chapter 4

Chapter 4

Methods of Inference

知識推論法

4. 知識推論法 S.S. Tseng & G.J. Hwang 2

4.1 Deductive and Induction 4.1 Deductive and Induction （演繹與歸（演繹與歸納）納）

• Deduction( 演繹 ): Logical reasoning in which conclus

ions must follow from their premises.

• Induction( 歸納 ): Inference from the specific case to t

he general.

• Intuition( 直觀 ): No proven theory.

• Heuristics( 啟發 ): Rules of thumb ( 觀測法 ) based up

on experience.

• Generate and test: Trial and error.


• Abduction( 反推 ): Reasoning back from a true c

onclusion to the premises that may have caused the co

nclusion.

• Autoepitemic( 自覺、本能 ): Self-knowledge

• Nonmonotonic( 應變知識 ): previous knowledge

may be incorrect when new evidence is obtained

• Analogy( 類推 ): based on the similarities to anot

her situation


Syllogism （三段論）• Syllogism （三段論） is simple, well-understood

branch of logic that can be completely proven.– Premise( 前提 ): Anyone who can program is intellige

nt

– Premise( 前提 ): John can program

– Conclusion( 結論 ): Therefore, John is intelligent.

• In general, a syllogism is any valid deductive argument having two premises and a conclusion.


Categorical Syllogism( 定言三段論 )

形態概要意思AEIO

所有 S為 P沒有 S為 P某些 S為 P某些 S不為 P

完全肯定完全否定部分肯定部分否定

定言命題的型態


•三段論的標準形態（ Standard form ）大前提：所有 M 為 P

小前提：所有 S 為 M

結論：所有 S 為 P

- P- P 代表結論的「謂詞」代表結論的「謂詞」 (Predicate)(Predicate) ，又稱為「大詞」（，又稱為「大詞」（ Major termMajor term ））- S- S 代表結論的「主詞」代表結論的「主詞」 (Subject)(Subject) ，又稱作「小詞」（，又稱作「小詞」（ Minor termMinor term ）。）。- - 含有大詞的前提稱為「大前提」（含有大詞的前提稱為「大前提」（ Major premiseMajor premise ）；）；- - 含有小詞的前提稱為「小前提」（含有小詞的前提稱為「小前提」（ Minor premiseMinor premise ）。）。- M- M 稱為「中詞」（稱為「中詞」（ Middle termMiddle term ））


Mood( 模式 )• Patterns of Categorical Statement

• 4 種 AAA 模式

Figure-1 Figure-2 Figure-3 Figure-4

大前提 MP PM MP PM

小前提 SM SM MS MS

形態 AAA-1 AAA-2 AAA-3 AAA-4

大前提 MP PM MP PM

小前提 SM SM MS MS


• ex: AAA-1– 所有 M 為 P

所有 S 為 M ∴ 所有 S 為 P

• We use decision procedure( 決策程序 ) to prove the validity of syllogistic argument

• The decision procedure for syllogisms can be done using Venn Diagrams( 維思圖 )


• ex: Decision procedure for Syllogism AEE-1

所有 M 為 P

沒有 S 為 M

∴ 沒有 S 為 P

S P

M

S P

M M

S P

(a)維思圖 (b)大前提後 (c)小前提後


General Rule under “some” quantifiers

1. If a class is empty, it is shaded.2. Universal statement, A and E, are always drawn

before particular ones.3. If a class has at least one member, mark it with a *.4. If a statement does not specify in which of two

adjacent classed an object exists, place a * on the line between the classes.

