chapter9 - Inference in First-Order Logic CHAPTER 9 HASSAN...

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CHAPTER 9 HASSAN KHOSRAVI SPRING2011 Inference in First-Order Logic
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Outline Reducing first-order inference to propositional inference Unification Generalized Modus Ponens Forward chaining Backward chaining Resolution
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Universal instantiation (UI) Notation: Subst({v/g}, α ) means the result of substituting g for v in sentence α 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 ), {x/John} King ( Richard ) Greedy ( Richard ) Evil ( Richard ), {x/Richard} King ( Father ( John )) Greedy ( Father ( John )) Evil ( Father ( John )), {x/Father(John)}
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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 ) where C 1 is a new constant symbol, called a Skolem constant Existential and universal instantiation allows to “propositionalize” any FOL sentence or KB EI produces one instantiation per EQ sentence UI produces a whole set of instantiated sentences per UQ sentence
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Reduction to propositional form Suppose the KB contains the following: x King(x) Greedy(x) Evil(x) Father(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
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Reduction continued Every FOL KB can be propositionalized so as to preserve entailment A ground sentence is entailed by new KB iff entailed by original KB Idea for doing inference in FOL: propositionalize KB and query apply resolution-based inference return result Problem: with function symbols, there are infinitely many ground terms, e.g., Father ( Father ( Father ( John ))), etc
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Reduction continued Theorem: Herbrand (1930). If a sentence α is entailed by a FOL KB, it is entailed by a finite subset of the propositionalized 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 Example x King(x) Greedy(x) Evil(x) Father(x) King(John) Greedy(Richard) Brother(Richard,John) Query Evil(X)?
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Depth 0 Father(John) Father(Richard) King(John) Greedy(Richard) Brother(Richard , John) King(John) Greedy(John) Evil(John) King(Richard) Greedy(Richard) Evil(Richard) King(Father(John)) Greedy(Father(John)) Evil(Father(John)) King(Father(Richard)) Greedy(Father(Richard)) Evil(Father(Richard)) Depth 1 Depth 0 + Father(Father(John)) Father(Father(John)) King(Father(Father(John))) Greedy(Father(Father(John))) Evil(Father(Father(John)))
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Problems with Propositionalization Problem: works if α is entailed, loops if α is not entailed Propositionalization generates lots of irrelevant sentences
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chapter9 - Inference in First-Order Logic CHAPTER 9 HASSAN...

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