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chapter09 - Inference in first-order logic Chapter 9...

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Inference in first-order logic Chapter 9 Chapter 9 1
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Outline Reducing first-order inference to propositional inference Unification Generalized Modus Ponens Forward and backward chaining Logic programming Resolution Chapter 9 2
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A brief history of reasoning 450 b.c. Stoics propositional logic, inference (maybe) 322 b.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 odel complete algorithm for FOL 1930 Herbrand complete algorithm for FOL (reduce to propositional) 1931 odel ¬∃ complete algorithm for arithmetic 1960 Davis/Putnam “practical” algorithm for propositional logic 1965 Robinson “practical” algorithm for FOL—resolution Chapter 9 3
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Universal instantiation (UI) Every instantiation of a universally quantified sentence is entailed by it: v α Subst ( { v/g } , α ) for any variable v and ground term g E.g., 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 )) . . . Chapter 9 4
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Existential instantiation (EI) For any sentence α , variable v , and constant symbol k that does not appear elsewhere in the knowledge base : v α Subst ( { v/k } , α ) E.g., x Crown ( x ) OnHead ( x, John ) yields Crown ( C 1 ) OnHead ( C 1 , John ) provided C 1 is a new constant symbol, called a Skolem constant Another example: from x d ( x y ) /dy = x y we obtain d ( e y ) /dy = e y provided e is a new constant symbol Chapter 9 5
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Existential instantiation 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 Chapter 9 6
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Reduction to propositional inference Suppose the KB contains just 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 : proposition symbols are King ( John ) , Greedy ( John ) , Evil ( John ) , King ( Richard ) etc. Chapter 9 7
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Reduction contd. Claim: a ground sentence * is entailed by new KB iff entailed by original KB Claim: every FOL KB can be propositionalized so as to preserve entailment Idea: propositionalize KB and query, apply resolution, return result Problem: with function symbols, there are infinitely many ground terms, e.g., Father ( Father ( Father ( John ))) Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositional KB Idea: For n = 0 to do create a propositional KB by instantiating with depth- n terms see if α is entailed by this KB Problem: works if α is entailed, loops if α
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