5. If an area has been shaded, no * can be put in it.


ex: Decision procedure for Syllogism IAI-1

某些 P 為 M

所有 M 為 S

∴ 某些 S 為 PS P

M(b) P M一些是

S P

M

＊

(a) M S所有是


4.2 State and problem spaces( 狀態與問題空間 )

• Tree （樹狀結構） : nodes, edges• Directed or undirected

• Digraph （雙向圖） : a graph with directed edges

• Lattice （晶格） : a directed acyclic graph


• A useful method of describing the behavior of an object is to define a graph called the state space. [state( 狀態 ) and action( 行動 )]– Initial state

– Operator

– State space

– Path

– Goal test

– Path cost


Finite State Machine（有限狀態機器）

• Determining valid strings WHILE,WRITE, and BEGIN

開始

H I LE

ITR

W

G IEB

not N

not Inot G

not E

錯誤

成功N


Finding solution in problem space

• State space （狀態空間） can be thought as a problem space （問題空間） .

• Finding the solution to a problem in a problem space involves finding a valid path from start to success( answer).

• The state space for the Monkey and Bananas Problem

• Traveling salesman problem （旅行推銷員問題）

• Graph algorithms, AND-OR Trees, etc.


Ex: Monkey and Bananas Problem

• 假設：– 房子裡有一懸掛的香蕉– 房子裡只有一張躺椅跟一把梯子– 猴子無法直接拿到香蕉

• 指示：– 跳下躺椅– 移動梯子– 把梯子移到香蕉下的位置– 爬上梯子– 摘下香蕉

• 初始狀態：– 猴子在躺椅上

猴子在躺椅上

猴子在地板上

躺椅位於香蕉下

躺椅不位於香蕉下

猴子不在梯子上

跳下躺椅觀察到躺椅在香蕉下

觀察到躺椅不在香蕉下

觀察到猴子不在梯子上

移動躺椅

猴子在梯子上

梯子不位於香蕉下觀察到梯子

不在香蕉下

觀察到猴子不在梯子上

梯子在香蕉下

觀察到梯子在香蕉下

移動猴子

移動梯子

猴子在梯子上摘下香蕉

猴子成功得到香蕉

爬上梯子


Ex: Travel Salesman Problem（旅行推銷員問題） A

BD

C

(a) 旅行推銷員的問題描述

C

B

A

D

A

C

A BDB

B C DBA

C

C

DA B

A BDC

B C

D

CA

(b) ( )搜尋路徑粗線是解答路徑


Ill-structured problem( 非結構化問題 )

• Ill-structured problems （非結構化問題） have uncertainties associated with it.– Goal not explicit– Problem space unbounded– Problem space not discrete– Intermediate states difficult to achieve– State operators unknown– Time constraint


Ex: 旅遊代理人特徵客戶的反應

目標不明顯我在想到底要去哪裡

問題空間範圍未被介定我不確定要去哪裡

問題狀態不是離散的我只是想去旅遊，目的地並不重要

中間的狀態不易實行我沒有足夠的錢去

狀態的可用運算元未知我不知道怎麼可以籌到錢

時間限制我必須儘快出發


4.3 Rules of Inference( 規則式推論 )

• Syllogism （三段論） addresses only a small portion of the possible logic statements.

• Propositional logicp q

p______

q

Inference is called direct reasoning ( 直接推論 ), modus ponens ( 離斷率 ), law of detachment ( 分離律 ) , and assuming the antecedent ( 假設前提 ).


p 　　 q 　　 p→q 　　 (p→q)p 　　 (p→q) p→q

T 　　 T 　　　 T 　　　　 T 　　　　　　 T

T 　　 F 　　　 F 　　　　 F 　　　　　　 T

F 　　 T 　　　 T 　　　　 F 　　　　　　 T

F 　　 F 　　　 T 　　　　 F 　　　　　　 T

Truth table for Truth table for Modus PonenseModus Ponense （（離斷率））


Law of Inference 　　　　　　　　　　　 Schemata

1.Law of Detachment 　　　　　　　　　　

2.Law of the Contrapositive

3. Law of Modus Tollens

4.Chain Rules(Law of the Syllogism)

