agent that reason logically 지식표현. agent that reason logically2 knowledge base a set of...

Agent that reason Agent that reason logicallylogically

지식표현

Agent that reason logically 2

Knowledge BaseKnowledge Base

A set of representations of facts about the worldKnowledge representation language tell : what has been told to the knowledge base pr

eviously ask : a question and the answer

Inference : what follows from what the KB has been TelledBackground knowledge : a knowledge base which may initially containedSentence : individual representation of a fact


Knowledge baseKnowledge base

The knowledge level :: saying what it knows to KB “Golden Gates Bridge links San Francisco and Marin CountryThe logical level :: the knowledge is encoding into sentences Links(GGBridge, SF, Marin)The implementation level :: the level that runs on the agent architecture (data structures to represent knowledge or facts)


KnowledgeKnowledge

declarative/procedural love(john, mary). can_fly(X) :- bird(X), not(can_fly(X)), !.

learning : general knowledge about the environment given a series of perceptsCommonsense knowledge


Specifying the Specifying the environmentenvironment

Figure 6.2 A typical wumpus world


Domain specific Domain specific knowledgeknowledge

Domain specific knowledge In the squares directly adjacent to a pit, the

agent will perceive a breeze

Commonsense knowledge logical reasoning stench(1,2) & ~setnch(2,1)

~wumpus(2,2) wumpus(1,3) stench(2,1) & stench(2,3) & stench(1,4)


Inference in Wumpus world(I)Inference in Wumpus world(I)

4,13,12,11,1

4,23,22,21,2

4,33,32,31,3

4,43,42,41,4

OKOK

A = AgentB = BreezeG = Glitter, GoldOK = Safe squareP = PitS = StenchV = VisitedW = Wumpus

4,13,12,11,1

4,23,22,21,2

4,33,32,31,3

4,43,42,41,4

OKOK

OK

AA

BV

P ?

Figure 6.3 The first step taken by the agent in the wumpus world.(a) The initial situation, after percept [None, None, None, None, None]. (b) After one move, with percept [None, Breeze, None, None, None].


Inference in Wumpus world(II)Inference in Wumpus world(II)

4,13,12,11,1

4,23,22,21,2

4,33,32,31,3

4,43,42,41,4

OKOK

A = AgentB = BreezeG = Glitter, GoldOK = Safe squareP = PitS = StenchV = VisitedW = Wumpus

4,13,12,11,1

4,23,22,21,2

4,33,32,31,3

4,43,42,41,4

OKOK

OK

A

VV

V

Figure 6.4 Two later stages in the progress of the agent.(a) After the third move, with percept [Stench, None, None, None, None]. (b) After the fifth move, with percept [Stench, Breeze, Glitter, None, None].

V VB

OK OK

W!

BOK

VS

P !

P ?

P ?

AW !S GB


Representation, Representation, Reasoning, and LogicReasoning, and Logic

Syntax : the possible configurations that constitute sentencesSemantics : the facts in the world to which the sentences refer


The logical reasoningThe logical reasoning

Aentences Sentence

Facts Fact

Sem

antics

Entails

Follows

Representation

World

Sem

antics

Figure 6.5 The connection between sentences and facts is provided by the semantics of the language. The property of one fact following from some other facts is mirrored by the property of one sentence being entailed by some other sentences. Logical inference generates new sentences that are entailed by existing sentences.


Inference IInference I

Entailment :: generation of new sentences that are necessarily true, given that the old sentences are true

Soundness, truth-preserving :: An inference procedure that generates only entailed sentences modus ponens <-> abduction

KB├i , is derived from KB by I

Proof :: a sound inference procedure


Inference IIInference II

Completeness :: an inference procedure that can find a proof for any sentence that is entailed

Proof :: specifying the reasoning steps that are sound

Valid :: if and only if all possible interpretations in all possible worldsTautologies, analytic sentences :: valid sentences

Satisfiable :: if and only if there is some interpretation in some world for which it is true

