Predicate Logic or FOLPropositional Logic can’t saySyntaxTermGoldbach’s ConjectureSemanticsRepresenting World in FOLSlide 8Negating QuantifiersMore TranslationsUnificationMost General Unifier (MGU)Occurs CheckingModeling with Definite Clauses: at most one positive literalProve: West is a criminalForward ChainingResolution gives forward chainingBackward ChainingResolution yeilds Backward ChainingResolution is non-directionalFOL -> Conjuctive Normal FormSkolemizationResolution in CNFResultsLimitationsSituation Calculus/PlanningBlocks World ExampleMore ExtensionsTimeSpace and moreExample QuestionsExpert Systems: Engineering ApproachMycin: by Shortliffe 1976SoyBean Disease DiagnosisPredicate Logic or FOLChapter 8Propositional Logic can’t say•If X is married to Y, then Y is married to X.•If X is west of Y, and Y is west of Z, then X is west of Z.•And a million other simple things.•Fix: –extend representation: add predicates–Extend operator(resolution): add unificationSyntax•See text for formal rules.•All of propositional + quantifiers, predicates, functions, and constants.•Variables can take on values of constants or terms.•Term = reference to object•Variables not allowed to be predicates.–E.G. What is the relationship between Bill and Hillary?•Text Notation: variables lower case, constants upper•Prolog Notation: variables are upper case, etcTerm•A term with no variables is a ground term.•Composite Objection: function of terms or primitives–Convenience: we don’t want to name all objects–e.g. nounphrase(det(the),adj(tall),noun(tree)).–E.g. leftLeg(John).–Successor of 1 may be s(1), but we write 2.–Successor of 2 s(s(1)), but we write 3.Goldbach’s Conjecture•For all n, if integer(n), even(n), greater(n,2) then there exists p1, p2, integer(p1), integer(p2), prime(p1),prime(p2), and equals(n,sum(p1,p2)).•Quantifiers: for all, there exists•Predicates: integer, greater, prime, even, equals.•Constants: 2•Functions: sum.Semantics•Validity = true in every model and every interpretation. •Interpretation = mapping of constants, predicates, functions into objects, relations, and functions.•For Goldbach wrt to standard integer model: interpretation = mapping n to an even integer. (Context).Representing World in FOL•All kings are persons. goes to?•for all x, King(x) & Person(x). •for all x, King(x) => Person(x).Representing World in FOL•All kings are persons.•for all x, King(x) => Person(x). OK.•for all x, King(x) & Person(x). Not OK.–this says every object is a king and a person.•In Prolog: person(X) :- king(X).•Everyone Likes icecream.•for all x, Likes(x, icecream).Negating Quantifiers•~ there exist x, P(x)•~ for all x, P(x)For all x, Likes(x,Icecream)No one likes liver.For all x, not Likes(x,Liver)•For all x, ~P(x)•There exists x, ~P(x)•There does not exist an x, not Likes(x,Icecream)•Not there exists x, Iikes(x,Liver).More Translations•Everyone loves someone.•There is someone that everyone loves.•Everyone loves their father.•See text.•For all x, there is a y(x) such that Loves(x,y(x)).•There is an M such that for all x, Loves(x,M).•M is skolem constant•For all x, Loves(x,Father(x)).•Father(x) is skolem function.Unification•If p and q are logical expressions, then Unify(p,q) = Substitution List S if using S makes p and q identical or fail.•Standardize apart: before unifying, make sure that p and q contain different variable names.Most General Unifier (MGU)•f(X,g(Y)) unifies with f(g(Z),U) with substitutions {X/g(a), Y/b, U/g(b), Z/b}.•But also if {X/g(Z), U/g(Y)}. •The MGU is unique up to renaming of variables. •All other unifiers are unify with the MGU.•Use Prolog with = for unification.Occurs Checking•When unifying a variable against a complex term, the complex term should not contain the same variable, else non-match.•Prolog doesn’t check this.• Ex. f(X,X) and f(Y,g(Y)) should not unify.Modeling with Definite Clauses: at most one positive literal1. It is a crime for an american to sell weapons to a hostile country.1’. American(x)&Weapons(y)&Hostile(z) & Sell(x,y,z) => Criminal (x).2. The country Nono has some missiles. There exists x Owns(Nono,x)&Missile(x). 2’. Missile(M1). … Skolem Constant introduction2’’. Owns(Nono,M1).Prove: West is a criminal3. All of its missiles where sold to it by Colonel West.3’. Missile(x)&Owns(Nono,x) => Sells(West,x,Nono).4’. Missile(x) => Weapon(x). .. “common sense”5’. Enemy(x,America) => Hostile(x).6’. American(West).7’. Enemy(Nono,American).Forward Chaining•Start with facts and apply rules until no new facts appear. Apply means use substitutions.•Iteration 1: using facts. •Missile(M1),American(West), Owns(Nono,M1), Enemy(Nono,America)•Derive: Hostile(Nono), Weapon(M1), Sells(West,M1,Nono).•Next Iteration: Criminal(West).•Forward chaining ok if few facts and rules, but it is undirected.Resolution gives forward chaining•Enemy(x,America) =>Hostile(x)•Enemy(Nono,America)•|- Hostile(Nono)•Not Enemy(x,America) or Hostile(x)•Enemy(Nono,America)•Resolve by {x/Nono}•To Hostile(Nono)Backward Chaining•Start with goal, Criminal(West) and set up subgoals. This ends when all subgoals are validated.•Iteration 1: subgoals American(x), Weapons(y) and Hostile(z).•Etc. Eventually all subgoals unify with facts.Resolution yeilds Backward Chaining•A(x) &W(y)&H(z)& S(x,y,z) =>C(x)•-A(x) or –W(y) or –H(z) or –S(x,y,z) or C(x).•Add goal –C(West).•Yields –A(West) or -W(y) or –H(z) or-S(West,y,z). Etc.Resolution is non-directional•Both a power (inference representation) and a weakness (no guidance in search)•-a or –b or –c or d or e equals•a,b,c =>d or e and•a,b,c, -d => e etc.•Prolog forces directionality and results in an incomplete theorem prover.FOL -> Conjuctive Normal Form•Similar to process for propositional logic, but•Use negations rules for quantifiers•Standarize variables apart•Universal quantification is implicit.•Skolemization: introduction of constants and functions to remove existential quantifiers.Skolemization•Introduction of constants or functions when removing existential quantifier.•There exists an x such that P(x) becomes: P(A) for some new constant symbol A.•Everyone has someone who loves him •For all x, Loves(F(x),x) where F(x) is a new function.Resolution in CNF•Just like
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