Methods of ProofProof methodsNormal FormExample: Conversion to CNFResolutionResolution AlgorithmResolution exampleHorn ClausesTry it YourselvesForward chainingForward chaining exampleSlide 12Slide 13Slide 14Slide 15Slide 16Slide 17Backward chainingBackward chaining exampleSlide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Forward vs. backward chainingModel CheckingThe DPLL algorithmThe WalkSAT algorithmHard satisfiability problemsSlide 34Slide 35Inference-based agents in the wumpus worldExpressiveness limitation of propositional logicSlide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47SummaryMethods of ProofChapter 7, second half.http://www.symantec.com/specprog/university/Proof methods•Proof methods divide into (roughly) two kinds:Application of inference rules:Legitimate (sound) generation of new sentences from old.–Resolution–Forward & Backward chaining Model checkingSearching through truth assignments.•Improved backtracking: Davis--Putnam-Logemann-Loveland (DPLL)•Heuristic search in model space: Walksat.Normal FormWe first rewrite into conjunctive normal form (CNF). |:KBequivalent to KB unsatifi ableaa=��We like to prove:KB a��A “conjunction of disjunctions”(A B) (B C D)ClauseClauseliterals• Any KB can be converted into CNF.• In fact, any KB can be converted into CNF-3 using clauses with at most 3 literals.Example: Conversion to CNFB1,1 (P1,2 P2,1)1. Eliminate , replacing α β with (α β)(β α).(B1,1 (P1,2 P2,1)) ((P1,2 P2,1) B1,1)2. Eliminate , replacing α β with α β.(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)3. Move inwards using de Morgan's rules and double-negation:(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)4. Apply distributive law ( over ) and flatten:(B1,1 P1,2 P2,1) (P1,2 B1,1) (P2,1 B1,1)( )a b a b� � =� ��Resolution•Resolution: inference rule for CNF: sound and complete!( )( )( )A B CAB C� ��- - - - - - - - - - - -\ �“If A or B or C is true, but not A, then B or C must be true.”( )( )( )A B CA D EB C D E� �� � �- - - - - - - - - - -\ � � �“If A is false then B or C must be true, or if A is truethen D or E must be true, hence since A is either true or false, B or C or D or E must be true.” ( )( )( )A BA BB B B�� �- - - - - - - -ں\\\\\\\\Simplification• The resolution algorithm tries to prove:• Generate all new sentences from KB and the query.• One of two things can happen:1. We find which is unsatisfiable. I.e. we can entail the query.2. We find no contradiction: there is a model that satisfies the sentence (non-trivial) and hence we cannot entail the query.Resolution Algorithm|KB equivalent toKB unsatisfi ableaa=��P P��KB a��Resolution example•KB = (B1,1 (P1,2 P2,1)) B1,1 •α = P1,2KB a��False inall worldsTrue!Horn Clauses• Resolution can be exponential in space and time.• If we can reduce all clauses to “Horn clauses” resolution is linear in space and timeA clause with at most 1 positive literal.e.g. • Every Horn clause can be rewritten as an implication with a conjunction of positive literals in the premises and a single positive literal as a conclusion.e.g.• 1 positive literal: definite clause• 0 positive literals: Fact or integrity constraint: e.g. • Forward Chaining and Backward chaining are sound and complete with Horn clauses and run linear in space and time.A B C�� ��B C A� �( ) ( )A B A B False��غ��Try it Yourselves•7.9 page 238: (Adapted from Barwise and Etchemendy (1993).) If the unicorn is mythical, then it is immortal, but if it is not mythical, then it is a mortal mammal. If the unicorn is either immortal or a mammal, then it is horned. The unicorn is magical if it is horned.•Derive the KB in normal form.•Prove: Horned, Prove: Magical.Forward chaining•Idea: fire any rule whose premises are satisfied in the KB,–add its conclusion to the KB, until query is found•Forward chaining is sound and complete for Horn KBAND gateOR gateForward chaining example“AND” gate“OR” GateForward chaining exampleForward chaining exampleForward chaining exampleForward chaining exampleForward chaining exampleForward chaining exampleBackward chainingIdea: work backwards from the query q•check if q is known already, or•prove by BC all premises of some rule concluding q•Hence BC maintains a stack of sub-goals that need to be proved to get to q.Avoid loops: check if new sub-goal is already on the goal stackAvoid repeated work: check if new sub-goal1. has already been proved true, or2. has already failedBackward chaining exampleBackward chaining exampleBackward chaining exampleBackward chaining examplewe need P to proveL and L to prove P.Backward chaining exampleBackward chaining exampleBackward chaining exampleBackward chaining exampleBackward chaining exampleBackward chaining exampleForward vs. backward chaining•FC is data-driven, automatic, unconscious processing,–e.g., object recognition, routine decisions•May do lots of work that is irrelevant to the goal •BC is goal-driven, appropriate for problem-solving,–e.g., Where are my keys? How do I get into a PhD program?•Complexity of BC can be much less than linear in size of KBModel CheckingTwo families of efficient algorithms:•Complete backtracking search algorithms: DPLL algorithm•Incomplete local search algorithms–WalkSAT algorithmThe DPLL algorithmDetermine if an input propositional logic sentence (in CNF) issatisfiable. This is just backtracking search for a CSP.Improvements:1. Early terminationA clause is true if any literal is true.A sentence is false if any clause is false.2. Pure symbol heuristicPure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A B), (B C), (C A), A and B are pure, C is impure. Make a pure symbol literal true. (if there is a model for S, then making a pure symbol true is also a model).3 Unit clause heuristicUnit clause: only one literal in the clauseThe only literal in a unit clause must be true.Note: literals can become a pure symbol or a unit clause when other literals obtain truth values. e.g. ( ) ( )A True A BA pure� � � �=The WalkSAT
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