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Chapter 4

Chapter 4. Methods of Inference 知識推論法. 4.1 Deductive and Induction （演繹與歸納）. Deduction( 演繹 ): Logical reasoning in which conclusions must follow from their premises. Induction( 歸納 ): Inference from the specific case to the general. Intuition( 直觀 ): No proven theory.

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Chapter 4

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  1. Chapter 4 Methods of Inference 知識推論法

  2. 4.1 Deductive and Induction （演繹與歸納） • Deduction(演繹): Logical reasoning in which conclusions must follow from their premises. • Induction(歸納): Inference from the specific case to the general. • Intuition(直觀): No proven theory. • Heuristics(啟發): Rules of thumb (觀測法) based upon experience. • Generate and test: Trial and error. S.S. Tseng & G.J. Hwang

  3. Abduction(反推): Reasoning back from a true conclusion to the premises that may have caused the conclusion. • 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 another situation S.S. Tseng & G.J. Hwang

  4. 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 intelligent • 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. S.S. Tseng & G.J. Hwang

  5. Categorical Syllogism(定言三段論) 定言命題的型態 S.S. Tseng & G.J. Hwang

  6. 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代表結論的「謂詞」(Predicate)，又稱為「大詞」（Major term） - S代表結論的「主詞」(Subject)，又稱作「小詞」（Minor term）。 - 含有大詞的前提稱為「大前提」（Major premise）； - 含有小詞的前提稱為「小前提」（Minor premise）。 - M稱為「中詞」（Middle term） S.S. Tseng & G.J. Hwang

  7. Mood(模式) • Patterns of Categorical Statement • 4種AAA模式 S.S. Tseng & G.J. Hwang

  8. 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) S.S. Tseng & G.J. Hwang

  9. ex: Decision procedure for Syllogism AEE-1 所有M為P 沒有S為M ∴沒有S為P S.S. Tseng & G.J. Hwang

  10. 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. S.S. Tseng & G.J. Hwang

  11. ex: Decision procedure for Syllogism IAI-1 某些P為M 所有M為S ∴某些S為P S.S. Tseng & G.J. Hwang

  12. 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 S.S. Tseng & G.J. Hwang

  13. 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 S.S. Tseng & G.J. Hwang

  14. Finite State Machine（有限狀態機器 ） • Determining valid strings WHILE,WRITE, and BEGIN S.S. Tseng & G.J. Hwang

  15. 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. S.S. Tseng & G.J. Hwang

  16. 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 S.S. Tseng & G.J. Hwang

  17. Ex: Travel Salesman Problem（旅行推銷員問題） S.S. Tseng & G.J. Hwang

  18. 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 S.S. Tseng & G.J. Hwang

  19. Ex:旅遊代理人 S.S. Tseng & G.J. Hwang

  20. 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 logic p  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). S.S. Tseng & G.J. Hwang

  21. Truth table for Modus Ponense（離斷率） pqp→q(p→q)p(p→q)  p→q TTTTT TFFFT FTTFT FFTFT S.S. Tseng & G.J. Hwang

  22. p→q p ∴q 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→q ∴~q→~p p→q ~q ∴~p P→q q→r ∴p→r pq ~p ∴q pq ~q ∴p ~(~p) ∴p S.S. Tseng & G.J. Hwang

  23. 7.De Morgan’s Law 8.Law of Simplification 9.Law of Conjunction 10.Law of Disjunctive Addition 11. Law of Conjunctive Argument ~(pq) ∴~p  ~q ~(pq) ∴~p ~q ~(pq) ∴~q pq ∴p p q ∴pq p ∴pq ~(pq) q ∴~p ~(pq) p ∴~q Table 3.8 Some Rules of Inference for Propositional Logic S.S. Tseng & G.J. Hwang

  24. Resolution in propositional Logic（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. S.S. Tseng & G.J. Hwang

  25. (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. S.S. Tseng & G.J. Hwang

  26. Given AxiomsConverted 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 S.S. Tseng & G.J. Hwang

  27. ～ｐ～ｑｒ ～r p ～p ～ｑ ～q ～t ｑ t ～t Resolution in Propositional Logic S.S. Tseng & G.J. Hwang

  28. Resolution 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). S.S. Tseng & G.J. Hwang

  29. Translating： x [ R ( x ) → L ( x ) ] x [ D ( x ) → ~L ( x ) ] x [ D ( x ) ＆ I ( x ) ] To prove: x [ I ( x ) & ~ R ( x ) ] S.S. Tseng & G.J. Hwang

  30. (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)ORL(x) C2=~D(y)OR~L(y) C3=D(A) C4=I(A) C5= ~I(z)ORR(z) S.S. Tseng & G.J. Hwang

  31. 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) ] S.S. Tseng & G.J. Hwang

  32. 4.4 Inference Chain (推斷鏈) D3 A2 D2 A1 B C D1 E Solution inference + inference +… + inference Chain Initial facts backward chaining forward chaining Infer from initial facts to solutions Assume that some solution is true, and try to prove the assumption by finding the required facts S.S. Tseng & G.J. Hwang

