Agent that reason logically

지식표현

Knowledge Base

A set of representations of facts about the world Knowledge representation language  tell : what has been told to the knowledge base previously  ask : a question and the answer Inference : what follows from what the KB has been Telled Background knowledge : a knowledge base which may initially contained Sentence : individual representation of a fact Agent that reason logically 2

Knowledge base

The knowledge level :: saying what it knows to KB  “Golden Gates Bridge links San Francisco and Marin Country The 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) Agent that reason logically 3

Knowledge

  declarative/procedural love(john, mary).

can_fly(X) :- bird(X), not(can_fly(X)), !.

learning : general knowledge about the environment given a series of percepts Commonsense knowledge Agent that reason logically 4

Specifying the environment

Figure 6.2 A typical wumpus world Agent that reason logically 5

Domain specific knowledge

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) Agent that reason logically 6

Inference in Wumpus world(I)

1,4 1,3 1,2 2,4 2,3 2,2 3,4 3,3 3,2 1,1 OK A 2,1 OK 3,1 4,4 4,3 4,2 4,1 A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = Stench V = Visited W = Wumpus 1,4 1,3 2,4 2,3 3,4 3,3 1,2 2,2 P ?

1,1 OK V OK 2,1 A OK B 3,2 3,1 4,4 4,3 4,2 4,1

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

].

Agent that reason logically 7

Inference in Wumpus world(II)

1,4 2,4 3,4 1,3 W!

2,3 3,3 1,2 A 2,2 1,1 OK V OK OK 2,1 B V OK 3,2 3,1 4,4 4,3 4,2 4,1 A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = Stench V = Visited W = Wumpus 1,4 2,4 P ?

3,4 4,4 1,3 W !

1,2 S V 1,1 OK V OK 2,3 A S G B 2,2 V OK 2,1 B V OK 3,3 P ?

3,2 3,1 P !

4,3 4,2 4,1

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

].

Agent that reason logically 8

Representation, Reasoning, and Logic

Syntax : the possible configurations that constitute sentences Semantics : the facts in the world to which the sentences refer Agent that reason logically 9

The logical reasoning

Representation World Aentences Entails Facts Follows Sentence Fact

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.

Agent that reason logically 10

Inference I

Entailment :: true generation of new sentences that are necessarily true, given that the old sentences are Soundness, truth-preserving :: sentences  An inference procedure that generates only entailed modus ponens <-> abduction KB ├ i  ,  is derived from KB by I Proof :: a sound inference procedure Agent that reason logically 11

Inference II

Completeness :: Proof :: an inference procedure that can find a proof for any sentence that is entailed specifying the reasoning steps that are sound Valid :: if and only if all possible interpretations in all possible worlds Tautologies, 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 Agent that reason logically 12

Logics

Boolean logic   Symbols represent whole propositions (facts) Boolean connectives First-order logic   objects, predicates connectives, quantifiers Agent that reason logically 13

Wrong logical reasoning

FIRST 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!

Agent that reason logically 14

Ontological and epistemological commitments

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 Agent that reason logically 15

Commitments

Formal languages and their and ontological and epistemological commitments Language Propositional logic First-order logic Temporal logic Probability theory Fuzzy logic Ontological Commitment (What exists in the world) Epistemological Commitment (What an agent believes about facts) facts facts, objects, relations times facts degree of truth true/false/unknown true/false/unknown true/false/unknown degree of belief 0…1 degree of belief 0…1 Agent that reason logically 16

Propositional Logic

logical constant : true/false propositional symbols : P, Q parentheses : (P & Q) logical connectives : &(conjuction), v(disjunction), ->(implication), < >(equivalence), ~(negation) Agent that reason logically 17

Grammar

Sentence



AtomicSentence | ComplexSentence AtomicSentence



True

|

False

| P | Q | R | … ComplexSentence

 (

Sentence

) |

Sentence Connective Sentence |



Sentence Connective

  |  |  | 

Figure 6.8

A BNF (Backus-Naur Form) grammar of sentences in propositional logic.

Agent that reason logically 18

Semantics

Truth table showing validity of a complex sentence

P H P



H

(P  H)  ┐

H

((P  H)  ┐ H) 

P

False False True True False True False True False True True True False False True False True True True True Agent that reason logically 19

Validity and Inference

Truth tables for five logical connectives P Q ┐ P P  Q P  Q P  Q P  Q 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 Agent that reason logically 20

Models

Any world in which a sentence is true under a particular interpretation Entailment :: a sentence knowledge base KB if the models of the KB are all models of   is entailed by a The set of models of P & Q is the intersection of the models of P and the models of Q Agent that reason logically 21

       Inference Rules for propositional logic 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.)  =>   ,  1   2   1   2     …   n  i  1 ,  2 , …,  n  1   2   i …   n …   n Unit Resolution: (From a disjunction, if one of the disjuncts is false, then you can infer the other one is true.)    ,    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   =>    , =>   =>  Figure 6.13 Seven inference for propositional logic. The unit resolution rule is a special case of the resolution Agent that reason logically 22

Complexity of propositional inference

NP-complete Monotonicity  If KB 1 ╞  then (KB 1 ∪ KB 2 ) ╞ Horn clause logic   polynomial time complexity P 1 ∧ P 2 ∧ ….

