does the circuit actually add properly? (circuit verification) 1. Assemble the relevant knowledge – Composed of wires and gates; Types of gates (AND, OR, XOR, NOT) – Irrelevant: size, shape, color, cost of gates 1. Decide on a vocabulary Dr M M Awais CS 331 35 FOL 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 – CS 331 Dr M M Awais G type(G) = XOR signal(out(1, G)) = 1 36 FOL 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(out(1, x 1), in(2, a 2)) connected(out(1, o 2), in(1, o 1)) connected(out(1, a 1), in(2, o 1)) connected(out(1, x 2), out(1, c 1)) connected(out(1, o 1), out(2, c 1)) connected(in(1, c 1), in(1, x 1)) connected(in(1, c 1), in(1, a 1)) connected(in(2, c 1), in(2, x 1)) connected(in(2, c 1), in(2, a 1)) connected(in(3, c 1), in(2, x 2)) connected(in(3, c 1), in(1, a 2)) CS 331 Dr M M Awais 37 FOL 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. Awais CS 331 Dr M M base 38 FOL 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 FOL 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 