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Outline

Outline. Recap Knowledge Representation I Textbook: Chapters 6, 7, 9 and 10. Some KR Languages. Propositional Logic Predicate Calculus Frame Systems Rules with Certainty Factors Bayesian Belief Networks Influence Diagrams Semantic Networks Concept Description Languages

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Outline

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  1. Outline • Recap • Knowledge Representation I • Textbook: Chapters 6, 7, 9 and 10

  2. Some KR Languages • Propositional Logic • Predicate Calculus • Frame Systems • Rules with Certainty Factors • Bayesian Belief Networks • Influence Diagrams • Semantic Networks • Concept Description Languages • Nonmonotonic Logic

  3. In Fact… • All popular knowledge representation systems are equivalent to (or a subset of) • Logic (Propositional Logic or Predicate Calculus) • Probability Theory

  4. Propositional Logic • Syntax • Atomic sentences: P, Q, … • Connectives:  , , ,  • Semantics • Truth Tables • Inference • Modus Ponens • Resolution • DPLL • GSAT • Resolution • Complexity

  5. Notation } Implication (syntactic symbol) • Sound  implies = • Complete = implies      = Inference Entailment

  6. Propositional Logic: SEMANTICS Q Q Q T T F F T F T T T F T T T T F F P P P F F F F T F F T T T T P P  Q P  Q P  Q  P F • Multiple interpretations • Assignment to each variable either T or F • Assignment of T or F to each connective via defns Note: (P  Q) equivalent to  P  Q

  7. FOL Definitions • Constants: a,b, dog33. • Name a specific object. • Variables: X, Y. • Refer to an object without naming it. • Functions: father-of • Mapping from objects to objects. • Terms: father-of(father-of(dog33)) • Refer to objects • Atomic Sentences: in(father-of(dog33), food6) • Can be true or false • Correspond to propositional symbols P, Q

  8. Terminology • Literal u or u, where u is a variable • Clause disjunction of literals • Formula, , conjunction of clauses • (u) take  and set all instances of u true; simplify • e.g. =((P, Q)(R, Q)) then (Q)=P • Pure literal var appearing in a formula either as a negative literal or a positive literal (but not both) • Unit clause clause with only one literal

  9. Definitions • valid = tautology = always true • satisfiable = sometimes true • unsatisfiable = never true 1) smoke  smoke 2) smoke  fire 3) (smoke  fire)  (smoke fire) 4) smoke  fire fire smoke smokevalid smoke firesatisfiable  (smoke fire)  (smoke fire) (smoke fire)   smoke firevalid valid

  10. Inference • Backward Chaining (Goal Reduction) • Based on rule of modus ponens • If know P1 ...  Pn and know (P1 ...  Pn )=> Q • Then can conclude Q • Resolution (Proof by Contradiction) • GSAT

  11. Student-Prof Example • Some students like all professors. No student likes any tough professors. Thus, no professor is tough.

  12. Unification and Substitution • Substitution • a set of pairs s={x=a, y=b} • Instance of a substitution • F=p(x,y,f(a)), Fs=applying s on F={p(a,b,f(a)} • Replacement is simultaneous t={x=a,y=x} • Composition of Substitutions st=? • Unifier: a substitution that makes two expressions the same • Most General Unifier: MGU is a smallest unifier; • Example: unify p(f(x), h(y), a) and p(f(x), z, a)

  13. Normal Forms (Chapter 9, page 281) • CNF = Conjunctive Normal Form • Conjunction of disjuncts (each disjunct = “clause”) (P  Q)  R (P  Q)  R (P  Q)  R P Q  R (P Q)  R (P  R)  (Q  R)

  14. Removing Existential • Skolem Constants (page 281) • Skolem Functions (page 282)

  15. Conversion to Normal Form • Remove implications • Move negation inwards • Standardize variables • Move quantifiers left • Skolemization (every body has a heart) • Distribute and, or’s • Clausal Form

  16. Resolution A  B  C, C  D  E A  B  D  E • Refutation Complete • Given an unsatisfiable KB in CNF, • Resolution will eventually deduce the empty clause • Proof by Contradiction • To show = Q • Show  {Q} is unsatisfiable!

  17. Resolution Refutation Procedure • Page 281 of text • Negating theorem • Normal Form Conversion • Derive an empty clause • Answer Extraction

  18. Student-Prof Example • FOL sentences • Conclusion clause: negate • Use refutation to prove.

  19. Finding Answers • Father’s father is a grandfarther • John is Ken’s father • Larry is Joh’s father • Question: who is Ken’s grandfather?

  20. Application: Logic Programming • Prolog (page 304) • Sequence of sentences • Horn clauses • Queries • Negation as failure • Distinct names = distinct objects • Built-in predicates for math, etc. • Example: membership function

  21. Logic Programming (page 304) • Defining membership • member(X, [X|L]). • member(X, [Y|L]) :- member(X,L). • How does Logic Programming Systems find answers?

  22. Graphically represent the following Birds are animals Mammals are animals Penguins are birds Cats are mammals Birds fly Penguins don’t fly Animals are alive Animals don’t fly Birds have two legs Mammals have 4 legs Semantic Networks have Properties Subset links Member links Semantic Networks (page 317)

  23. GSAT [1992] Procedure GSAT (CNF formula: , max-restarts, max-climbs) For i := 1 to max-restarts do A := randomly generated truth assignment for j := 1 to max-climbs do if A satisfies  then return yes A := random choice of one of best successors to A ;; successor means only 1 (var,val) changes from A ;; best means making the most clauses true

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