Logics of Rational Agency Lecture 3

Transcription

1 Logics of Rational Agency Lecture 3 Eric Pacuit Tilburg Institute for Logic and Philosophy of Science Tilburg Univeristy ai.stanford.edu/~epacuit July 29, 2009 Eric Pacuit: LORI, Lecture 3 1

2 Plan for the Course Introduction, Motivation and Background Lecture 2: Lecture 3: Lecture 4: Lecture 5: Basic Ingredients for a Logic of Rational Agency Logics of Rational Agency and Social Interaction, Part I Logics of Rational Agency and Social Interaction, Part II Conclusions and Eric Pacuit: LORI, Lecture 3 2

3 Plan for the Course Logics of Rational Agency Eric Pacuit: LORI, Lecture 3 3

4 Basic Ingredients informational attitudes (eg., knowledge, belief, certainty) group notions (eg., common knowledge and coalitional ability) time, actions and ability motivational attitudes (eg., preferences) normative attitudes (eg., obligations) Eric Pacuit: LORI, Lecture 3 4

5 Basic Ingredients informational attitudes (eg., knowledge, belief, certainty) group notions (eg., common knowledge and coalitional ability) time, actions and ability motivational attitudes (eg., preferences) normative attitudes (eg., obligations) Eric Pacuit: LORI, Lecture 3 4

6 Temporal Logics t 0 t 1 t 2 t 3 Eric Pacuit: LORI, Lecture 3 5

7 Computational vs. Behavioral Structures q 0 x = 1 q 1 x = 2 Eric Pacuit: LORI, Lecture 3 6

8 Computational vs. Behavioral Structures q 0 x = 1 q 1 x = 2 Eric Pacuit: LORI, Lecture 3 6

9 Computational vs. Behavioral Structures q 0 x = 1 q 0 q 0 q 0 q 0 q 1 q 1 x = 2 q 0 q 0 q 0 q 0 q 0 q 1 q 0 q 1 q 0 q 0 q 1 q 1. Eric Pacuit: LORI, Lecture 3 7

10 Temporal Logics Eric Pacuit: LORI, Lecture 3 8

11 Temporal Logics Linear Time Temporal Logic: Reasoning about computation paths: ϕ: ϕ is true some time in the future. A. Pnuelli. A Temporal Logic of Programs. in Proc. 18th IEEE Symposium on Foundations of Computer Science (1977). Eric Pacuit: LORI, Lecture 3 8

12 Temporal Logics Linear Time Temporal Logic: Reasoning about computation paths: ϕ: ϕ is true some time in the future. A. Pnuelli. A Temporal Logic of Programs. in Proc. 18th IEEE Symposium on Foundations of Computer Science (1977). Branching Time Temporal Logic: Allows quantification over paths: ϕ: there is a path in which ϕ is eventually true. E. M. Clarke and E. A. Emerson. Design and Synthesis of Synchronization Skeletons using Branching-time Temproal-logic Specifications. In Proceedings Workshop on Logic of Programs, LNCS (1981). Eric Pacuit: LORI, Lecture 3 8

13 Temporal Logics q 0 x = 1 q 0 P x=2 q 0 q 0 q 0 q 1 q 1 x = 2 q 0 q 0 q 0 q 0 q 0 q 1 q 0 q 1 q 0 q 0 q 1 q 1. Eric Pacuit: LORI, Lecture 3 9

14 Temporal Logics q 0 x = 1 q 0 P x=2 q 0 q 0 q 0 q 1 q 1 x = 2 q 0 q 0 q 0 q 0 q 0 q 1 q 0 q 1 q 0 q 0 q 1 q 1. Eric Pacuit: LORI, Lecture 3 9

15 Temporal Logics q 0 x = 1 q 0 P x=2 q 0 q 0 q 0 q 1 q 1 x = 2 q 0 q 0 q 0 q 0 q 0 q 1 q 0 q 1 q 0 q 0 q 1 q 1. Eric Pacuit: LORI, Lecture 3 9

16 Many Agents The previous model assumes there is one agent that controls the transition system. Eric Pacuit: LORI, Lecture 3 10

