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 Nonmonotonic Logic

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

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

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

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  smoke valid  smoke  fire satisfiable  (  smoke  fire)  (  smoke   fire) valid (smoke   fire)   smoke   fire valid

Inference Backward Chaining (Goal Reduction) –Based on rule of modus ponens –If know P 1 ...  P n and know (P 1 ...  P n )=> Q –Then can conclude Q Resolution (Proof by Contradiction) GSAT

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

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)

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)

Removing Existential Skolem Constants (page 281) Skolem Functions (page 282)

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

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!

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

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

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

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

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

Semantic Networks (page 317) 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

GSAT 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 [1992]