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![Resolution Inference
• Two clauses,
• which are assumed to be standardized apart so that they share no variables,
• can be resolved if they contain complementary literals.
• Complementary Literals - if one unifies with the negation of the other
• Where UNIFY(li, ¬mj) = θ
• [Animal(F(x)) ∨ Loves(G(x), x)] and [¬Loves(u, v) ∨ ¬Kills(u, v)] can be
resolved.
• By eliminating the complementary literals Loves(G(x), x) and ¬Loves(u,
v), with unifier θ = {u/G(x), v/x} - Binary Resolution Rule.
• Resolvent Clause - [Animal(F(x)) ∨ ¬Kills(G(x), x)]](https://image.slidesharecdn.com/knowledgerepresentationandreasoning-230409150648-f587ef12/85/Knowledge-Representation-and-Reasoning-pptx-21-320.jpg)
![• A. ∀ x [∀ y Animal(y) ⇒ Loves(x, y)] ⇒ [∃ y Loves(y, x)]
• B. ∀ x [∃ z Animal(z) ∧ Kills(x, z)] ⇒ [∀ y ¬Loves(y, x)]
• C. ∀ x Animal(x) ⇒ Loves(Jack, x)
• D. Kills(Jack, Tuna) ∨ Kills(Curiosity, Tuna)
• E. Cat(Tuna)
• F. ∀ x Cat(x) ⇒ Animal(x)
• ¬G. ¬Kills(Curiosity, Tuna)](https://image.slidesharecdn.com/knowledgerepresentationandreasoning-230409150648-f587ef12/85/Knowledge-Representation-and-Reasoning-pptx-23-320.jpg)
![Resolution Inference
• Two clauses,
• which are assumed to be standardized apart so that they share no variables,
• can be resolved if they contain complementary literals.
• Complementary Literals - if one unifies with the negation of the other
• Where UNIFY(li, ¬mj) = θ
• [Animal(F(x)) ∨ Loves(G(x), x)] and [¬Loves(u, v) ∨ ¬Kills(u, v)] can be
resolved.
• By eliminating the complementary literals Loves(G(x), x) and ¬Loves(u,
v), with unifier θ = {u/G(x), v/x} - Binary Resolution Rule.
• Resolvent Clause - [Animal(F(x)) ∨ ¬Kills(G(x), x)]](https://image.slidesharecdn.com/knowledgerepresentationandreasoning-230409150648-f587ef12/75/Knowledge-Representation-and-Reasoning-pptx-21-2048.jpg)
![• A. ∀ x [∀ y Animal(y) ⇒ Loves(x, y)] ⇒ [∃ y Loves(y, x)]
• B. ∀ x [∃ z Animal(z) ∧ Kills(x, z)] ⇒ [∀ y ¬Loves(y, x)]
• C. ∀ x Animal(x) ⇒ Loves(Jack, x)
• D. Kills(Jack, Tuna) ∨ Kills(Curiosity, Tuna)
• E. Cat(Tuna)
• F. ∀ x Cat(x) ⇒ Animal(x)
• ¬G. ¬Kills(Curiosity, Tuna)](https://image.slidesharecdn.com/knowledgerepresentationandreasoning-230409150648-f587ef12/75/Knowledge-Representation-and-Reasoning-pptx-23-2048.jpg)
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Knowledge Representation and Reasoning.pptx
- 1.
- 2. Knowledge Based Reasoning •Goal of AI • AI Systems vs Application Programs. • Path Finding Problem • Matrix Multiplication • Mimic Human Problem solving • Knowledge and Experience. Knowledge Based Reasoning exploiting memory, experience and rules of thumb Problem Solution Model Based Reasoning by first principles using search methods
- 3. Logic and Inferences •Symbolic representation • Formal Logic • Eg: • All men are mortal • Socrates is a man • Socrates is mortal. (inference) • Entailment • Given a collection of true statements(premises) • What other statements are logically entailed premises. • Properties of Logic – Soundness, Completeness and Consistency.
- 4. First order Logic •sufficiently expressive to represent a good deal of our commonsense knowledge.
- 6. Basic Elements -Symbols • Objects, Relations and Functions • Constant Symbols – Richard, John • predicate symbols – Brother, onHead, King • Function symbols – leftleg
- 7. Terms • A termis a logical expression that refers to an object. • Constant symbols are therefore terms. • Eg: King John’s left leg - LeftLeg(John) • A complex term is formed by a function symbol followed by a parenthesized list of terms as arguments to the function symbol.
- 8. Atomic sentences • Anatomic sentence (or atom for short) is formed from a predicate symbol optionally followed by a parenthesized list of terms. • Eg: Brother(Richard, John) • Married(Father(Richard), Mother(John)) • An atomic sentence is true in a given model if the relation referred to by the predicate symbol holds among the objects referred to by the arguments.