5.Law of Disjunctive Inference

6.Law of the Double Negation

p→qp ∴q

p→q∴~q→~p

p→q~q∴~p

pq~q∴p

P→qq→r∴p→r

~(~p)∴p

pq~p∴q


7.De Morgan’s Law

8.Law of Simplification

9.Law of Conjunction

10.Law of Disjunctive Addition

11. Law of Conjunctive Argument

Table 3.8 Some Rules of Inference for Propositional Logic

~(pq)∴~p ~q

~(pq)∴~p ~q

~(pq)∴~q

pq∴p

pq∴pq

p∴pq

~(pq)p∴~q

~(pq)q∴~p


Resolution in propositional LogicResolution in propositional Logic（命題邏輯（命題邏輯分解）

F: Rules or facts known to be TRUE

S: A conclusion to be Proved

• Convert all the propositions of F to clause form.

2. Negate S and convert the result to clause form. Add it

　 to the set of clauses obtained in step 1.

3. Repeat until either a contradiction is found or no

progress can be made:

(1) Select two clauses.

　 Call these the Parent clauses.


(2) Resolve them together.

　 The resulting clause, called the resolvent, will be the

　 disjunction of all of the literals of both of the parent

　 clauses with the following exception ： If there are any

　 pairs of literals L and ~L. Such that one of the parent

　 clauses contains L and the other contaions ~L, then

　 eliminate both L and ~L from the resolvent.

(3) If the resolvent is the empty clause, then a

　 contradiction has been found. If it is not, then add it

　 to the set of clauses available to the procedure.


Given Axioms Converted to Clause Form

p p

(p q) r 　～ p ～ q r

(s t) q ～ s q 　

～ t q

t 　 t

1.

2.

3.

4.

5.

p = 下雨 q = 騎車s = 路線熟悉t = 路途遠r = 穿雨衣


～ｐ～ｑｒ

～ t ｑ

Resolution in Propositional Logic

～ p ～ｑ

～ r

p

～ q

～ tt


Resolution with quantifiersResolution with quantifiers

Example （ from Nilsson ）：

Whoever can read (R) is literate (L).

Dolphins (D) aren’t literate (~L).

Some dolphins (D) are intelligent (I).

To prove ： Some who are intelligent (I) can’t read (~R).


Translating ：

To prove: 　　 x [ I ( x ) & ~ R ( x ) ]

x [ R ( x ) → L ( x ) ]

x [ D ( x ) → ~L ( x ) ]

x [ D ( x ) ＆ I ( x ) ]


(1) － (4) ：

　　 x [~ R ( x ) OR L ( x ) ] & 　 y [ ~ D ( y )

　　 OR ~ L ( y ) ] & D ( A ) & I ( A ) &

　　　 z [~ I ( z ) OR R ( z ) ]

(5) － (9) ：

　　 C1=~R(x) 　 OR 　 L(x)

　　 C2=~D(y) 　 OR 　 ~L(y)

　　 C3=D(A)

　　 C4=I(A)

　　 C5= ~I(z) 　 OR 　 R(z)


• The second order logic can have quantifiers that range over function and predicate symbols

• If P is any predicate of one document– Then– x =y (for every P [P(x) P(y) ]


4.4 Inference Chain ( 推斷鏈 ) 　　　　　　　　 D3

A2 D2

A1 B C D1 E Solution

inference + inference +… + inference

Chain

Infer from initialfacts to solutions

Infer from initialfacts to solutions

Assume that some solution is true, and try to provethe assumption by findingthe required facts

Assume that some solution is true, and try to provethe assumption by findingthe required facts

forwardchaining

backwardchaining

Initial facts


　 Rule1: elephant(x) mammal(x)

　 Rule2: mammal(x) animal(x)

Fact ： John is an elephant.

　 elephant (John) is true 　　　 X=John (Unification)

　 elephant(x) mammal(x)

　　　　　 X’=X=John

　 mammal(x’) animal(x’)

•　 Forward Chaining （前向鏈結） :

Mammal(John) is trueMammal(John) is true

animal(John) is trueanimal(John) is true• Unification （變數替代）

The process of finding substitutions for variables to make arguments match.


