A brief history of reasoning 450 b.c. Stoics propositional logic, - PowerPoint PPT Presentation

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) Inference in first-order logic 1847 Boole propositional logic (again) 1879 Frege first-order logic 1922 Wittgenstein proof by truth tables 1930 G¨ odel ∃ complete algorithm for FOL 1930 Herbrand complete algorithm for FOL (reduce to propositional) Chapter 9 1931 G¨ odel ¬∃ complete algorithm for arithmetic 1960 Davis/Putnam “practical” algorithm for propositional logic 1965 Robinson “practical” algorithm for FOL—resolution Chapter 9 1 Chapter 9 3 Outline Universal instantiation (UI) ♦ Reducing first-order inference to propositional inference Every instantiation of a universally quantified sentence is entailed by it: ∀ v α ♦ Unification Subst ( { v/g } , α ) ♦ Generalized Modus Ponens for any variable v and ground term g ♦ Forward and backward chaining E.g., ∀ x King ( x ) ∧ Greedy ( x ) ⇒ Evil ( x ) yields ♦ Logic programming King ( John ) ∧ Greedy ( John ) ⇒ Evil ( John ) ♦ Resolution King ( Richard ) ∧ Greedy ( Richard ) ⇒ Evil ( Richard ) King ( Father ( John )) ∧ Greedy ( Father ( John )) ⇒ Evil ( Father ( John )) . . . Chapter 9 2 Chapter 9 4

Reduction to propositional inference Existential instantiation (EI) Suppose the KB contains just the following: For any sentence α , variable v , and constant symbol k that does not appear elsewhere in the knowledge base : ∀ x King ( x ) ∧ Greedy ( x ) ⇒ Evil ( x ) ∃ v α King ( John ) Greedy ( John ) Subst ( { v/k } , α ) Brother ( Richard, John ) E.g., ∃ x Crown ( x ) ∧ OnHead ( x, John ) yields Instantiating the universal sentence in all possible ways, we have Crown ( C 1 ) ∧ OnHead ( C 1 , John ) King ( John ) ∧ Greedy ( John ) ⇒ Evil ( John ) provided C 1 is a new constant symbol, called a Skolem constant King ( Richard ) ∧ Greedy ( Richard ) ⇒ Evil ( Richard ) King ( John ) Another example: from ∃ x d ( x y ) /dy = x y we obtain Greedy ( John ) d ( e y ) /dy = e y Brother ( Richard, John ) The new KB is propositionalized: proposition symbols are provided e is a new constant symbol King ( John ) , Greedy ( John ) , Evil ( John ) , King ( Richard ) etc. Chapter 9 5 Chapter 9 7 Existential instantiation contd. Reduction contd. Claim: a ground sentence ∗ is entailed by new KB iff entailed by original KB UI can be applied several times to add new sentences; the new KB is logically equivalent to the old Claim: every FOL KB can be propositionalized so as to preserve entailment EI can be applied once to replace the existential sentence; Idea: propositionalize KB and query, apply resolution, return result the new KB is not equivalent to the old, but is satisfiable iff the old KB was satisfiable 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 is not entailed Theorem: Turing (1936), Church (1936), entailment in FOL is semidecidable Chapter 9 6 Chapter 9 8

Problems with propositionalization Unification We can get the inference immediately if we can find a substitution θ Propositionalization seems to generate lots of irrelevant sentences. such that King ( x ) and Greedy ( x ) match King ( John ) and Greedy ( y ) E.g., from ∀ x King ( x ) ∧ Greedy ( x ) ⇒ Evil ( x ) θ = { x/John, y/John } works King ( John ) Unify ( α, β ) = θ if αθ = βθ ∀ y Greedy ( y ) Brother ( Richard, John ) it seems obvious that Evil ( John ) , but propositionalization produces lots of p q θ facts such as Greedy ( Richard ) that are irrelevant Knows ( John, x ) Knows ( John, Jane ) { x/Jane } Knows ( John, x ) Knows ( y, OJ ) With p k -ary predicates and n constants, there are p · n k instantiations Knows ( John, x ) Knows ( y, Mother ( y )) Knows ( John, x ) Knows ( x, OJ ) With function symbols, it gets much much worse! Chapter 9 9 Chapter 9 11 Unification Unification We can get the inference immediately if we can find a substitution θ We can get the inference immediately if we can find a substitution θ such that King ( x ) and Greedy ( x ) match King ( John ) and Greedy ( y ) such that King ( x ) and Greedy ( x ) match King ( John ) and Greedy ( y ) θ = { x/John, y/John } works θ = { x/John, y/John } works Unify ( α, β ) = θ if αθ = βθ Unify ( α, β ) = θ if αθ = βθ p q θ p q θ Knows ( John, x ) Knows ( John, Jane ) Knows ( John, x ) Knows ( John, Jane ) { x/Jane } Knows ( John, x ) Knows ( y, OJ ) Knows ( John, x ) Knows ( y, OJ ) { x/OJ, y/John } Knows ( John, x ) Knows ( y, Mother ( y )) Knows ( John, x ) Knows ( y, Mother ( y )) Knows ( John, x ) Knows ( x, OJ ) Knows ( John, x ) Knows ( x, OJ ) Chapter 9 10 Chapter 9 12

