Inference in First-Order Logic

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2. Inference in First-Order Logic • Inference in FOL can be done by converting the KB to propositional logic and using propositional inference. Basically, we need convert sentences with quantifiers to corresponding sentences without quantifiers

3. Recall • xP is true in a model iff P is true with x being each possible object in the model • x King(x)  Person(x) equivalent to • Richard the LionHeart is a king Richard the LionHeart is a person • King John is a king  King John is a person • Richard’s left leg is a king  Richard’s left leg is a person • John’s left leg is a king  John’s left leg is a person • The crown is a king  The crown is a person • Roughly speaking, equivalent to conjunction of instantiations of P

4. Universal instantiation(a.k.a. universal elimination) • If xP(x) is true, then P(C) is true, where C is any constant in the domain of x • Example: x eats(Ziggy, x)  eats(Ziggy, IceCream) • The variable symbol can be replaced by any ground term, i.e., any constant symbol or function symbol applied to ground terms only

5. Recall • xP is true in a model iff P is true with x being some possible object in the model • x Crown(x)OnHead(x, John)equivalent to • (Richard the LionHeart is a crown  Richard the LionHeart is on John’s head) (King John is a crown  King John is on John’s head) (Richard’s left leg is a crown  Richard’s left leg is on John’s head) (John’s left leg is a crown John’s left leg is on John’s head)  (The crown is a crown  the crown is on John’s head) • Roughly speaking, equivalent to a disjunction of instantiations of P

6. Existential instantiation(a.k.a. existential elimination) • From xP(x) infer P(C) • Example: • x eats(Ziggy, x)  eats(Ziggy, Stuff) • Note that the variable is replaced by a brand-new constant not occurring in this or any other sentence in the KB • Convenient to use this to reason about the unknown object, rather than constantly manipulating the existential quantifier

7. Reduction to propositional form • Suppose the KB contains the following x King(x)  Greedy(x)  Evil(x) King(John) Greedy(John) Brother(Richard,John) • Instantiating the universal sentence in all possible ways, we have King(John)  Greedy(John)  Evil(John) King(Richard)  Greedy(Richard)  Evil(Richard) King(John) Greedy(John) Brother(Richard,John) • The new KB is propositionalized, propositional symbols are King(John), Greedy(John), Evil(John), King(Richard), etc

8. Reasoning in First-Order Logic • Example • The law says that it is a crime for a Gaul to sell potion formulas to hostile nations • The country Rome, and enemy of Gaul, has acquired some potion formulas, and all of its formulas were sold to it by a Druid Traitorix • Traitorix is a Gaul • Is Traitorix a criminal? (DONE IN CLASS)

9. Inference Approaches in FOL • Forward or Backward Chaining • Use Generalized Modus Ponens to add new atomic sentences • Requires KB to be in form of first-order definite clauses • Previous example used Forward Chaining • Resolution-based inference • Note that all of these methods are generalizations of their propositional equivalents