Forward Chaining （前向推論）Rule1 ： A1 and B1 C1

Rule2 ： A2 and C1 D2

Rule3 ： A3 and B2 D3

Rule4 ： C1 and D3 G

Facts ： A1 is true, A2 is true , A3 is true, B1 is true, B2 is true

　　 {A1, A2, A3, B1, B2} match {r1, r3}

fire r1 {A1, A2, A3, B1, B2, C1} match {r1, r2, r3}

fire r2 {A1, A2, A3, B1, B2, C1, D2} match{r1, r2, r3}

fire r3 {A1, A2, A3, B1, B2, C1 D2, D3} match{r1, r2, r3, r4}

fire r4 {A1, A2, A3, B1, B2, C1 D2, D3, G }GOAL


Backward Chaining ( 反向推論 )

2. Assume G is true

Verify C1 and D3

Verify A3 and B2

Verify A1 and B1

1. Assume G’ is true

Verify C1 and D4

Verify A1 and B1

rule1 : A1 and B1 C1

rule2 : A2 and C1 D2

rule3 : A3 and B2 D3

rule4 : C1 and D3 Grule5 : C1 and D4 G’

OK OK

D4 is unknown, ask user.

If D4 is FALSE, give up.

D4 is unknown, ask user.

If D4 is FALSE, give up.

R5

R1

R4

R3

OKOK

OK OK

facts : A1, A2, B1, B2, A3


AA22 AA33

CC11

AA11 BB11

BB22

DD22

GG

DD33

?

GOALGOAL


• Good application of forward chainingGood application of forward chaining （前向鏈結）（前向鏈結）

Goal

Broad and Not Deep　　　 ortoo many possible goals

Facts


• Good application of backward chainingGood application of backward chaining （後向鏈（後向鏈結）結）

Narrow and Deep

Facts

GOALS


• Forward Chaining （前向鏈結）　 Planning

　 Monitoring

　 Control

　 Data-driven

　 Explanation not facilitated

• Backward chaining （後向鏈結）　 Diagnosis

　 Goal-driven

　 Explanation facilitated


Analogy• Try to relate old situations as guides to new

ones• Consider tic-tac-toe with values as a magic

square (15 game)» 6 1 8» 7 5 3» 2 9 4

• 18 game from set {2,3,4,5,6,7,8,9,10}• 21 game from set {3,4,5,6,7,8,9,10,11}


Nonmonotonic reasoning

• In nonmonotonic system, the theorems do not necessarily increase as the number of axioms increases.

• As a very simple example, suppose there is a fact that asserts the time. As soon as time changes by a second, the old fact is no longer valid.


4.5 Reasoning Under Uncertainty( 不確定性推論 )

• Uncertainty can be considered as the lack of adequate infor

mation to make a decision.

• Classical probability, Bayescian probability, Dempster-S

hafer theory, and Zadeh’s fuzzy theory.

• In the MYCIN and PROSPECTOR systems conclusion are

arrived at even when all the evidence needed to absolutely

prove the conclusion is not known.


Many different types of error can contribute to uncertainty

ExampleTurn the value off Turn value-1Turn value-1 offValue is stuck Value is not stuckTurn value-1 to 5Turn value-1 to 5.4Turn value-1 to 5.4 or 6 or 0Value-1 setting is 5.4 or 5.5 or 5.1Value-1 setting is 7.5Value-1 is not stuck because it’s never been stuck beforeOutput is normal and so value-1 is in good condition

ErrorAmbiguousIncompleteIncorrectFalse positive ( 接受錯誤值 )False negative ( 拒絕正確值 )ImpreciseInaccurateUnreliableRandom errorSystematic errorInvalid induction

Invalid deduction

ReasonWhat value?Which way?Correct is onValue is not stuckValue is stuckCorrect is 5.4Correct is 9.2Equipment errorStatistical fluctuation ( 波動 )Mis-calibration ( 刻度 )

Value is stuck

Value is stuck in open position


• A hypothesis is an assumption to be tested.• Type 1 error (false positive) means acceptance of a hypothes

is when it is not true.• Type 2 error (false negative) means rejection of a hypothesis

when it is true.