Unification Generalized Modus Ponens (GMP) We can get the inference immediately if we can find a substitution θ such that King ( x ) and Greedy ( x ) match King ( John ) and Greedy ( y ) ∀ x King ( x ) ∧ Greedy ( x ) ⇒ Evil ( x ) King ( John ) θ = { x/John, y/John } works Greedy ( y ) Unify ( α, β ) = θ if αθ = βθ Brother ( Richard, John ) p 1 ′ , p 2 ′ , . . . , p n ′ , ( p 1 ∧ p 2 ∧ . . . ∧ p n ⇒ q ) ′ θ = p i θ for all i p q θ where p i qθ Knows ( John, x ) Knows ( John, Jane ) { x/Jane } Knows ( John, x ) Knows ( y, OJ ) { x/OJ, y/John } p 1 ′ is King ( John ) Knows ( John, x ) Knows ( y, Mother ( y )) { y/John, x/Mother ( John ) } p 1 is King ( x ) p 2 ′ is Greedy ( y ) p 2 is Greedy ( x ) Knows ( John, x ) Knows ( x, OJ ) θ is { x/John, y/John } q is Evil ( x ) qθ is Evil ( John ) GMP used with KB of definite clauses ( exactly one positive literal) All variables assumed universally quantified Chapter 9 13 Chapter 9 15 Unification Soundness of GMP We can get the inference immediately if we can find a substitution θ Need to show that such that King ( x ) and Greedy ( x ) match King ( John ) and Greedy ( y ) ′ , . . . , p n ′ , ( p 1 ∧ . . . ∧ p n ⇒ q ) | p 1 = qθ θ = { x/John, y/John } works provided that p i ′ θ = p i θ for all i Unify ( α, β ) = θ if αθ = βθ Lemma: For any definite clause p , we have p | = pθ by UI 1. ( p 1 ∧ . . . ∧ p n ⇒ q ) | = ( p 1 ∧ . . . ∧ p n ⇒ q ) θ = ( p 1 θ ∧ . . . ∧ p n θ ⇒ qθ ) p q θ 2. p 1 ′ , . . . , p n ′ | = p 1 ′ ∧ . . . ∧ p n ′ | = p 1 ′ θ ∧ . . . ∧ p n ′ θ Knows ( John, x ) Knows ( John, Jane ) { x/Jane } Knows ( John, x ) Knows ( y, OJ ) { x/OJ, y/John } 3. From 1 and 2, qθ follows by ordinary Modus Ponens Knows ( John, x ) Knows ( y, Mother ( y )) { y/John, x/Mother ( John ) } Knows ( John, x ) Knows ( x, OJ ) fail Standardizing apart eliminates overlap of variables, e.g., Knows ( z 17 , OJ ) Chapter 9 14 Chapter 9 16

Example knowledge base Example knowledge base contd. The law says that it is a crime for an American to sell weapons to hostile . . . it is a crime for an American to sell weapons to hostile nations: nations. The country Nono, an enemy of America, has some missiles, and American ( x ) ∧ Weapon ( y ) ∧ Sells ( x, y, z ) ∧ Hostile ( z ) ⇒ Criminal ( x ) all of its missiles were sold to it by Colonel West, who is American. Nono . . . has some missiles Prove that Col. West is a criminal Chapter 9 17 Chapter 9 19 Example knowledge base contd. Example knowledge base contd. . . . it is a crime for an American to sell weapons to hostile nations: . . . it is a crime for an American to sell weapons to hostile nations: American ( x ) ∧ Weapon ( y ) ∧ Sells ( x, y, z ) ∧ Hostile ( z ) ⇒ Criminal ( x ) Nono . . . has some missiles, i.e., ∃ x Owns ( Nono, x ) ∧ Missile ( x ) : Owns ( Nono, M 1 ) and Missile ( M 1 ) . . . all of its missiles were sold to it by Colonel West Chapter 9 18 Chapter 9 20