first order predicate logic ppt

inference in first order logic in artificial intelligence ppt and theory of first order logic in artificial intelligence ppt
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Dr.BenjaminClark,United States,Teacher
Published Date:21-07-2017
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Inference in first-order logic Chapter 9 (Please turn your mobile devices to silent. Thanks) CS3243 - Inference 1 Last Time  First Order Logic  Reasons about objects, predicates I  ntroduces equality and quantifiers  Brief excursion into Prolog T   o be finished and related to more in depth today CS3243 - Inference 2 Outline  Reducing first-order inference to propositional inference  Unification  Generalized Modus Ponens  Forward chaining  Backward chaining  Resolution CS3243 - Inference 3 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)) … CS3243 - Inference 4 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 ) ∧ OnHead(C ,John) 1 1 provided C is a new constant symbol, called a 1 Skolem constant CS3243 - Inference 5 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. CS3243 - Inference 6 Blank spaces to fill in on this slide Propositionalization   Every FOL KB can be propositionalized so as to preserve entailment   Convert it to propositional logic   A ground sentence is entailed by new KB iff entailed by original KB   Idea: propositionalize KB and query, apply resolution, return result   But there’s a problem: with function symbols, there are infinitely many ground terms:  e.g., Father(Father(Father(John))) CS3243 - Inference 7 Blank spaces to fill in on this slide Propositionalization, continued Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositionalized KB Remind you of any other Idea: For n = 0 to ∞ do algorithm? 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 for FOL is semidecidable (algorithms exist that say yes to every entailed sentence, but no algorithm exists that also says no to every non- entailed sentence.) CS3243 - Inference 8 Problems with propositionalization   Propositionalization seems to generate lots of irrelevant sentences.   E.g., from: ∀x King(x) ∧ Greedy(x) ⇒ Evil(x) King(John) ∀y Greedy(y) Brother(Richard,John)   it seems obvious that Evil(John), but propositionalization produces lots of facts such as Greedy(Richard) that are irrelevant k   With p k-ary predicates and n constants, there are p·n instantiations. 2   E.g., p=1 k=2 n=3, Rel(_,_).  3 × 3 = 3 = 9 possibilities CS3243 - Inference 9 Blank spaces to fill in on this slide Unification   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) θ = x/John,y/John works   Unify(α,β) = θ if αθ = βθ p q θ Knows(John,x) Knows(John,Jane) x/Jane Knows(John,x) Knows(y,OJ) x/OJ,y/John 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 ,OJ) 17 CS3243 - Inference 10 Unification  To unify Knows(John,x) and Knows(y,z), θ = y/John, x/z or θ = y/John, x/John, z/John  The first unifier is more general than the second.  There is a single most general unifier (MGU) that is unique up to renaming of variables. MGU = y/John, x/z CS3243 - Inference 11 The unification algorithm CS3243 - Inference 12 The unification algorithm CS3243 - Inference 13 Blank spaces to fill in on this slide Let’s do one together Knows(John,x) Knows(y,Mother(y)) CS3243 - Inference 14 Generalized Modus Ponens (GMP) p ', p ', … , p ', ( p ∧ p ∧ … ∧ p ⇒q) 1 2 n 1 2 n where p 'θ = p θ for all i i i qθ p ' is King(John) p is King(x) 1 1 p ' is Greedy(y) p is Greedy(x) 2 2 θ is x/John,y/John q is Evil(x) q θ is Evil(John)   GMP used with KB of definite clauses (exactly one positive literal)   n.b. recall Horn form allows at most one positive literal (less restrictive)   All variables assumed universally quantified CS3243 - Inference 15 Let’s do an example with a KB   The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.   Prove that Colonel West is a criminal CS3243 - Inference 16 The example KB in FOL ... 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 ) and Missile(M ) 1 1 … all of its missiles were sold to it by Colonel West Missile(x) ∧ Owns(Nono,x) ⇒ Sells(West,x,Nono) Missiles are weapons: Missile(x) ⇒ Weapon(x) An enemy of America counts as "hostile“: Enemy(x,America) ⇒ Hostile(x) West, who is American … American(West) The country Nono, an enemy of America … Enemy(Nono,America) CS3243 - Inference 17 Forward chaining algorithm CS3243 - Inference 18 Forward chaining proof CS3243 - Inference 19 Forward chaining proof CS3243 - Inference 20