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Artificial Intelligence Lecture No. 10

Artificial Intelligence Lecture No. 10 . Dr. Asad Ali Safi ​ Assistant Professor, Department of Computer Science,  COMSATS Institute of Information Technology (CIIT) Islamabad, Pakistan. Summary of Previous Lecture. A knowledge-based agent The Wumpus World Syntax , semantics

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Artificial Intelligence Lecture No. 10

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  1. Artificial IntelligenceLecture No. 10 Dr. Asad Ali Safi ​ Assistant Professor, Department of Computer Science,  COMSATS Institute of Information Technology (CIIT) Islamabad, Pakistan.

  2. Summary of Previous Lecture • A knowledge-based agent • The Wumpus World • Syntax , semantics • Entailment • Logic as a KR language

  3. Today’s Lecture • Logic • Propositional logic • Pros and cons of propositional logic • First-order logic • Syntax of FOL: Basic elements • Atomic/complex sentences • Connections between Quantifiers

  4. No independent access to the world • The reasoning agent often gets its knowledge about the facts of the world as a sequence of logical sentences and must draw conclusions only from them without independent access to the world. • Thus it is very important that the agent’s reasoning is sound!

  5. LANGUAGES

  6. What is logic? • We can also think of logic as an “algebra” for manipulating only two values: true (T) and false (F) • We will cover: • Propositional logic--the simplest kind

  7. Propositional logic • Propositional logic consists of: • The logical values true and false (T and F) • Propositions: “Sentences,” which • Are atomic (that is, they must be treated as indivisible units, with no internal structure), and • Have a single logical value, either true or false • Operators, both unary and binary; when applied to logical values, yield logical values • The usual operators are and, or, not, and implies

  8. Propositional logic: Syntax • Propositional logic is the simplest logic – illustrates basic ideas • The proposition symbols P1, P2 etc are sentences • If S is a sentence, ¬S is a sentence (negation, not) • If S1 and S2 are sentences, S1 ∧ S2 is a sentence (conjunction, AND) • If S1 and S2 are sentences, S1 ∨ S2 is a sentence (disjunction, OR) • If S1 and S2 are sentences, S1 ⇒ S2 is a sentence (implication, IMPLIES) • If S1 and S2 are sentences, S1 ⇔ S2 is a sentence (biconditional)

  9. Truth tables • Logic, like arithmetic, has operators, which apply to one, two, or more values (operands) • A truth table lists the results for each possible arrangement of operands • Order is important: x op y may or may not give the same result as y op x • The rows in a truth table list all possible sequences of truth values for n operands, and specify a result for each sequence • Hence, there are 2n rows in a truth table for n operands

  10. Unary operators • There are four possible unary operators: • Only the last of these (negation) is widely used (and has a symbol,¬ ,for the operation

  11. Useful binary operators • Here are the binary operators that are traditionally used: • Notice in particular that material implication (⇒) only approximately means the same as the English word “implies” • Any other binary operators can be constructed from a combination of these (along with unary not, ¬)

  12. Logical expressions • All logical expressions can be computed with some combination of and (∧), or (∨ ), and not (¬) operators • For example, logical implication can be computed this way: • Notice that ¬X ∨ Y is equivalent to X ⇒ Y

  13. Pros and cons of propositional logic  Propositional logic is declarative  Propositional logic allows partial/disjunctive/negated information • (unlike most data structures and databases) • Propositional logic is compositional: • meaning of B1,1 P1,2 is derived from meaning of B1,1 and of P1,2  Meaning in propositional logic is context-independent • (unlike natural language, where meaning depends on context)  Propositional logic has very limited expressive power • (unlike natural language) • E.g., cannot say "pits cause breezes in adjacent squares“ • except by writing one sentence for each squar

  14. First-order logic • While 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, … “Evil King John ruled England in 1200.” • Object: John, England, 1200; Relation; ruled; Properties: evil, king

  15. Logics in General • Ontological Commitment: • What exists in the world — TRUTH • PL : facts hold or do not hold. • FOL : objects with relations between them that hold or do not hold • Epistemological Commitment: • What an agent believes about facts — BELIEF

  16. Syntax of FOL: Basic elements • Constant Symbols: • Stand for objects • e.g., KingJohn, 2, UCI,... • Predicate Symbols • Stand for relations • E.g., Brother(Richard, John), greater_than(3,2)... • Function Symbols • Stand for functions • E.g., Sqrt(3), LeftLegOf(John),...