Unsatisfiable :: a sentence that is not satisfiable


Logics Logics

Boolean logic Symbols represent whole propositions

(facts) Boolean connectives

First-order logic objects, predicates connectives, quantifiers


Wrong logical reasoningWrong logical reasoningFIRST VILLAGER: We have found a witch. May we burn her?ALL: A witch! Burn her!BEDEVERE: Why do you think she is a witch?SECOND VILLAGER: She turned me into a newt.BEDEVERE: A newt?SECOND VILLAGER (after looking at himself for some time): I got better.ALL: Burn her anyway.BEDEVERE: Quiet! Quiet! There are ways of telling whether she is a witch.BEDEVERE: Tell me … What do you do with witches?ALL: Burn them.BEDEVERE: And what do you burn, apart from witches?FOURTH VILLAGER: … Wood?BEDEVERE: So why do witches burn?SECOND VILLAGER: (pianissimo) Because they’re made of wood?BEDEVERE: Good.ALL: I see. Yes, of course.BEDEVERE: So how can we tell if she is made of wood?FIRST VILLAGER: Make a bridge out of her.BEDEVERE: Ah … but can you not also make bridges out of stone?ALL: Yes, of course … um … er …BEDEVERE: Does wood sink in water?ALL: No, no, it floats. Throw her in the pond.BEDEVERE: Wait. Wait … tell me, what also floats on water?ALL: Bread? No, no no. Apples … gravy … very small rocks …BEDEVERE: No, no no.KING ARTHUR: A duck!(They all turn and look at ARTHUR. BEDEVERE looks up very impressed.)BEDEVERE: Exactly. So … logically …FIRST VILLAGER (beginning to pick up the thread): If she .. Weight the same as a duck … she’s made of wood.BEDEVERE: And therefore?ALL: A witch!


Ontological and Ontological and epistemological epistemological commitmentscommitments

Ontological commitments :: to do with the nature of reality Propositional logic(true/false),

Predicate logic, Temporal logic Epistemological commitments :: to do with the possible states of knowledge an agent can have using various types of logic degree of belief fuzzy logic


Commitments Commitments

Language Ontological Commitment(What exists in the world)

Epistemological Commitment(What an agent believes about facts)

Propositional logicFirst-order logicTemporal logicProbability theoryFuzzy logic

factsfacts, objects, relationstimesfactsdegree of truth

true/false/unknowntrue/false/unknowntrue/false/unknowndegree of belief 0…1degree of belief 0…1

Formal languages and their and ontological and epistemological commitments


Propositional LogicPropositional Logic

logical constant : true/false propositional symbols : P, Q parentheses : (P & Q) logical connectives : &(conjuction), v(disjunction), ->(implication), <->(equivalence), ~(negation)


SemanticsSemantics

P H PH (P H) ┐H ((P H) ┐H)P

FalseFalseTrueTrue

FalseTrueFalseTrue

FalseTrueTrueTrue

FalseFalseTrueFalse

TrueTrueTrueTrue

Truth table showing validity of a complex sentence


Validity and InferenceValidity and Inference

Truth tables for five logical connectives

P Q ┐P PQ PQ PQ PQ

False False True True

False True False True

True True False False

False False False True

False True True True

True True False True

True False False True


ModelsModels

Any world in which a sentence is true under a particular interpretationEntailment :: a sentence is entailed by a knowledge base KB if the models of the KB are all models of

The set of models of P & Q is the intersection of the models of P and the models of Q


Inference Rules for propositional Inference Rules for propositional logiclogic

Modus Ponens or Implication-Elimination: (From an implication and the premise of the implication, you can infer the conclusion.)

And-Elimination: (From a conjunction, you can infer any of the conjuncts.)

And-Introduction: (From a list of sentences, you can infer their conjunction.)

Or-Introduction: (From a sentence, you can infer its disjunction with anything else at all.)

Double-Negation Elimination: (From a doubly negated sentence, you can infer a positive sentence.)

Unit Resolution: (From a disjunction, if one of the disjuncts is false, then you can infer the other one is true.)

Figure 6.13 Seven inference for propositional logic. The unit resolution rule is a special case of the resolution rule, which in turn is a special case of the full resolution rule for first-order logic discussed in Chapter 9.