17 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Eric Pacuit: LORI, Lecture 3 10

18 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Example: Suppose that there are two agents: a server (s) and a client (c). The client asks to set the value of x and the server can either grant or deny the request. Assume the agents make simultaneous moves. Eric Pacuit: LORI, Lecture 3 10

19 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Example: Suppose that there are two agents: a server (s) and a client (c). The client asks to set the value of x and the server can either grant or deny the request. Assume the agents make simultaneous moves. set1 set2 deny grant Eric Pacuit: LORI, Lecture 3 10

20 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Example: Suppose that there are two agents: a server (s) and a client (c). The client asks to set the value of x and the server can either grant or deny the request. Assume the agents make simultaneous moves. deny grant set1 q 0 q 0, q 1 q 0 set2 q 0 q 1, q 1 q 1 Eric Pacuit: LORI, Lecture 3 11

21 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Example: Suppose that there are two agents: a server (s) and a client (c). The client asks to set the value of x and the server can either grant or deny the request. Assume the agents make simultaneous moves. deny grant set1 q q q 0 q 0, q 1 q 0 set2 q q q 0 q 1, q 1 q 1 Eric Pacuit: LORI, Lecture 3 12

22 From Temporal Logic to Strategy Logic Coalitional Logic: Reasoning about (local) group power. [C]ϕ: coalition C has a joint action to bring about ϕ. M. Pauly. A Modal Logic for Coalition Powers in Games. Journal of Logic and Computation 12 (2002). Eric Pacuit: LORI, Lecture 3 13

23 From Temporal Logic to Strategy Logic Coalitional Logic: Reasoning about (local) group power. [C]ϕ: coalition C has a joint action to bring about ϕ. M. Pauly. A Modal Logic for Coalition Powers in Games. Journal of Logic and Computation 12 (2002). Alternating-time Temporal Logic: Reasoning about (local and global) group power: A ϕ: The coalition A has a joint action to ensure that ϕ will remain true. R. Alur, T. Henzinger and O. Kupferman. Alternating-time Temproal Logic. Jouranl of the ACM (2002). Eric Pacuit: LORI, Lecture 3 13

24 Multi-agent Transition Systems, deny (P x=1 [s]p x=1 ) (P x=2 [s]p x=2 ) q 0 x = 1 set2, grant set1, grant q 1 x = 2, deny Eric Pacuit: LORI, Lecture 3 14

25 Multi-agent Transition Systems, deny P x=1 [s]p x=2 q 0 x = 1 set2, grant set1, grant q 1 x = 2, deny Eric Pacuit: LORI, Lecture 3 14

26 Multi-agent Transition Systems, deny P x=1 [s, c]p x=2 q 0 x = 1 set2, grant set1, grant q 1 x = 2, deny Eric Pacuit: LORI, Lecture 3 14

27 Preference (Modal) Logics x, y objects x y: x is at least as good as y Eric Pacuit: LORI, Lecture 3 15

28 Preference (Modal) Logics x, y objects x y: x is at least as good as y 1. x y and y x (x y) 2. x y and y x (y x) 3. x y and y x (x y) 4. x y and y x (x y) Eric Pacuit: LORI, Lecture 3 15

29 Preference (Modal) Logics x, y objects x y: x is at least as good as y 1. x y and y x (x y) 2. x y and y x (y x) 3. x y and y x (x y) 4. x y and y x (x y) Properties: transitivity, connectedness, etc. Eric Pacuit: LORI, Lecture 3 15

30 Preference (Modal) Logics Modal betterness model M = W,, V Eric Pacuit: LORI, Lecture 3 16

31 Preference (Modal) Logics Modal betterness model M = W,, V Preference Modalities ϕ: there is a world at least as good (as the current world) satisfying ϕ M, w = ϕ iff there is a v w such that M, v = ϕ Eric Pacuit: LORI, Lecture 3 16

32 Preference (Modal) Logics Modal betterness model M = W,, V Preference Modalities ϕ: there is a world at least as good (as the current world) satisfying ϕ M, w = ϕ iff there is a v w such that M, v = ϕ M, w = ϕ iff there is v w and w v such that M, v = ϕ Eric Pacuit: LORI, Lecture 3 16

