1 Soundness and Completeness zKB |- S: S is provable from KB. zA proof procedure is sound if: yIf KB |- S, then KB |= S. yThat is, the procedure produces only correct consequences. zA proof procedure is complete if: yIf KB |= S, then KB |- S. yThat is, the procedure produces all the consequences. zIdeally, the procedure should be sound and complete. (Ideals are nice in theory).

2 Knock Knock Logic zWho’s there? yJoe  Mike, Sally zBackground knowledge: yMike => Sally y  Sally   Rita zHence?

3 Modus Ponens zFrom A and A  B, infer B. zA and B can be any sentence. zModus ponens with a few axiom schemas is sound and complete: y A  (B  A) yA  (B  C)  ((A  B)  (A  C)) y(  A   B)  (B  A) yMore in the book.

4 Some Useful Equivalences zP  Q is equivalent to:  P  Q z  (P  Q) is equivalent to:  P   Q z  (P  Q) is equivalent to:  P   Q

5 Normal Forms zCNF = Conjunctive Normal Form zConjunction of disjuncts (each disjunct = “clause”) (P  Q)  R (P  Q)  R  (P  Q)  R  P   Q  R (  P   Q)  R (  P  R)  (  Q  R)

6 Resolution A  B  C,  C  D   E A  B  D   E zRefutation Complete yGiven an unsatisfiable KB in CNF, yResolution will eventually deduce the empty clause zProof by Contradiction yTo show   = Q yShow   {  Q} is unsatisfiable!

7 Knock Knock Resolution yJoe  Mike, ySally, y  Mike  Sally, y  Sally   Rita

8 Resolution Example prove P (A B C) (B) (  B D) (  C A D) (  D P Q) (  Q)

9 Computational Complexity zDetermining satisfiability is NP-complete. zEven when all clauses have at most 3 literals. zHence, also validity and entailment testing are NP-complete. zBut, some recent progress is encouraging! zIf all clauses have at most 2 literals, it is polynomial. zBut if the KB is in DNF, satisfiability is polynomial. yWhat does this tell us about transforming a CNF into a DNF knowledge base?

10 Horn Clauses zIf every sentence in KB is of the form: Then Modus Ponens is –Polynomial time, and –Complete! A  B  C ...  F  Z equivalently  A   B   C ...   F  Z Clause means a big disjunction At most one positive literal

11 Horn Rule Inference zBackward or forward chaining. yP  Q  S yP1  Q  S1 yR1  R2  Q yR1, R2, P.

12 Limitations of Prop. Logic zCumbersome for large domains: yMan-Abraham, Man-Isaac, Man-Jacob yWoman-Sara, Woman-Rachel, Woman-Leah yMan-Abraham  Human-Abraham yWoman-Sara  Human-Sara zCannot deal with infinite domains. zWe’d like to say: yAbraham, Sara etc. are objects. yfor all X, Man(X)  Human(X) yfor all n, Integer(n)  Integer(n+1).

13 First Order Logic (FOPC) zWe identify the objects in our domain. yAbraham, Sara, Isaac, Rachel, yFather-of(Isaac), Mother-of(Isaac). zPredicates specify properties of objects, and tuples of objects: yMan(Abraham), Woman(Sara), yMarried(Abraham, Sara). zQuantified formulas: y  X Man(X)  Human(X) y  X  Y Loves(Y,X).

14 FOL Definitions zConstants: a,b, dog33, Abraham. yName a specific object. zVariables: X, Y. yRefer to an object without naming it. zFunctions: dad-of yMapping from objects to objects. zTerms: father-of(mother-of(dog33)) yRefer to objects zAtomic Sentences: in(father-of(dog33), h1) yCan be true or false yCorrespond to propositional symbols P, Q

15 More Definitions zLogical connectives: , ,   zQuantifiers: y  Forall y  There exists zExamples yAbraham is a man. yAll professors wear glasses. yEvery person is loved by someone who isn’t their mother.

16 Quantifier / Connective Interaction z  x E(x)  G(x) yequivalent to  x E(x)   x G(x)? z  x E(x)  G(x) yequivalent to  x E(x)   x G(x)? z  x E(x)  G(x) z  x E(x)  G(x) z  x E(x)  G(x) E(x) == “x is an elephant” G(x) == “x has the color grey”

17 Nested Quantifiers: Order matters! zExamples yEvery dog has a tail ySomeone is loved by everyone  x  y P(x,y)   y  x P(x,y)

18 If your thesis is entirely vacuous, add a few formulas in predicate calculus. - famous disgruntled advisor

19 FOPC Semantics zAn interpretation includes: yA non-empty universe of discourse, O yA mapping from the constants to elements of O. yFor every function symbol of arity n, a mapping from O n to O. yFor every predicate symbol of arity n, a subset of O n. zWe can now define the truth value of every well formed formula.

20 Unification zUseful for first order inference  a,b city(a)  city(b)  connected(a,b) city(kent) city(seattle) zAlso for compilation zEmphasize variables with ? zUnify(x, y) return mgu yUnify(city(?a), city(kent)) returns ?a/kent

21 Unification Examples zUnify(road(?a, kent), road(seattle, ?b)) zUnify(road(?a, ?a), road(seattle, kent)) zUnify(f(g(?x, dog), ?y)), f(g(cat, ?y), dog) zUnify(f(g(?x)), f(?x))

22 Skolemization z  d  t dog(d)  connected(d, t) z  x  y person(y)  loves(y, x)