1. ייצוג מידע ודרכי החלטה

2. Logics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language Semantics define the “meaning” of sentences; – i.e., define truth of a sentence in a world E.g., the language of arithmetic – x+2 ≥ y is a sentence; x2+y > is not a sentence 2 Logic in general

3. Propositional logic is the simplest logic – illustrates basic ideas The proposition symbols P 1, P 2, etc. are sentences – If S is a sentence,  S is a sentence (negation) – If S 1 and S 2 are sentences, S 1  S 2 is a sentence (conjunction) – If S 1 and S 2 are sentences, S 1  S 2 is a sentence (disjunction) – If S 1 and S 2 are sentences, S 1  S 2 is a sentence (implication) Implication also is Not S 1  S 2 – If S 1 and S 2 are sentences, S 1  S 2 is a sentence (biconditional) 3 Propositional logic: Syntax

4. Rules for evaluating truth with respect to a model m:  S is true iff S is false S 1  S 2 is true iff S 1 is true and S 2 is true S 1  S 2 is true iff S 1 is true or S 2 is true S 1  S 2 is true iffS 1 is false or S 2 is true i.e., is false iffS 1 is true and S 2 is false S 1  S 2 is true iffS 1  S 2 is true and S 2  S 1 is true Simple recursive process evaluates an arbitrary sentence, e.g.,  P 1,2  (P 2,2  P 3,1 ) = true  (false  true) = true  true = true 4 Propositional logic: Semantics

5. 5 Truth tables for connectives

6. More examples Show that A  B ≡ (A → B) Λ (B → A) t → w) Λ ~ w] → ~ t Show that: [(t → w) Λ ~ w] → ~ t p → q) Λ q → r)] → p → r) Show that: [(p → q) Λ (q → r) ] → (p → r)

7. Law of Modus Tollens Given: t → w t → w  w ~ w Prove:  t  ~ t t → w) Λ ~ w] → ~ t or [(t → w) Λ ~ w] → ~ t Set up a truth table to prove!

8. t → w) Λ ~ w] → ~ t] Prove [(t → w) Λ ~ w] → ~ t] tw~t~w t → w t → w) Λ ~ w (t → w) Λ ~ w t → w) Λ ~ w ] → ~ t [(t → w) Λ ~ w ] → ~ t

9. t → w) Λ ~ w] → ~ t Prove [(t → w) Λ ~ w] → ~ t tw~t~w t → w t → w) Λ ~ w (t → w) Λ ~ w t → w) Λ ~ w → ~ t (t → w) Λ ~ w → ~ t TTFFTFT TFFTFFT FTTFTFT FFTTTTT t → w) Λ ~ w] → ~ t is a Tautology therefore a valid argument! [(t → w) Λ ~ w] → ~ t is a Tautology therefore a valid argument!

10. p → q) Λ q → r)] → p → r) [(p → q) Λ (q → r) ] → (p → r) Chain Rule (Law of Syllogism) pqr p → q q → r p → q) Λ q → r) (p → q) Λ (q → r) p → r See above

11. p → q) Λ q → r)] → p → r) [(p → q) Λ (q → r) ] → (p → r) Chain Rule ( Law of Syllogism) pqr p → q q → r p → q) Λ q → r) (p → q) Λ (q → r) p → r See above TTT TTF TFT TFF FTT FTF FFT FFF

12. p → q) Λ q → r)] → p → r) [(p → q) Λ (q → r) ] → (p → r) Chain Rule (Law of Syllogism) pqr p → q q → r p → q) Λ q → r) (p → q) Λ (q → r) p → r See above TTTTTTTT TTFTFFFT TFTFTFTT TFFFTFFT FTTTTTTT FTFTFFTT FFTTTTTT FFFTTTTT

13. Chain Rule Example p : You study q  r q : You pass r : You get a surprise p  q P 1: P 2: If you study, then you will pass. If you pass, then you will get a surprise.