  33. Forward Chaining（前向鏈結）: 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’) • Unification（變數替代） The process of finding substitutions for variables to make arguments match. Mammal(John) is true animal(John) is true S.S. Tseng & G.J. Hwang

  34. 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 S.S. Tseng & G.J. Hwang

  35. rule1 : A1 and B1 C1 rule2 : A2 and C1 D2 rule3 : A3 and B2 D3 rule4 : C1 and D3 G rule5 : C1 and D4 G’ Backward Chaining (反向推論) facts : A1, A2, B1, B2, A3 1. Assume G’ is true Verify C1 and D4 Verify A1 and B1 2. Assume G is true Verify C1 and D3 Verify A3 and B2 Verify A1 and B1 R5 R4 R1 R3 OK OK OK OK D4 is unknown, ask user. If D4 is FALSE, give up. OK OK S.S. Tseng & G.J. Hwang

  36. A1 B1 A2 A3 B2 C1 D2 D3 ? G GOAL S.S. Tseng & G.J. Hwang

  37. Good application of forward chaining（前向鏈結） Goal Facts Broad and Not Deep or too many possible goals S.S. Tseng & G.J. Hwang

  38. Good application of backward chaining（後向鏈結） Narrow and Deep GOALS Facts S.S. Tseng & G.J. Hwang

  39. 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 S.S. Tseng & G.J. Hwang

  40. 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} S.S. Tseng & G.J. Hwang

  41. 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. S.S. Tseng & G.J. Hwang

  42. 4.5Reasoning Under Uncertainty(不確定性推論) • Uncertainty can be considered as the lack of adequate information to make a decision. • Classical probability, Bayescian probability, Dempster-Shafer 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. S.S. Tseng & G.J. Hwang

  43. Reason What value? Which way? Correct is on Value is not stuck Value is stuck Correct is 5.4 Correct is 9.2 Equipment error Statistical fluctuation (波動) Mis-calibration (刻度) Value is stuck Value is stuck in open position Example Turn the value off Turn value-1 Turn value-1 off Value is stuck Value is not stuck Turn value-1 to 5 Turn value-1 to 5.4 Turn value-1 to 5.4 or 6 or 0 Value-1 setting is 5.4 or 5.5 or 5.1 Value-1 setting is 7.5 Value-1 is not stuck because it’s never been stuck before Output is normal and so value-1 is in good condition Error Ambiguous Incomplete Incorrect False positive (接受錯誤值) False negative (拒絕正確值) Imprecise Inaccurate Unreliable Random error Systematic error Invalid induction Invalid deduction Many different types of error can contribute to uncertainty S.S. Tseng & G.J. Hwang

  44. A hypothesis is an assumption to be tested. • Type 1 error (false positive) means acceptance of a hypothesis 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 S.S. Tseng & G.J. Hwang

  45. 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 S.S. Tseng & G.J. Hwang

  46. Deductive errors p→q q 　∴ p If John is a father, than John is a man John is a man ∴ John is a father S.S. Tseng & G.J. Hwang

  47. 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 the probability that if it crashes, it is Brand X? non-Brand X? S.S. Tseng & G.J. Hwang

  48. Decision Tree for the Disk drive Crashes Act Prior P(Hi) Choose Brand X P(X)=0.8 Don’t Choose Brand X P(X’)=0.2 Conditional P(E | Hi ) No Crash P(C’ | X’)=0.5 Crash P(C | X’)=0.5 Crash P(C | X)=0.75 No Crash P(C’ | X)=0.25 P(C’∩X’)=0.1 Joint -P(E ∩ Hi ) =P(E | Hi ) P( Hi ) P(C ∩X’)=0.1 P(C’ ∩ X)=0.2 P(C ∩X)=0.6 0.1 0.1 0.2 0.6 Posterior P(H i | E) = P (E ∩ Hi) iP(E∩Hi) P(X’ | C’) = P(X’ | C) = P(X | C’) = P(X | C) = 0.1+0.2 0.1+0.6 0.2+0.1 0.6+0.1 = 1 / 3 = 1 / 7 = 2 / 3 = 6 / 7 S.S. Tseng & G.J. Hwang

  49. P(Hi | E) = = j j 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 = S.S. Tseng & G.J. Hwang

  50. Hypothetical Reasoning and Backward Induction. Probabilities Prior Subjective Opinion of Site - P (Hi) No Oil P(O’)=0.4 Oil P(O)=0.6 P(+)=P(+∩O)+P(+∩O’)=0.48+0.04=0.52 P(-)=P(-∩O)+P(-∩O’)=0.12+0.36=0.48 -Test P(- | O’) =0.9 +Test P(+ | O’) =0.1 -Test P(- | O) =0.2 +Test P(+ | O) =0.8 Conditional Seismic Test Result P(-∩O’) =0.36 P(+∩O’) =0.04 P(-∩O) =0.12 P(+∩O) =0.48 Joint -P(E∩H) =P(E | Hi) P(Hi) S.S. Tseng & G.J. Hwang

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