1. 1 Logical Inference Algorithms CS 171/271 (Chapter 7, continued) Some text and images in these slides were drawn from Russel & Norvig’s published material

2. 2 Inference Rules Modus Ponens   ,   And-Elimination     Logical Equivalences

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4. 4 Validity and Satisfiability A sentence is valid if it is true in all models KB ╞  iff (KB   ) is valid (deduction theorem) A sentence is satisfiable if it is true in some model KB ╞  iff (KB   ) is unsatisfiable (proof by contradiction)  is satisfiable iff  is not valid  is valid iff  is unsatisfiable

5. 5 Resolution Inference Rule Simple case: a  b,  b  c a  c (b and  b are complementary literals that are eliminated) General case: replace a and c with disjunctions of any number of literals

6. 6 Conjunctive Normal Form Any sentence can be converted to a logically equivalent sentence that is a conjunction of disjunctions of literals Ands of or-clauses This can be done by repeated applications of biconditional elimination, implication elimination and distributivity Motivation: if KB is in CNF, can devise an inference algorithm based on resolution

7. 7 Algorithm Using Resolution Convert (KB   ) to conjunctive normal form (CNF) Get pairs of clauses and eliminate complementary literals if they exist If an empty clause results, (KB   ) is unsatisfiable, which means KB ╞  Proof by contradiction

8. 8 Resolution

9. 9 Resolution Example KB = (B 1,1  (P 1,2  P 2,1 ))  B 1,1 α =  P 1,2

10. 10 Restricted Knowledge Bases If sentences in the KB are of a particular form, inference may turn out to be easier, simpler, quicker Full power of resolution not really needed in many practical situations Case in point: Horn Clauses

11. 11 Horn Clauses Horn-Clause Clause of or-ed literals where at most one literal is positive Can be converted to an implication Example: (  a   b  c )  ( a  b  c ) Can use Modus Ponens and chaining in an entailment procedure

12. 12 Forward Chaining Algorithm Assume KB contains single (positive) symbols known to be true implications Implications with premises that contain the symbols yield new symbols once premise has been satisfied Continue until q (the query symbol) is encountered

13. 13 Forward Chaining

14. 14 Forward-Chaining Example

15. 15 Backward-Chaining Variant of chaining that starts with target query q Look for implications that conclude q Take note of its premises If one of those premises can be shown true (also by backward chaining), then q is true Goal-directed reasoning

16. 16 Analysis of Inference Algorithms Soundness Completeness Time Complexity

17. 17 Improvement to Model Checking DPLL algorithm Same as Model Enumeration with some improvements: Early termination Pure symbol heuristic Unit clause heuristic

18. 18 DPLL (Backtracking)

19. 19 An Inference Agent in the Wumpus World KB initially contains sets of sentences that: State absence of pit in room [1,1] State absence of wumpus in room [1,1] State how a breeze arises State how a stench arises Knows there is exactly one wumpus At least one wumpus Of two squares, one should not have wumpus 155 sentences with 64 distinct symbols

20. 20 An Inference Agent in the Wumpus World On each percept: TELL status of stench or breeze Grab if glitter is perceived ASK if there is a provably safe square, or at least a possible safe square, then go there May need a list of actions to go there

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