33 Preference (Modal) Logics 1. ϕ ϕ 2. ϕ ϕ 3. ϕ ψ ( ψ (ψ ϕ)) 4. ϕ ϕ Theorem The above logic (with Necessitation and Modus Ponens) is sound and complete with respect to the class of preference models. J. van Benthem, O. Roy and P. Girard. Everything else being equal: A modal logic approach to ceteris paribus preferences. JPL, Eric Pacuit: LORI, Lecture 3 17

34 Preference Modalities ϕ ψ: the state of affairs ϕ is at least as good as ψ (ceteris paribus) G. von Wright. The logic of preference. Edinburgh University Press (1963). Eric Pacuit: LORI, Lecture 3 18

35 From worlds to sets and back Lifting X Y if y Y x X : x y Eric Pacuit: LORI, Lecture 3 19

36 From worlds to sets and back Lifting X Y if y Y x X : x y A(ϕ ψ) Eric Pacuit: LORI, Lecture 3 19

37 From worlds to sets and back Lifting X Y if y Y x X : x y A(ϕ ψ) X Y if y Y x X : x y A(ϕ [ ] ψ) Eric Pacuit: LORI, Lecture 3 19

38 From worlds to sets and back Lifting X Y if y Y x X : x y A(ϕ ψ) X Y if y Y x X : x y A(ϕ [ ] ψ) Deriving P 1 >> P 2 >> P 3 >> >> P n x > y iff x and y differ in at least one P i and the first P i where this happens is one with P i x and P i y F. Liu and D. De Jongh. Optimality, belief and preference Eric Pacuit: LORI, Lecture 3 19

39 Introduction, Motivation and Background Basic Ingredients for a Logic of Rational Agency Lecture 3: Lecture 4: Lecture 5: Logics of Rational Agency and Social Interaction, Part I Logics of Rational Agency and Social Interaction, Part II Conclusions and Eric Pacuit: LORI, Lecture 3 20

40 Once a semantics and language are fixed, then standard questions can be asked: eg. develop a proof theory, completeness, decidability, model checking. Eric Pacuit: LORI, Lecture 3 21

41 How should we compare the different logical systems? Embedding one logic in another: Eric Pacuit: LORI, Lecture 3 22

42 How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Eric Pacuit: LORI, Lecture 3 22

43 How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Compare different models for a fixed language: Eric Pacuit: LORI, Lecture 3 22

44 How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Compare different models for a fixed language: Alternating-Time Temporal Logics: Three different semantics for the ATL language. V. Goranko and W. Jamroga. Comparing Semantics of Logics for Multiagent Systems. KRA, Eric Pacuit: LORI, Lecture 3 22

45 How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Compare different models for a fixed language: Alternating-Time Temporal Logics: Three different semantics for the ATL language. V. Goranko and W. Jamroga. Comparing Semantics of Logics for Multiagent Systems. KRA, Comparing different frameworks: Eric Pacuit: LORI, Lecture 3 22

46 How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Compare different models for a fixed language: Alternating-Time Temporal Logics: Three different semantics for the ATL language. V. Goranko and W. Jamroga. Comparing Semantics of Logics for Multiagent Systems. KRA, Comparing different frameworks: eg. PDL vs. Temporal Logic, PDL vs. STIT, STIT vs. ATL, etc. Eric Pacuit: LORI, Lecture 3 22

47 How should we merge the different logical systems? Eric Pacuit: LORI, Lecture 3 23

48 How should we merge the different logical systems? Combining logics is hard! D. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev. Many Dimensional Modal Logics: Theory and Applications Eric Pacuit: LORI, Lecture 3 23

49 How should we merge the different logical systems? Combining logics is hard! D. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev. Many Dimensional Modal Logics: Theory and Applications Theorem ϕ ϕ is provable in combinations of Epistemic Logics and PDL with certain cross axioms ( [a]ϕ [a] ϕ) (and full substitution). R. Schmidt and D. Tishkovsky. On combinations of propositional dynamic logic and doxastic modal logics. JOLLI, Eric Pacuit: LORI, Lecture 3 23