=> ,

i

1 2 … n

1 2 … n

1, 2, …, n

1 2 … n

i

,

,

=>

=> , =>

Resolution: (This is the most difficult. Because cannot be both true and false, one of the other disjucts must be true in one of the premises. Or equivalently, implication is transitive.)

or equivalently


Complexity of Complexity of propositional inferencepropositional inference

NP-complete Monotonicity If KB1╞ then (KB1 ∪ KB2) ╞

Horn clause logic polynomial time complexity P1∧P2∧….∧Pn ⇒ Q


Wumpus worldWumpus world

Initial state ~S1,1 ~B1,1 ~S2,1 B2,1 S1,2 ~B1,2

Rule R1: ~S1,1 -> ~W1,1 & ~W1,2 & ~W2,1 R2: ~S2,1 -> ~W1,1 & ~W2,1 & ~W2,2 & ~W3,1 R3: ~S1,2 -> ~W1,1 & ~W1,2 & ~W2,2 & ~W1,3 R4: S1,2 -> W1,3 V W1,2 V W2,2 V W1,2


Finding the wumpusFinding the wumpus

Inference process Modus ponens : ~S1,1 and R1 ~W1,1 & ~W1,2 & ~W2,1 And-Elimination ~W1,1 ~W1,2 ~W2,1 Modus ponens and And-Elimination: ~W2,2 ~W2,1 ~W3,1 Modus ponens S1,2 and R4 W1,3 V W1,2 V W2,2 V W1,1


Inference process(cont.)Inference process(cont.)

unit resolution ~W1,1 and W1,3 V W1,2 V W2,2 V W1,1

W1,3 V W1,2 V W2,2 unit resolution ~W2,2 and W1,3 V W1,2 V W2,2

W1,3 V W1,2 unit resolution ~W1,2 and W1,3 V W1,2 W1,3


Translating knowledge Translating knowledge into actioninto action

A1,1 & EastA & W2,1 -> ~Forward EastA :: facing east

Propositional logic is not powerful enough to solve the wumpus problem easily


숙제숙제

6.3, 6.6, 6.7, 6.9, 6.10, 6.12, 6.15, 6.16

First-order LogicFirst-order Logic


Limitation of Limitation of propositional logicpropositional logic

A very limited ontology

to need to the representation power

first-order logic


First-order logicFirst-order logic

A stronger set of ontological commitments

A world in FOL consists of objects, properties,

relations, functions Objects people, houses, number, colors, Bill ClintonRelations brother of, bigger than, owns, loveProperties red, round, bogus, primeFunctions father of, best friend, third inning of


ExamplesExamples

“One plus two equals three” objects :: one, two, three, one plus two Relation :: equal Function :: plus

“Squares neighboring the wumpus are smelly Objects :: wumpus, square Property :: smelly Relation :: neighboring


First order logicsFirst order logics

Objects 와 relations시간 , 사건 , 카테고리 등은 고려하지 않음영역에 따라 자유로운 표현이 가능함 ‘ king’ 은 사람의 property 도 될 수 있고 , 사람과 국가를 연결하는 relation 이 될 수도 있다일차술어논리는 잘 알려져 있고 , 잘 연구된 수학적 모형임


예예

Constant symbols :: A, B, John, Predicate symbols :: Round, BrotherFunction symbols :: Cosine, FatherOfTerms :: King John, Richard’s left legAtomic sentences :: Brother(Richard,John), Married(FatherOf(Richard), MotherOf(John))

Complex sentences :: Older(John,30)=>~younger(John,30)


Quantifiers Quantifiers

World = {a, b, c}Universal quantifier (∀)

∀x Cat(x) => Mammal(x) Cat(a) => Mammal(a) & Cat(a) => Mammal(a) &

Cat(a) => Mammal(a) Existential quantifier (∃)

∃x Sister(x, Sopt) & Cat(x)


Nested quantifiersNested quantifiers

∀x,y Parent(x,y) => Child(y,x)

∀x,y Brother(x,y) => Sibling(y,x)

