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Predicate Logic for ArtifiCIAL iNTELLIGENCE
- 1. 1 Chapter 7 Propositional andPredicate Logic
- 2. 2 What is ArtificialIntelligence? z A more difficult question is: What is intelligence? z This question has puzzled philosophers, biologists and psychologists for centuries. z Artificial Intelligence is easier to define, although there is no standard, accepted definition. 評判 搜尋解答 邏輯推理 記憶 控制 學習 創作 weak sub? strong Fuzzy,NN,GA In my opinion: 星艦戰將與 蝸蟲的學習
- 3. 3 Chapter 7 Contents zWhat is Logic? z Logical Operators z Translating between English and Logic z Truth Tables z Complex Truth Tables z Tautology z Equivalence z Propositional Logic z Deduction z Predicate Calculus z Quantifiers ∀ and ∃ z Properties of logical systems z Abduction and inductive reasoning z (Modal logic)
- 4. 4 What is Logic? zReasoning about the validity of arguments. z An argument is valid if its conclusions follow logically from its premises – even if the argument doesn’t actually reflect the real world: „ All lemons are blue „ Mary is a lemon „ Therefore, Mary is blue.
- 5. 5 Logical Operators z AndΛ z Or V z Not z Implies → (if… then…) z Iff ↔ (if and only if)
- 6. 6 Translating between Englishand Logic z Facts and rules need to be translated into logical notation. z For example: „ It is Raining and it is Thursday: „ R Λ T „ R means “It is Raining”, T means “it is Thursday”.
- 7. 7 Translating between Englishand Logic z More complex sentences need predicates. E.g.: „ It is raining in New York: „ R(N) „ Could also be written N(R), or even just R. z It is important to select the correct level of detail for the concepts you want to reason about.
- 8. 8 Truth Tables z Tablesthat show truth values for all possible inputs to a logical operator. z For example: z A truth table shows the semantics of a logical operator.
- 9. 9 Complex Truth Tables zWe can produce truth tables for complex logical expressions, which show the overall value of the expression for all possible combinations of variables:
- 10. 10 Tautology (恆真式) z Theexpression A v ¬A is a tautology. z This means it is always true, regardless of the value of A. z A is a tautology: this is written ╞ „ A tautology is true under any interpretation. „ An expression which is false under any interpretation is contradictory. z 一切數學證明的動作來自: ╞ (P ^(PÆQ) Æ Q) 或者強調那是動作的結果: P ^(PÆQ)├ Q
- 11. 11 Equivalence z Two expressionsare equivalent if they always have the same logical value under any interpretation: „A Λ B ≡ B Λ A z Equivalences can be proven by examining truth tables.
- 12. 12 Some Useful Equivalences zA v A ≡ A z A Λ A ≡ A z A Λ (B Λ C) ≡ (A Λ B) Λ C z A v (B v C) ≡ (A v B) v C z A Λ (B v C) ≡ (A Λ B) v (A Λ C) z A Λ (A v B) ≡ A z A v (A Λ B) ≡ A z A Λ true ≡ A A Λ false ≡ false z A v true ≡ true A v false ≡ A
- 13. 13 Propositional Logic (命題邏輯) zPropositional logic is a logical system. z It deals with propositions. „ Inference (what results from assumptions?) „ Reasoning (is it true or not?) z Propositional Calculus is the language we use to reason about propositional logic. z A sentence in propositional logic is called a well-formed formula (wff). „ Wff: English Æ logic sentence
- 14. 14 Propositional Logic z Thefollowing are wff’s: z P, Q, R… z true, false z (A) z ¬A z A Λ B z A v B z A → B z A ↔ B 定義 PÆQ ≡ ¬ P V Q (真值表比對) e.g. Tall ^ Strong Æ Ball_Player ≡ ¬ (Tall ^ Strong) V Ball_Player ≡ ¬ Tall V ¬ Strong V Ball_Player
- 15. 15 Chapter 7 Contents zWhat is Logic? z Logical Operators z Translating between English and Logic z Truth Tables z Complex Truth Tables z Tautology z Equivalence z Propositional Logic z Deduction z Predicate Calculus z Quantifiers ∀ and ∃ z Properties of logical systems z Abduction and inductive reasoning z (Modal logic) Recall:
- 16. 16 Deduction z The processof deriving a conclusion from a set of assumptions. z Use a set of rules, such as: A A → B B (modus ponens… 拉丁文:推論法) z If we deduce a conclusion C from a set of assumptions, we write: z {A1, A2, …, An} ├ C 7.11(p.191 ~ p.195)
- 17.
- 18. 18 Predicate Calculus (述語推算) zPredicate Calculus extends the syntax of propositional calculus with predicates and quantifiers: „ P(X) – P is a predicate. z First Order Predicate Calculus (FOPC) allows predicates to apply to objects or terms, but not functions or predicates.
- 19. 19 Quantifiers ∀ and∃ z ∀ - For all: „ ∀xP(x) is read “For all x’es, P (x) is true”. z ∃ - There Exists: „ ∃x P(x) is read “there exists an x such that P(x) is true”. z Relationship between the quantifiers: „ ∃x P(x) ≡ ¬(∀x)¬P(x) „ “If There exists an x for which P holds, then it is not true that for all x P does not hold”. ∃x Like(x, War) ≡ ¬(∀x) ¬Like(x, War)
- 20. 20 Deduction over FOPC--- Search z Dog(X) ^ Meets(X,Y)^Dislikes(X,Y) Æ Barks_at(X,Y) z Close_to(Z, DormG) Æ Meets(Snoopy, Z) z Man(W) Æ Dislikes(Snoopy, W) z Man(John), Dog(Snoopy), Close_to(John,DormG) z Dog(X) ^ Meets(X,Y)^Dislikes(X,Y) Æ Barks_at(X,Y) z Close_to(Z, DormG) Æ Meets(Snoopy, Z) z Dislikes(Snoopy, John) z Dog(Snoopy), Close_to(John,DormG) {John/W} Barks_at(Snoopy,John) ??
- 21. 21 Deduction over FOPC--- Goal Tree Barks_at(Snoopy,John)? Dog(Snoopy) Meets(Snoopy,John) Dislikes(Snoopy,John) Close_to(John,DormG) Man(John) Other_Resons Yes Yes Yes {Snoopy/X} {John/Y} {John/Y} {John/Z} {John/W} Recall:
- 22. 22 Properties of LogicalSystems z Soundness(可靠): Is every theorem valid? z Completeness(週延): Is every tautology a theorem? z Decidability(可推導): Does an algorithm exist that will determine if a wff is valid? z Monotonicity(不受破壞): Can a valid logical proof be made invalid by adding additional premises or assumptions?
- 23. 23 Abduction and InductiveReasoning z Abduction: B A → B A z Not logically valid, BUT can still be useful. z In fact, it models the way humans reason all the time: „ Every non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a penguin. z Not valid reasoning, but likely to work in many situations. 過度演繹
- 24. 24 Modal logic z Modallogic is a higher order logic. z Allows us to reason about certainties, and possible worlds. z If a statement A is contingent then we say that A is possibly true, which is written: ◊A z If A is non-contingent, then it is necessarily true, which is written: A Skip: cf. “fuzzy logic” … to appear later