50 Merging logics of rational agency Reasoning about information change (knowledge and time/actions) Knowledge, beliefs and certainty Epistemizing logics of action and ability: knowing how to achieve ϕ vs. knowing that you can achieve ϕ Entangling knowledge and preferences Planning/intentions (BDI) Eric Pacuit: LORI, Lecture 3 24

51 Example Ann would like Bob to attend her talk; however, she only wants Bob to attend if he is interested in the subject of her talk, not because he is just being polite. There is a very simple procedure to solve Ann s problem: have a (trusted) friend tell Bob the time and subject of her talk. Is this procedure correct? Eric Pacuit: LORI, Lecture 3 25

52 Example Ann would like Bob to attend her talk; however, she only wants Bob to attend if he is interested in the subject of her talk, not because he is just being polite. There is a very simple procedure to solve Ann s problem: have a (trusted) friend tell Bob the time and subject of her talk. Is this procedure correct? Yes, if 1. Ann knows about the talk. 2. Bob knows about the talk. 3. Ann knows that Bob knows about the talk. 4. Bob does not know that Ann knows that he knows about the talk. 5. And nothing else. Eric Pacuit: LORI, Lecture 3 25

53 Example A, B A, B P s B P t P means The talk is at 2PM. Eric Pacuit: LORI, Lecture 3 26

54 Example A, B A, B P s B P t P means The talk is at 2PM. M, s = K A P K B P Eric Pacuit: LORI, Lecture 3 26

55 Example A, B A, B P s B P t P means The talk is at 2PM. M, s = K A P K B P Eric Pacuit: LORI, Lecture 3 26

56 Example A A, B A, B B w 1 P P w 2 P s B P t w 3 P B A P w 4 Eric Pacuit: LORI, Lecture 3 26

57 Epistemic Temporal Logic R. Parikh and R. Ramanujam. A Knowledge Based Semantics of Messages. Journal of Logic, Language and Information, 12: , 1985, FHMV. Reasoning about Knowledge. MIT Press, Eric Pacuit: LORI, Lecture 3 27

58 The Playground t = 0 e 2 e 4 e 2 e 1 e 2 t = 1 e 1 e 5 e 2 e 6 e 4 e 2 e 7 t = 2 e 1 e 3 e 7 e 3 e 1 e 3 t = 3 Eric Pacuit: LORI, Lecture 3 28

59 The Playground t = 0 e 2 e 4 e 2 e 1 e 2 t = 1 e 1 e 5 e 2 e 6 e 4 e 2 e 7 t = 2 e 1 e 3 e 7 e 3 e 1 e 3 t = 3 Eric Pacuit: LORI, Lecture 3 28

60 The Playground t = 0 j e 2 e 4 e 2 e 1 e 2 t = 1 t = 2 e 1 e 5 e 2 e 6 i e 4 e 2 j e 7 e 1 e 3 i e 7 e 3 i e 1 e 3 t = 3 Eric Pacuit: LORI, Lecture 3 28

61 The Playground : Notation Eric Pacuit: LORI, Lecture 3 29

62 The Playground : Notation Let Σ be any set. The elements of Σ are called events. Eric Pacuit: LORI, Lecture 3 29

63 The Playground : Notation Let Σ be any set. The elements of Σ are called events. Given any set X, X is the set of finite strings over X and X ω the set of infinite strings over X. Elements of Σ Σ ω will be called histories. Eric Pacuit: LORI, Lecture 3 29

64 The Playground : Notation Let Σ be any set. The elements of Σ are called events. Given any set X, X is the set of finite strings over X and X ω the set of infinite strings over X. Elements of Σ Σ ω will be called histories. Given H Σ Σ ω, len(h) is the length of H. Eric Pacuit: LORI, Lecture 3 29

