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17-kb-systems_fol

17-kb-systems_fol - First-Order Logic Idea Foundations of...

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1 Foundations of Artificial Intelligence First-Order Logic CS472 – Fall 2007 Thorsten Joachims First-Order Logic Idea: Don’t treat propositions as “atomic” entities. First-Order Logic: Objects: cs472, fred, ph219, emptylist … Relations/Predicates: is_Man(fred), Located(cs472, ph219) … Note: Relations typically correspond to verbs Functions: Pair(search,Pair(learning,Pair(kbsystems, emptylist))) – Connectives: , , ¬ , , – Quantifiers: • Universal: x: ( is_Man(x) is_Mortal(x) ) • Existential: y: ( is_Father(y, fred) ) Example: Representing Facts in First-Order Logic 1. Lucy* is a professor 2. All professors are people. 3. Fuchs is the dean. 4. Deans are professors. 5. All professors consider the dean a friend or don’t know him. 6. Everyone is a friend of someone. 7. People only criticize people that are not their friends. 8. Lucy criticized Fuchs. * Name changed for privacy reasons. Example: Proof Knowledge base: is-prof(lucy) x ( is-prof(x) is-person(x) ) is-dean(fuchs) x (is-dean(x) Æ is-prof(x)) x ( y ( is-prof(x) is-dean(y) is-friend-of(y,x) ¬ knows(x, y) ) ) x ( y ( is-friend-of (y, x) ) ) x (Vy (is-person(x) is-person(y) criticize (x,y) ¬ is-friend-of (y,x))) criticize(lucy,fuchs) Question: Is Fuchs no friend of Lucy? ¬ is-friend-of(fuchs,lucy) Knowledge Engineering 1. Identify the task. 2. Assemble the relevant knowledge. 3. Decide on a vocabulary of predicates, functions, and constants. 4. Encode general knowledge about the domain. 5. Encode a description of the specific problem instance. 6. Pose queries to the inference procedure and get answers. 7. Debug the knowledge base. Inference Procedures: Theoretical Results There exist complete and sound proof procedures for propositional and FOL. Propositional logic Use the definition of entailment directly. Proof procedure is exponential in n , the number of symbols. In practice, can be much faster… Polynomial-time inference procedure exists when KB is expressed as Horn clauses : where the P i and Q are non-negated atoms. First-Order logic Godel’s completeness theorem showed that a proof procedure exists… But none was demonstrated until Robinson’s 1965 resolution algorithm . Entailment in first-order logic is semidecidable .
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