  17. Syntax of FOL: Basic elements • Constants KingJohn, 2, UCI,... • Predicates Brother, >,... • Functions Sqrt, LeftLegOf,... • Variables x, y, a, b,... • Connectives , , , ,  • Equality = • Quantifiers , 

  18. Relations • Some relations are properties: they state some fact about a single object: Round(ball), Prime(7). • n-ary relations state facts about two or more objects: Married(John,Mary), LargerThan(3,2). • Some relations are functions: their value is another object: Plus(2,3), Father(Dan).

  19. Atomic sentences Term =function (term1,...,termn) or constant or variable • A logical expression that refers to an object • LeftLegOf(Richard) • There are 2 kinds of terms: • constant symbols: Table, Computer • function symbols: LeftLeg(Pete), Sqrt(3), Plus(2,3) etc Atomic sentence = predicate (term1,...,termn) or term1 = term2 • An Atomic sentence is formed from a predicate symbol followed by list of terms. • Examples: LargerThan(2,3) is false. Brother_of(Mary,Pete) is false. Married(Father(Richard), Mother(John)) could be true or false • Note: Functions do not state facts and form no sentence: • Brother(Pete) refers to John (his brother) and is neither true nor false. • Brother_of(Pete,Brother(Pete)) is True. Function Binary relation

  20. Complex Sentences • We make complex sentences with connectives (just like in propositional logic). property binary relation function objects connectives

  21. More Examples • Brother(Richard, John)  Brother(John, Richard) • King(Richard)  King(John) • King(John) =>  King(Richard) • LessThan(Plus(1,2) ,4)  GreaterThan(1,2) (Semantics are the same as in propositional logic)

  22. Variables • Person(John) is true or false because we give it a single argument ‘John’ • We can be much more flexible if we allow variables which can take on values in a domain. e.g., all persons x, all integers i, etc. • E.g., can state rules like Person(x) => HasHead(x) or Integer(i) => Integer(plus(i,1)

  23. Universal Quantification  •  means “for all” • Allows us to make statements about all objects that have certain properties • Can now state general rules:  x King(x) => Person(x)  x Person(x) => HasHead(x) • i Integer(i) => Integer(plus(i,1)) Note that • x King(x)  Person(x) is not correct! This would imply that all objects x are Kings and are People  x King(x) => Person(x) is the correct way to say this

  24. Existential Quantification  • x means “there exists an x such that….” (at least one object x) • Allows us to make statements about some object without naming it • Examples: x King(x) x Lives_in(John, Castle(x)) iInteger(i) GreaterThan(i,0) Note that  is the natural connective to use with  (And => is the natural connective to use with  )

  25. Combining Quantifiers  x  y Loves(x,y) • For everyone (“all x”) there is someone (“y”) who loves them • y  x Loves(x,y) - there is someone (“y”) who loves everyone Clearer with parentheses:  y (  x Loves(x,y) )

  26. Connections between Quantifiers • Asserting that all x have property P is the same as asserting that does not exist any x that don’t have the property P  x Likes(x, 271 class)   x Likes(x, 271 class) In effect: -  is a conjunction over the universe of objects -  is a disjunction over the universe of objects Thus, DeMorgan’s rules can be applied

  27. De Morgan’s Law for Quantifiers Generalized De Morgan’s Rule De Morgan’s Rule Rule is simple: if you bring a negation inside a disjunction or a conjunction, always switch between them (or and, and  or).

  28. Summery of Today’s Lecture • logic • Propositional logic • Pros and cons of propositional logic • First-order logic • Syntax of FOL: Basic elements • Atomic/complex sentences • Connections between Quantifiers

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