65 The Playground : Notation Let Σ be any set. The elements of Σ are called events. Given any set X, X is the set of finite strings over X and X ω the set of infinite strings over X. Elements of Σ Σ ω will be called histories. Given H Σ Σ ω, len(h) is the length of H. Given H, H Σ Σ ω, we write H H if H is a finite prefix of H. Eric Pacuit: LORI, Lecture 3 29

66 The Playground : Notation Let Σ be any set. The elements of Σ are called events. Given any set X, X is the set of finite strings over X and X ω the set of infinite strings over X. Elements of Σ Σ ω will be called histories. Given H Σ Σ ω, len(h) is the length of H. Given H, H Σ Σ ω, we write H H if H is a finite prefix of H. FinPre(H) = {H H H such that H H } is the set of finite prefixes of the elements of H. Eric Pacuit: LORI, Lecture 3 29

67 The Playground : Notation Let Σ be any set. The elements of Σ are called events. Given any set X, X is the set of finite strings over X and X ω the set of infinite strings over X. Elements of Σ Σ ω will be called histories. Given H Σ Σ ω, len(h) is the length of H. Given H, H Σ Σ ω, we write H H if H is a finite prefix of H. FinPre(H) = {H H H such that H H } is the set of finite prefixes of the elements of H. ɛ is the empty string and FinPre ɛ (H) = FinPre(H) {ɛ}. Eric Pacuit: LORI, Lecture 3 29

68 History-based Frames Definition Let Σ be any set of events. A set H Σ Σ ω is called a protocol provided FinPre ɛ (H) H. A rooted protocol is any set H Σ Σ ω where FinPre(H) H. Eric Pacuit: LORI, Lecture 3 30

69 History-based Frames Definition Let Σ be any set of events. A set H Σ Σ ω is called a protocol provided FinPre ɛ (H) H. A rooted protocol is any set H Σ Σ ω where FinPre(H) H. Definition An ETL frame is a tuple Σ, H, { i } i A where Σ is a (finite or infinite) set of events, H is a protocol, and for each i A, i is an equivalence relation on the set of finite strings in H. Eric Pacuit: LORI, Lecture 3 30

70 History-based Frames Definition Let Σ be any set of events. A set H Σ Σ ω is called a protocol provided FinPre ɛ (H) H. A rooted protocol is any set H Σ Σ ω where FinPre(H) H. Definition An ETL frame is a tuple Σ, H, { i } i A where Σ is a (finite or infinite) set of events, H is a protocol, and for each i A, i is an equivalence relation on the set of finite strings in H. Some assumptions: 1. If Σ is assumed to be finite, then we say that F is finitely branching. 2. If H is a rooted protocol, F is a tree frame. Eric Pacuit: LORI, Lecture 3 30

71 Formal Languages Pϕ (ϕ is true sometime in the past), F ϕ (ϕ is true sometime in the future), Y ϕ (ϕ is true at the previous moment), Nϕ (ϕ is true at the next moment), N e ϕ (ϕ is true after event e) K i ϕ (agent i knows ϕ) and C B ϕ (the group B A commonly knows ϕ). Eric Pacuit: LORI, Lecture 3 31

72 History-based Models An ETL model is a structure H, { i } i A, V where H, { i } i A is an ETL frame and V : At 2 finite(h) is a valuation function. Formulas are interpreted at pairs H, t: H, t = ϕ Eric Pacuit: LORI, Lecture 3 32

73 Truth in a Model H, t = Pϕ iff there exists t t such that H, t = ϕ H, t = F ϕ iff there exists t t such that H, t = ϕ H, t = Nϕ iff H, t + 1 = ϕ H, t = Y ϕ iff t > 1 and H, t 1 = ϕ H, t = K i ϕ iff for each H H and m 0 if H t i H m then H, m = ϕ H, t = Cϕ iff for each H H and m 0 if H t H m then H, m = ϕ. where is the reflexive transitive closure of the union of the i. Eric Pacuit: LORI, Lecture 3 33