• Error of measurement– Precision

• The millimeter( 公釐 ) ruler is more precise than centimeter ruler.

– accuracy


Error & Induction

The process of induction is the opposite of deduction

　 The fire alarm goes off ( 響起 )

∴ There is a fire.

An even stronger argument is

　 The fire alarm goes off & I smell smoke

∴ There is a fire.

Although this is a strong argument, it is not proof that there is a fire.

My clothes are burning


Deductive errors

　 p→q

　 q

　∴ p

If John is a father, than John is a man

John is a man

∴ John is a father


Baye’s Theorem （貝氏定理）• Conditional probability （條件機率） , P(A | B) , states the

probability of event B occurred. Crash= Brand X(0.6)+ Not

X(0.1)=0.7

• P( X|C) =

　　　　 P( C | X) P(X) = (0.75)(0.8) = 6

P(C) 0.7 7

• Suppose you have a drive and don’t know its brand, what is t

he probability that if it crashes, it is Brand X? non-Brand X?


　　　　　　　 P(E∩Hi) 　　　 P(E | H i) P(Hi)

　　　　　　　 P(E ∩Hj) 　　 P(E | Hj) P(Hj)

　　　　　　 P(E | Hi)P(Hi)

　　　　　　　　 P(E)

• Bayes’ Theorem （貝氏定理） is commonly used for decision tree analysis of business and the social science.

• Used in Prospector expert system to decide favorite sites of mineral exploration

P(Hi | E) = =

=

j j


Hypothetical Reasoning and Backward Induction.

P(+)=P(+∩O)+P(+∩O’)=0.48+0.04=0.52

P(-)=P(-∩O)+P(-∩O’)=0.12+0.36=0.48

No Oil

P(O’)=0.4

Oil

P(O)=0.6

-Test

P(- | O’)

=0.9

+Test

P(+ | O’)

=0.1

-Test

P(- | O)

=0.2

+Test

P(+ | O)

=0.8

P(-∩O’)

=0.36

P(+∩O’)

=0.04

P(-∩O)

=0.12

P(+∩O)

=0.48Joint -P(E∩H)

=P(E | Hi) P(Hi)

Probabilities

Prior

Subjective Opinion

of Site - P (Hi)

Conditional

Seismic Test Result


-Test

P(-)=0.48

No Oil

P(O’|-)

= (9)(4)

0.48

= 3/4

P(-∩O)

=0.36

P(+∩O)

=0.04

P(-∩O)

=0.12

P(+∩O)

=0.48

Joint -P(E∩H)

=P(Hi | E) P(E)

Probabilities

Unconditional

P (E)

Posterior

of Site - P(Hi | E)

P ( E| Hi) P (Hi)

　　　 P(E)

+Test

P(+)=0.52

Oil

P(O|-)

= (2)(6)

0.48

= 1/4

No Oil

P(O’|+)

= (1)(4)

0.52

= 1/13

Oil

P(O|+)

= (8)(6)

0.52

= 12/13

=


• Oil release , if successful $1250000• Drilling expense -$200000• Seismic survey -$50000• Expected payoff (success)

• 846153=1000000 *12/13 – 1000000*1/13

• Fail• -500000= 1000000*1/4- 1000000*3/4

• Expected payoff (total)• 416000= 846153*0.52 – 50000 * 0.48


Temporal reasoning and Markov chain

• Temporal reasoning: reasoning about events that depend on time

• Temporal logic

• The system’s progression through a sequence of status is called a Stochastic process if it is probabilistic.


– P11 P12

– P21 P22

– Where Pmn is the probability of a transition from state m to state n.