∀x∃y Loves(x,y)

∃y∀x Loves(x,y)


De Morgan’s RuleDe Morgan’s Rule

∀x ~P ~∃x P ~P&~Q ~(P v Q)~∀x P ∃x ~P ~(P&Q) ~P v ~Q ∀x P ~∃x ~P P&Q ~(~P v ~ Q) ∃x P ~∀x ~P P v Q ~(~P&~Q)


EqualityEquality

Identity relation

Father(John) = Henry∃x,y Sister(Spot,x) & Sister(Spot,y)

& ~(x=y)≠ ∃x,y Sister(Spot,x) & Sister(Spot,y)


Higher-order logicHigher-order logic

∀x,y (x=y) (∀p p(x) p(y))

∀f,g (f=g) (∀x f(x) g(x))∀


-expression-expression

x,y x2 – y2

-expression can be applied to arguments to yield a logical term in the same way that a function can be

(x,y x2 – y2)(25,24) = 252-242 = 49x,y Gender(x) ≠Gender(y) & Address(x) = Address(y)


∃∃! (The uniqueness ! (The uniqueness quantifier)quantifier)

∃!x King(x)∃x King(x) & ∀y King(y) => x=y

world 를 고려하여 보여주면 => object 가 1, 2, 3 개일 때

{a} w0 king={}, w1 king={a} w1 만 model

{a,b} w0 king={}, w1 king={a}, w2 {b}, w3 {a,b} w1, w2 만 model


Representation of Representation of sentences by FOPLsentences by FOPL

One’s mother is one’s female parent ∀m,c Mother(c)=m Female(m) & Parent(m)

One’s husband is one’s male spouse ∀w,h Husband(h,w) Male(h) & Spouse(h,w)

Male and female are disjoint categories ∀x Male(x) ~Female(x)

A grandparent is a parent of one’s parent ∀g,c Grandparent(g,c) ∃p parent(g,p) &

parent(p,g)


Representation of Representation of sentences by FOPLsentences by FOPL

A sibling is another child of one’s parents ∀x,y Sibling(x,y) x≠y & ∃p Parent(p,x) & Parent(p,y)Symmetric relations

∀x,y Sibling(x,y) Sibling(y,x)


The domain of sets (I)The domain of sets (I)

The only sets are the empty set and those made by adjoining something to a set :

∀s Set(s) (s=EmptySet) v (∃x,s2 Set(s2) & s=Adjoin(x,s2))The empty set has no elements adjoined into it.

~∃x,s Adjoin(x,s)=EmptySetAdjoining an element already in the set has no effect

∀x,s Member(x,s) s=Adjoin(x,s)The only members of a set are the elements that were adjoined into it

∀x,s Member(x,s) ∃y,s2 (s=Adjoin(y,s2) & (x=y v Member(x,s)))


The domain of sets (II)The domain of sets (II)

A set is a subset of another if and only if all of the first set’s are members of the second set :

∀s1,s2 Subset(s1,s2)

(∀x Member(x,s1) => member(x,s2))Two sets are equal if and only if each is a subset of the other:

∀s1,s2 (s1=s2) (Subset(s1,s2) & Subset(s2,s1))


The domain of sets (III)The domain of sets (III)

An object is a member of the intersection of two sets if and only if it is a member of each of sets :

∀x,s1,s2 Member(x,Intersection(s1,s2)) Member(x,s1) & Member(x,s2)

An object is a member of the union of two sets if and only if it is a member of either set :

∀x,s1,s2 Member(x,Union(s1,s2)) Member(x,s1) v Member(x,s2)


Asking questions and Asking questions and getting answersgetting answers

Tell(KB, (∀m,c Mother(c)=m Female(m) &

Parent(m,c)))……Tell(KB, (Female(Maxi) & Parent(Maxi,Spot) & Parent(Spot,Boots)))

Ask(KB,Grandparent(Maxi,Boots)Ask(KB, ∃x Child(x, Spot))Ask(KB, ∃x Mother(x)=Maxi)Substitution, unification, {x/Boots}

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