14. Two sentences are logically equivalent iff true in same models: α ≡ β iff α ╞ β and β ╞ α 14 Logical equivalence

15. A sentence is satisfiable if it is true in some model e.g., A  B, C A sentence is unsatisfiable if it is true in no models e.g., A  A Disjunction normal form (DNF) : Only “Or” between Logic statements – ( A 1  B 1 )  (A 2  B 2 )  (A 3  B 3 ) Conjunction normal form (CNF) : Only “And” between Logic statements – ( A 1  B 1 )  (A 2  B 2 )  (A 3  B 3 ) 15 Satisfiability

16. Consider random 3-CNF sentences (randomly selected 3 distinct symbols, each negated with 50% probability), e.g., (  D   B  C)  (B   A   C)  (  C   B  E)  (E   D  B)  (B  E   C) m = number of clauses n = number of symbols (overall, in the KB) – Hard problems seem to cluster near m/n = 4.3 (critical point) – Lower ratio is less constrained, higher ratio is more constrained 16 Hard satisfiability problems

17. 17 Hard satisfiability problems Graph showing probability that a random 3-CNF sentence with n=50 symbols is satisfiable, as a function of the clause/symbol ratio m/n

18. 18 Other Logics…

19. ConstantsKingJohn, 2, HU,... PredicatesBrother, >,... FunctionsSqrt, LeftLegOf,... Variablesx, y, a, b,... Connectives , , , ,  Equality= Quantifiers ,  19 First Order Logic

20.  Everyone at HU is smart:  x At(x, HU)  Smart(x)  x P is true in a model m iff P is true with x being each possible object in the model Roughly speaking, equivalent to the conjunction of instantiations of P At(KingJohn, HU)  Smart(KingJohn)  At(Richard, HU)  Smart(Richard)  At(HU, HU)  Smart(HU) ... 20 Universal quantification

21.  Someone at TAU is smart:  x At(x, TAU)  Smart(x)  x P is true in a model m iff P is true with x being some possible object in the model Roughly speaking, equivalent to the disjunction of instantiations of P At(KingJohn, TAU)  Smart(KingJohn)  At(Richard, TAU)  Smart(Richard)  At(TAU, TAU)  Smart(TAU) ... 21 Existential quantification

22. Brothers are siblings  x  y Brother(x, y)  Sibling(x, y) “Sibling” is symmetric  x  y Sibling(x, y)  Sibling(y, x) One’s mother is one’s female parent  x  y Mother(x, y)  (Female(x)  Parent(x, y)) A first cousin is a child of a parent’s sibling  x  y FirstCousin(x, y)   p  ps Parent(p, x)  Sibling(ps, p)  Parent(ps, y) 22 Fun with sentences

23. The set domain:  s Set(s)  (s = {} )  (  x,s 2 Set(s 2 )  s = {x|s 2 })  x,s {x|s} = {}  x,s x  s  s = {x|s}  x,s x  s  [  y,s 2 } (s = {y|s 2 }  (x = y  x  s 2 ))]  s 1,s 2 s 1  s 2  (  x x  s 1  x  s 2 )  s 1,s 2 (s 1 = s 2 )  (s 1  s 2  s 2  s 1 )  x,s 1,s 2 x  (s 1  s 2 )  (x  s 1  x  s 2 )  x,s 1,s 2 x  (s 1  s 2 )  (x  s 1  x  s 2 ) 23 Using FOL

24. Examples http://people.umass.edu/partee/NZ_2006/M ore%20Answers%20for%20Practice%20in%20 Logic%20and%20HW%201.pdf http://people.umass.edu/partee/NZ_2006/M ore%20Answers%20for%20Practice%20in%20 Logic%20and%20HW%201.pdf

25. The set domain:  s Set(s)  (s = {} )  (  x,s 2 Set(s 2 )  s = {x|s 2 })  x,s {x|s} = {}  x,s x  s  s = {x|s}  x,s x  s  [  y,s 2 } (s = {y|s 2 }  (x = y  x  s 2 ))]  s 1,s 2 s 1  s 2  (  x x  s 1  x  s 2 )  s 1,s 2 (s 1 = s 2 )  (s 1  s 2  s 2  s 1 )  x,s 1,s 2 x  (s 1  s 2 )  (x  s 1  x  s 2 )  x,s 1,s 2 x  (s 1  s 2 )  (x  s 1  x  s 2 ) 25 Using FOL