S = { P1,P2, …, Pn} where P1+P2+…+Pn= 1

S2 = S1 T

S2 = [0.8,0.2] = [0.2,0.8]0.1 0.9

0.6 0.4


• Assume 10 percent of all people who now use Brand X drive will buy another Brand X when needed. 60 percent of people who don’t use Brand X will buy Brand X when they need a new drive. Over a period of time, how many people will use Brand X?S3 = [0.5,0.5], S4 = [0.35,0.65],S5 = [0.425,0.575], S6 = [0.3875,0.6125]S7 = [0.40625,0.59375], S8 = [0.396875,0.602125]

Steady state matrix


The odds of belief

• “The patient is covered with red spots”

• Proposition A: “The patient has measles”

• P(A|B) :(degree of belief that A is true, given B)is not necessarily a probability if the events and propositions can not be repeated or has a math basis.


• The odds on A against B given some event C are odds =P(A|C)/ P(B|C)

• If B = A’ – odds =P(A|C)/ P(A’|C) =P(A|C)/ (1-P(A|C) )

• Likelihood of P = 0.95– Odds = .95/(1-.95) = 19 to 1


Sufficiency and necessity

Bayes’ Theorem is

　　　　　　　 P(H|E) =

Negation P(H’|E) =

P(E | H)P(H)

P(E)

P(E | H’)P(H’) 　　　 P(E)


P(H | E) 　 P(E | H) P(H)

P(H’ | E) 　 P(E | H’) P(H’)

Defining the prior odds on H as

　　　　 P(H)

　　　　 P(H’)

　　　　　 P(H | E)

　　　　　 P(H’ | E)

O(H) =

＝

O(H | E) =


Likelihood ratio

　　　 P(E | H)

　　　 P(E | H’)

O(H | E) = LS O(H)

odds-likelihood form of Bayes’ Theorem.

The factor LS is also called likelihood of sufficiency

because if LS =∞ then the evidence E is logically s

ufficient for concluding that H is true.

LS=


LS 　　　　　　　　　 Effect on Hypothesis

0 　　　　　　　　　 H is false when E is true or

　　　　　　　　　　 E’ is necessary for concluding H

Small(0<LS<<1) 　　　 E is unfavorable for concluding H

1 　　　　　　　　　 E has no effect on belief of H

Large(1<<LS) 　　　　 E is favorable for concluding H

　　　　　　　　　 E is logically sufficient for H or

　　　　　　　　　　 Observing E means H must be true


LN 　　　　　　 Effect on Hypothesis

0 　　　　　　　 H is false when E is true or E is necessary for H

small(0<LN<<1) 　 Absence of E is unfavorable for concluding H

1 　　　　　　　 Absence of E has no effect on H

large(1<<LN) 　　 Absence of E is favorable of H

　　　　　　　 Absence of E is logically sufficient for H


Uncertainty in inference chains

• Uncertainty may be present in rules, evidence used by the rules, or both.


Expert Inconsistency

If LS > 1 then P(E | H’) < P(E | H)

1 – P(E | H’) > 1 – P(E | H)

　　　 1-P(E | H)

　　　 1-P(E | H’)

Case 1 ： LS>1 and LN <1

Case 2 ： LS<1 and LN >1

Case 3 ： LS= LN = 1

LN= < 1


Exercise• 考慮以下的事實與規則，試以前向鏈結和後向鏈結描述其推論過程。

事實 : A1, A2, A3, A4, B1, B2

規則 : R1: A1 and A3 --> C2

R2: A1 and B1 --> C1

R3: A2 and C2 --> D2

R4: A3 and B2 --> D3

R5: C1 and D2 --> G1

R6: B1 and B2 --> D4

R7: A1 and A2 and A3 --> D2

R8: C1 and D3 --> G2

R9: C2 and A4 --> G3

目標 : G1, G2 and G3

chapter 4

Documents