Lecture 13-14-Inference in First-order Logic

Lecture 13-14-Inference in First-order Logic - CS 561:...

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CS 561: Artificial Intelligence Instructor: Sofus A. Macskassy, macskass@usc.edu TAs: Nadeesha Ranashinghe ( nadeeshr@usc.edu ) William Yeoh ( wyeoh@usc.edu ) Harris Chiu ( chiciu@usc.edu ) Lectures: MW 5:00-6:20pm, OHE 122 / DEN Office hours: By appointment Class page: http://www-rcf.usc.edu/~macskass/CS561-Spring2010/ This class will use http://www.uscden.net/ and class webpage - Up to date information - Lecture notes - Relevant dates, links, etc. Course material: [AIMA] Artificial Intelligence: A Modern Approach, by Stuart Russell and Peter Norvig. (2nd ed) 1 CS561 - Lecture 13-14 - Macskassy - Spring 2010
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Inference in FOL [AIMA Ch. 9] Reducing first-order inference to propositional inference Unification Generalized Modus Ponens Forward and backward chaining Logic programming Resolution 2 CS561 - Lecture 13-14 - Macskassy - Spring 2010
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A brief history of reasoning 450b.c. Stoics propositional logic, inference (maybe) 322b.c. Aristotle “syllogisms” (inference rules), quantifiers 1565 Cardano probability theory (propositional logic+ uncertainty) 1847 Boole propositional logic (again) 1879 Frege first-order logic 1922 Wittgenstein proof by truth tables 1930 Gödel 9 complete algorithm for FOL 1930 Herbrand complete algorithm for FOL (reduce to propositional) 1931 Gödel :9 complete algorithm for arithmetic 1960 Davis/Putnam “practical” algorithm for propositional logic 1965 Robinson “practical” algorithm for FOL— resolution 3 CS561 - Lecture 13-14 - Macskassy - Spring 2010
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Universal instantiation (UI) Every instantiation of a universally quantified sentence is entailed by it: 8 v ® S UBST ({ v / g } ; ® ) for any variable v and ground term g E.g., 8 x King ( x ) ^ Greedy ( x ) ) Evil ( x ) yields King ( John ) ^ Greedy ( John ) ) Evil ( John ) King ( Richard ) ^ Greedy ( Richard ) ) Evil ( Richard ) King ( Father ( John )) ^ Greedy ( Father ( John )) ) Evil ( Father ( John )) 4 CS561 - Lecture 13-14 - Macskassy - Spring 2010
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Existential instantiation (EI) For any sentence ® , variable v , and constant symbol k that does not appear elsewhere in the knowledge base : 9 v ® S UBST ({ v / k } ; ® ) E.g., 9 x Crown ( x ) ^ OnHead ( x; John ) yields Crown ( C1 ) ^ OnHead ( C1; John ) provided C1 is a new constant symbol, called a Skolem constant Another example: from 9 x d ( x y ) =dy = x y we obtain d ( e y ) =dy = e y provided e is a new constant symbol 5 CS561 - Lecture 13-14 - Macskassy - Spring 2010
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Existential instantition contd. UI can be applied several times to add new sentences; the new KB is logically equivalent to the old EI can be applied once to replace the existential sentence; the new KB is not equivalent to the old, but is satisfiable iff the old KB was satisfiable 6 CS561 - Lecture 13-14 - Macskassy - Spring 2010
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Reduction to propositional inference Suppose the KB contains just the following: 8 x King ( x ) ^ Greedy ( x ) ) Evil ( x ) King ( John
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Lecture 13-14-Inference in First-order Logic - CS 561:...

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