- 9. Complex sentences • uselogical connectives to construct more complex sentences . ¬Brother(LeftLeg(Richard), John) Brother(Richard, John) ∧ Brother(John, Richard) King(Richard) ∨ King(John) ¬King(Richard) ⇒ King(John)
- 10. Quantifiers • To expressproperties of entire collections of objects, instead of enumerating the objects by name. • universal and existential • Universal quantification (∀) All kings are persons ∀ x King(x) ⇒ Person(x) • Variable • Ground Term – A term with no variables. • Existential quantification (∃) • A statement about “some” object. • ∃ x Crown(x) ∧ OnHead(x, John)
- 11. Nested quantifiers • expressmore complex sentences using multiple quantifiers. Brothers are siblings ∀ x ∀ y Brother(x, y) ⇒ Sibling(x, y) ∀ x, y Sibling(x, y) ⇔ Sibling(y, x) Everybody loves somebody ∀ x ∃ y Loves(x, y) There is someone who is loved by everyone ∃ y ∀ x Loves(x, y)
- 12. Equality • We canuse the equality symbol to signify that two term refer to the same object. Father(John) = Henry
- 13. Inference rules forquantifiers • ∀ x King(x) ∧ Greedy(x) ⇒ Evil(x) . • King(John) ∧ Greedy(John) ⇒ Evil(John) King(Richard) ∧ Greedy(Richard) ⇒ Evil(Richard) King(Father(John)) ∧ Greedy(Father(John)) ⇒ Evil(Father(John)) . • The rule of Universal Instantiation (UI) • We can infer any sentence obtained by substituting a ground term (a term without variables) for the variable. • SUBST(θ, α) • {x/John}, {x/Richard}, and {x/Father(John)}
- 14. Rule for ExistentialInstantiation • The variable is replaced by a single new constant symbol. • ∃ x Crown(x) ∧ OnHead(x, John) • Crown(C1) ∧ OnHead(C1, John) • C1 does not appear elsewhere in the knowledge base. • Skolem constant. • Skolemization • Universal Instantiation can be applied many times to produce many different consequences. • Existential Instantiation can be applied once, and then the existentially quantified sentence can be discarded. • ∃ x Kill(x, Victim) - Kill(Murderer, Victim)
- 15. UNIFICATION AND LIFTING •∀ x King(x) ∧ Greedy(x) ⇒ Evil(x) • King(John) • Greedy(John) • Generalized Modus Ponens • Lifted version of Modus Ponens
- 16. Unification • finding substitutionsthat make different logical expressions look identical. UNIFY(p, q) = θ where SUBST(θ, p) = SUBST(θ, q) . Eg: • AskVars(Knows(John, x)) • UNIFY(Knows(John, x), Knows(John, Jane)) = {x/Jane} • UNIFY(Knows(John, x), Knows(y, Bill)) = {x/Bill, y/John} • UNIFY(Knows(John, x), Knows(y, Mother(y))) = {y/John, x/Mother(John)} • UNIFY(Knows(John, x), Knows(x, Elizabeth)) = fail . • standardizing apart • UNIFY(Knows(John, x), Knows(x17, Elizabeth)) = {x/Elizabeth, x17/John} .
- 17. Forward Chaining • Propositionalforward chaining • start with the atomic sentences in the knowledge base and apply Modus Ponens in the forward direction, adding new atomic sentences, until no further inferences can be made. • First-order definite clauses • disjunctions of literals of which exactly one is positive. • Either an atomic sentence. • Or an implication whose antecedent is a conjunction of positive literals and whose consequent is a single positive literal. • Eg: • King(x) ∧ Greedy(x) ⇒ Evil(x) . • King(John) . • Greedy(y) .
- 18. Backward Chaining • workbackward from the goal, chaining through rules to find known facts that support the proof.
- 19. RESOLUTION • Conjunctive normalform for first-order logic • a conjunction of clauses, where each clause is a disjunction of literals. • Eliminate implications • Move ¬ inwards • Standardize variables • Skolemize • Drop universal quantifiers • Distribute ∨ over ∧
- 20. Conjunctive Normal Form •∀ x American(x) ∧ Weapon(y) ∧ Sells(x, y, z) ∧ Hostile(z) ⇒ Criminal(x) ¬American(x) ∨ ¬Weapon(y) ∨ ¬Sells(x, y, z) ∨ ¬Hostile(z) ∨ Criminal(x)
- 21. Resolution Inference • Twoclauses, • which are assumed to be standardized apart so that they share no variables, • can be resolved if they contain complementary literals. • Complementary Literals - if one unifies with the negation of the other • Where UNIFY(li, ¬mj) = θ • [Animal(F(x)) ∨ Loves(G(x), x)] and [¬Loves(u, v) ∨ ¬Kills(u, v)] can be resolved. • By eliminating the complementary literals Loves(G(x), x) and ¬Loves(u, v), with unifier θ = {u/G(x), v/x} - Binary Resolution Rule. • Resolvent Clause - [Animal(F(x)) ∨ ¬Kills(G(x), x)]
- 22. Example • Everyone wholoves all animals is loved by someone. • Anyone who kills an animal is loved by no one. • Jack loves all animals. • Either Jack or Curiosity killed the cat, who is named Tuna. • Did Curiosity kill the cat?
- 23. • A. ∀x [∀ y Animal(y) ⇒ Loves(x, y)] ⇒ [∃ y Loves(y, x)] • B. ∀ x [∃ z Animal(z) ∧ Kills(x, z)] ⇒ [∀ y ¬Loves(y, x)] • C. ∀ x Animal(x) ⇒ Loves(Jack, x) • D. Kills(Jack, Tuna) ∨ Kills(Curiosity, Tuna) • E. Cat(Tuna) • F. ∀ x Cat(x) ⇒ Animal(x) • ¬G. ¬Kills(Curiosity, Tuna)
- 24. • A1. Animal(F(x))∨ Loves(G(x), x) • A2. ¬Loves(x, F(x)) ∨ Loves(G(x), x) • B. ¬Loves(y, x) ∨ ¬Animal(z) ∨ ¬Kills(x, z) • C. ¬Animal(x) ∨ Loves(Jack, x) • D. Kills(Jack, Tuna) ∨ Kills(Curiosity, Tuna) • E. Cat(Tuna) • F. ¬Cat(x) ∨ Animal(x) • ¬G. ¬Kills(Curiosity, Tuna)