26. דרכים להחליט בפועל Fuzzy Logic MDP Game Theory

27. Copyright © 2002, 2004, Andrew W. Moore Applications of MDPs This extends the search algorithms of your first lectures to the case of probabilistic next states. Many important problems are MDPs…. … Robot path planning … Travel route planning … Elevator scheduling … Bank customer retention … Autonomous aircraft navigation … Manufacturing processes … Network switching & routing

28. The “Standard” Approach – MDP MDP model is a 4-tuple where: S is the set of all possible environment states. N is a group of agents. A i is the set of all possible joint actions applicable in the environment by agent i. Pr models dynamics – S x A x S  [0, 1] with Pr(s i, a, s j ) denotes the probability that action a executed in state s i, will transition to state s j. R is the reward function for agents’ possible actions.

29. Copyright © 2002, 2004, Andrew W. Moore Markov Decision Processes An MDP has… A set of states {s 1 ··· s N } A set of actions {a 1 ··· a M } A set of rewards {r 1 ··· r N } (one for each state) A transition probability function At each step: 0. Call current state S i 1. Receive reward r i 2. Choose action  {a 1 ··· a M } 3. If you choose action a k you’ll move to state S j with probability 4. All future rewards are discounted by 

30. John Nash, the person portrayed in “A Beautiful Mind”

31. Game theory: Payoff matrix A payoff matrix shows the payout to each player, given the decision of each player Action CAction D Action A10, 28, 3 Action B12, 410, 1 Person 1 Person 2

32. How do we find Nash equilibrium (NE)? Step 1: Pretend you are one of the players Step 2: Assume that your “opponent” picks a particular action Step 3: Determine your best strategy (strategies), given your opponent’s action – Underline any best choice in the payoff matrix Step 4: Repeat Steps 2 & 3 for any other opponent strategies Step 5: Repeat Steps 1 through 4 for the other player Step 6: Any entry with all numbers underlined is NE

33. Decision tree in a sequential game: Person 1 chooses first A B C Person 1 chooses yes Person 1 chooses no Person 2 chooses yes Person 2 chooses no 20, 20 5, 10 10, 5 10, 10

34. Slide 34 2 player zero-sum finite NONdeterministic games of perfect information The search tree now includes states where neither player makes a choice, but instead a random decision is made according to a known set of outcome probabilities. Game theory value of a state is the expected final value if both players are optimal. Let’s compute a matrix form of this! ( )-a ( )-chance ( )-b -20 +4 ( )-b ( )-chance +3 ( )-a +10 ( )-a -5 ( )-a p=0.8p=0.2 p=0.5

35. Slide 35 Minimax with Matrix Forms A can decide from this matrix which strategy is “best”. For each strategy, A considers the worst-case counter strategy by B. A chooses the row with the maximum minimum value. For A, the value of the game is this value. In this example A chooses A-II, and says game has value 3. When B decides which strategy is best, B searches for which column has the minimum maximum value. In this example, B chooses B-II, and says game has value 3. B-IB-IIB-III A-I73 A-II734 A-III222 A-IV222 Fundamental game theory result (proved by von Neumann): In a 2-player, zero-sum game of perfect information, Maximin==Minimax. And there always exists an optimal pure strategy for each player.

36. Fuzzy Logic  What is Fuzzy Logic?  Problem-solving control system methodology  Linguistic or "fuzzy" variables  Example: IF (process is too hot) AND (process is heating rapidly) THEN (cool the process quickly)

37. Approach  The Rule Matrix  Error (Columns)  Error-dot (Rows)  Input conditions (Error and Error-dot)  Output Response Conclusion (Intersection of Row and Column) -ve Error Zero Error +ve Error -ve Error- dot Zero Error- dot No change +ve Error- dot