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

PT

T

F

F

Q

PT

T

F

F

Q

PT

T

F

F

Q

PT

F

P Q P Q P Q P

Note: (P Q) equivalent to P Q

T

F F

F

F

T T

T

T

T TF

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 P1 ... Pn and know (P1 ... Pn )=> 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 assignmentfor 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]