∧ P n ⇒ Q  Agent that reason logically 23

Wumpus 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 Agent that reason logically 24

Finding the wumpus

Inference process     Modus ponens : ~ S 1,1 and R 1  ~ 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 Agent that reason logically 25

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 Agent that reason logically 26

Translating knowledge into action

A1,1 &

East

A

East

A :: & W2,1 -> ~ facing east

Forward

Propositional logic is not powerful enough to solve the wumpus problem easily Agent that reason logically 27

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

Agent that reason logically 28

First-order Logic

Limitation of propositional logic

A very limited ontology  to need to the representation power  first-order logic Agent that reason logically 30

First-order logic

A stronger set of ontological commitments A world in FOL consists of objects, properties, relations, functions Objects  Clinton people, houses, number, colors, Bill Relations  Properties  brother of, bigger than, owns, love red, round, bogus, prime Functions  father of, best friend, third inning of Agent that reason logically 31

Examples

“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 Agent that reason logically 32

First order logics

Objects 와 relations 시간 , 사건 , 카테고리 등은 고려하지 않음 영역에 따라 자유로운 표현이 가능함  ‘ king’ 은 사람의 국가를 연결하는 property 도 될 수 있고 , 사람과 relation 이 될 수도 있다 일차술어논리는 잘 알려져 있고 , 잘 연구된 수학적 모형임 Agent that reason logically 33

Syntax and Semantics

Sentence



AtomicSentence

|

Sentence Connective Sentence

|

Auantifier Variable,…Sentence

| 

Sentence

|

AtomicSentence



(Sentence) Predicate

(

Term

,…) |

Term=Term Term



Function

(

Term,…

) |

Constant

|

Variable Connective

 

Quantifier

 

Constant



Variable



A a

| |  | |

X

1

x

 |

s

|  | | |

John …

 |

Predicate



Function



Before Mother

| |

… HanColor LeftLegOf

|

Raining

|

…

|

…

Figure 7.1

The syntax of first-order logic (with equality) in BNF (Backus-Naur Form).

Agent that reason logically 34

예 Constant symbols :: A, B, John, Predicate symbols :: Round, Brother Function symbols :: Cosine, FatherOf Terms :: King John, Richard’s left leg Atomic sentences :: Brother(Richard,John), Married(FatherOf(Richard), MotherOf(John)) Complex sentences :: Older(John,30)=>~younger(John,30) Agent that reason logically 35

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) Agent that reason logically 36

Nested 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) Agent that reason logically 37

De Morgan’s Rule

∀ x ~P  ~ ∀ x P  ~ ∃ x P ~P&~Q  ∃ x ~P ~(P&Q)  ∀ x P  ~ ∃ x ~P P&Q  ~(P v Q) ~P v ~Q ~(~P v ~ Q) ∃ x P  ~ ∀ x ~P P v Q  ~(~P&~Q) Agent that reason logically 38

Equality

Identity relation Father(John) = Henry ≠ ∃ x,y Sister(Spot,x) & Sister(Spot,y) & ~(x=y) ∃ x,y Sister(Spot,x) & Sister(Spot,y) Agent that reason logically 39

Higher-order logic

∀ x,y (x=y)  ( ∀ p p(x)  p(y)) ∀ f,g (f=g)  ( ∀ x f(x)  g(x)) Agent that reason logically 40



-expression

 x,y x 2 – y 2  -expression can be applied to arguments to yield a logical term in the same way that a function can be (  x,y x 2 – y 2 )(25,24) = 25 2 -24 2 = 49  x,y Gender(x) ≠Gender(y) & Address(x) = Address(y) Agent that reason logically 41

∃

! (The uniqueness quantifier)

∃ !x King(x) ∃ x King(x) & ∀ y King(y) => x=y world 를 고려하여 보여주면 {a} w0  king={}, w1  => object 가 king={a}  1, 2, 3 개일 때 w1 만 model {a,b} w0  king={}, w1  king={a}, w2  {b}, w3  {a,b}  w1, w2 만 model Agent that reason logically 42

Representation of sentences 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) Agent that reason logically 43

Representation of sentences 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) Agent that reason logically 44

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)=EmptySet Adjoining an element already in the set has no effect ∀ x,s Member(x,s) into it  s=Adjoin(x,s) The only members of a set are the elements that were adjoined ∀ x,s Member(x,s)  ∃ y,s2 (s=Adjoin(y,s2) & (x=y v Member(x,s))) Agent that reason logically 45

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)) Agent that reason logically 46

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) Agent that reason logically 47

Asking questions and getting 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} Agent that reason logically 48