Artificial Intelligence Lecture 1-2: Representational Methods - PDF document

Artificial Intelligence Lecture 1-2: Representational Methods •Propositional Logic and Predicate Logic CS 231: LUMS Lahore Dr. M M Awais Lecture 1-2 Propositional Logic • Symbols P,Q,R,S …….. • Truth Symbols True (T), False (F) • Connectives � (and), � (or), � (Implication), � � (Equality, equivalence) � (not) Statements (Propositions) could be true/false FACTS are also called Atomic Proposition 2 1

Lecture 1-2 Are same Truth Table � P � Q � P � Q P � Q P Q T T F F T T T F F T F F F T T F T T F F T T T T 3 Lecture 1-2 Possible Sentences • P � Q Conjuncts • P � Q Disjuncts • P � Q P=Premise/Antecedent Q=conclusion/Consequent • � P • (P � Q) = ( � P � Q) Inter conversion 4 2

Lecture 1-2 Laws of Propositional Expressions • Demorgan’s Law � (P V Q)=( � P � � Q) • Distributive Law P V(Q � R)= (P � Q) � (P � R) • Commutative Law P � Q= Q � P • Associative Law (P � Q) � R=P � (Q � R) • Contrapositive Law P � Q= � P � � Q 5 Lecture 1-2 Inferencing • If simple facts are known to be true one can find the truth value for the expressions • Thus INTERPRETATIONS can be done. • Interpretation is the assignment of truth values to the sentences Symbols T/F Mapping 6 3

Lecture 1-2 Expressions for KR • Fact 1: Ali likes cakes P • Fact 2: Ali eats cakes Q • P � Q : Ali Likes cakes or eats cakes • P � Q : Ali likes cakes and eats ckes • � Q : Ali does not eat cakes • P � Q: If Ali likes cakes then he eats cakes • P � Q:????? – Above and vice versa 7 Lecture 1-2 Formal Logic Formal Logic • The most widely used formal logic method is FIRST-ORDER PREDICATE LOGIC Components : Alphabets Formal language Axioms Inference Rules 8 4

Lecture 1-2 Alphabets- -I I Alphabets 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 9 Lecture 1-2 Alphabets- -II II Alphabets Function: father(ali) A function that specifies the unique element, that is the father of Ali killer(X) x is a killer Have arity ‘n=1’ (number of arguments to the function) Predicate man(shahid) A predicate which gets TRUTH value equal to 1 (or represented by T) when the interpretation is true. Here Shahid is a man so the predicate is true . bigger(ali , father(babar)) Ali is bigger than the father of Babar. 10 5

Lecture 1-2 Alphabets- -III III Alphabets Connectives: ^ and v or ~ not Implication (when applied to representing logic consider implication sentences as IF-THEN rules) Quantification All persons can see There is a person who cannot see Universal quantifiers Existential quantifiers 11 Lecture 1-2 Examples 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) 12 6

Lecture 1-2 Examples Examples 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) ) 13 Lecture 1-2 Examples 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) 14 7

Lecture 1-2 Examples 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) 15 Lecture 1-2 Undecidable Predicate • For which exhaustive testing is required • Example: • ∀ X likes(zahra, X) • This sentence is computationally impossible to calculate 16 8

Lecture 1-2 Example: Robotic Arm • Represent the initial details of the systems • Generate sentences of descriptive nature and or implicative nature • Modify the facts using new sentences on(b,a) On( c,d) ontable(b) A C ontable(d) clear(a) B D clear(c) hand_empty 17 Lecture 1-2 Definitions • Logically Follows : X logically follows from a set of predicate calculus expressions S if every interpretation that satisfy S also satisfy X (X F S) • Satisfy: If S has value T under interpretation I then I satisfy S (interpretation that makes the sentence true) • Model: If I satisfies S for all variables then I is a model of S • Satisfiable: S is satisfiable iff there exists an interpretation and variable assignments that satisfy it 18 9

Lecture 1-2 S F X S: All birds fly S: Sparrow is bird X: Sparrow flies All humans are mortal Shahid is a human Shahis is mortal All birds fly Sparrow is bird Sparrow flies 19 Lecture 1-2 Definitions • Inconsistent: If a set of expressions are not satisfiable • Valid:If any expression has a value T for all possible interpertations • Sound: If an expression logically follows from another expression then the inferential rule is sound • Complete: When inferential rule produces every expression that logically follows a particular expression 20 10

Lecture 1-2 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 21 11