1

Soundness and Completeness

KB |- S: S is provable from KB.A proof procedure is sound if:

If KB |- S, then KB |= S. That is, the procedure produces only correct

consequences.A proof procedure is complete if:

If KB |= S, then KB |- S. That is, the procedure produces all the consequences.

Ideally, the procedure should be sound and complete. (Ideals are nice in theory).

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Knock Knock Logic

Who’s there? Joe Mike, Sally

Background knowledge: Mike => Sally

Sally Rita

Hence?

3

Modus Ponens

From A and A B, infer B.A and B can be any sentence.Modus ponens with a few axiom schemas

is sound and complete: A (B A) A (B C) ((A B) (A C)) ( A B) (B A) More in the book.

4

Some Useful Equivalences

P Q is equivalent to: P Q(P Q) is equivalent to: P Q

(P Q) is equivalent to: P Q

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Normal Forms

CNF = Conjunctive Normal FormConjunction of disjuncts (each disjunct =

“clause”)

(P Q) R

(P Q) R

(P Q) R P Q R

(P Q) R

(P R) (Q R)

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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!

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Knock Knock Resolution

Joe Mike, Sally,

Mike Sally,

Sally Rita

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

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Computational ComplexityDetermining satisfiability is NP-complete. Even when all clauses have at most 3 literals.Hence, also validity and entailment testing are NP-

complete.But, some recent progress is encouraging!If all clauses have at most 2 literals, it is polynomial.But if the KB is in DNF, satisfiability is polynomial.

What does this tell us about transforming a CNF into a DNF knowledge base?

10

Horn Clauses

If 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 mean

s a

big disjuncti

on

At most one

positive literal

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Horn Rule Inference

Backward or forward chaining. P Q S P1 Q S1 R1 R2 Q R1, R2, P.

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Limitations of Prop. Logic

Cumbersome for large domains: Man-Abraham, Man-Isaac, Man-Jacob Woman-Sara, Woman-Rachel, Woman-Leah Man-Abraham Human-Abraham Woman-Sara Human-Sara

Cannot deal with infinite domains.We’d like to say:

Abraham, Sara etc. are objects. for all X, Man(X) Human(X) for all n, Integer(n) Integer(n+1).

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First Order Logic (FOPC)

We identify the objects in our domain. Abraham, Sara, Isaac, Rachel, Father-of(Isaac), Mother-of(Isaac).

Predicates specify properties of objects, and tuples of objects: Man(Abraham), Woman(Sara), Married(Abraham, Sara).

Quantified formulas: X Man(X) Human(X) X Y Loves(Y,X).

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FOL DefinitionsConstants: a,b, dog33, Abraham.

Name a specific object. Variables: X, Y.

Refer to an object without naming it.Functions: dad-of

Mapping from objects to objects.Terms: father-of(mother-of(dog33))

Refer to objectsAtomic Sentences: in(father-of(dog33), h1)

Can be true or false Correspond to propositional symbols P, Q

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More DefinitionsLogical connectives: , , Quantifiers:

Forall There exists

Examples Abraham is a man.

All professors wear glasses.

Every person is loved by someone who isn’t their mother.

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Quantifier / Connective Interaction

x E(x) G(x) equivalent to x E(x) x G(x)?

x E(x) G(x) equivalent to x E(x) x G(x)?

x E(x) G(x)x E(x) G(x)x E(x) G(x)

E(x) == “x is an elephant”G(x) == “x has the color grey”

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Nested Quantifiers: Order matters!

Examples Every dog has a tail

Someone is loved by everyone

x y P(x,y) y x P(x,y)

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If your thesis is entirely vacuous,

add a few formulas in predicate

calculus.

- famous disgruntled advisor

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FOPC Semantics

An interpretation includes: A non-empty universe of discourse, O A mapping from the constants to elements of O. For every function symbol of arity n, a mapping

from O n to O. For every predicate symbol of arity n, a subset of

O n. We can now define the truth value of every

well formed formula.

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UnificationUseful for first order inference

a,b city(a) city(b) connected(a,b)city(kent)city(seattle)

Also for compilationEmphasize variables with ?Unify(x, y) return mgu

Unify(city(?a), city(kent)) returns ?a/kent

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Unification Examples

Unify(road(?a, kent), road(seattle, ?b))

Unify(road(?a, ?a), road(seattle, kent))

Unify(f(g(?x, dog), ?y)), f(g(cat, ?y), dog)

Unify(f(g(?x)), f(?x))

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Skolemization

d t dog(d) connected(d, t)

x y person(y) loves(y, x)