74 Truth in a Model H, t = Pϕ iff there exists t t such that H, t = ϕ H, t = F ϕ iff there exists t t such that H, t = ϕ H, t = Nϕ iff H, t + 1 = ϕ H, t = Y ϕ iff t > 1 and H, t 1 = ϕ H, t = K i ϕ iff for each H H and m 0 if H t i H m then H, m = ϕ H, t = Cϕ iff for each H H and m 0 if H t H m then H, m = ϕ. where is the reflexive transitive closure of the union of the i. Eric Pacuit: LORI, Lecture 3 33

75 t = 0 j e 2 e 4 e 2 e 1 e 2 t = 1 t = 2 e 1 e 5 e 2 e 6 i e 4 e 2 e 7 e 1 e 3 i e 7 e 3 i e 1 e 3 t = 3 Eric Pacuit: LORI, Lecture 3 34

76 Returning to the Example Ann would like Bob to attend her talk; however, she only wants Bob to attend if he is interested in the subject of her talk, not because he is just being polite. Eric Pacuit: LORI, Lecture 3 35

77 Returning to the Example Ann would like Bob to attend her talk; however, she only wants Bob to attend if he is interested in the subject of her talk, not because he is just being polite. There is a very simple procedure to solve Ann s problem: have a (trusted) friend tell Bob the time and subject of her talk. Eric Pacuit: LORI, Lecture 3 35

78 Returning to the Example Ann would like Bob to attend her talk; however, she only wants Bob to attend if he is interested in the subject of her talk, not because he is just being polite. There is a very simple procedure to solve Ann s problem: have a (trusted) friend tell Bob the time and subject of her talk. Is this procedure correct? Eric Pacuit: LORI, Lecture 3 35

79 t = 0 m 2PM m 3PM t = 1 m A C t m A C t t = 2 t = 3 m C B m C B t m C B t m C B t t

80 t = 0 m 2PM m 3PM t = 1 m A C t m A C t t = 2 t = 3 m C B m C B t m C B t m C B t t H, 3 = ϕ

81 t = 0 m 2PM m 3PM t = 1 m A C t m A C t t = 2 t = 3 m C B m C B t m C B t m C B t t Bob s uncertainty: H, 3 = K B P 2PM

82 t = 0 m 2PM m 3PM t = 1 m A C t t t = 2 t = 3 m C B t t t m C B Bob s uncertainty + Protocol information : H, 3 = K B P 2PM

83 t = 0 m 2PM m 3PM t = 1 m A C t t t = 2 t = 3 m C B t t t m C B Bob s uncertainty + Protocol information : H, 3 = K B K A K B P 2PM

84 t = 0 m 2PM m 3PM t = 1 m A C t t t = 2 t = 3 m C B t t t m C B Bob s uncertainty + Protocol information : H, 3 = K B K A K B P 2PM

85 t = 0 m 2PM m 3PM t = 1 m A C t t t = 2 t = 3 m C B t t t m C B Bob s uncertainty + Protocol information : H, 3 = K B K A K B P 2PM

86 t = 0 m 2PM m 3PM t = 1 m A C t t t = 2 t = 3 m C B t t t m C B Bob s uncertainty + Protocol information : H, 3 = K B K A K B P 2PM

87 Parameters of the Logical Framework Eric Pacuit: LORI, Lecture 3 37

88 Parameters of the Logical Framework 1. Expressivity of the formal language. Does the language include a common knowledge operator? A future operator? Both? Eric Pacuit: LORI, Lecture 3 37

89 Parameters of the Logical Framework 1. Expressivity of the formal language. Does the language include a common knowledge operator? A future operator? Both? 2. Structural conditions on the underlying event structure. Do we restrict to protocol frames (finitely branching trees)? Finitely branching forests? Or, arbitrary ETL frames? Eric Pacuit: LORI, Lecture 3 37

90 Parameters of the Logical Framework 1. Expressivity of the formal language. Does the language include a common knowledge operator? A future operator? Both? 2. Structural conditions on the underlying event structure. Do we restrict to protocol frames (finitely branching trees)? Finitely branching forests? Or, arbitrary ETL frames? 3. Conditions on the reasoning abilities of the agents. Do the agents satisfy perfect recall? No miracles? Do they agents know what time it is? Eric Pacuit: LORI, Lecture 3 37

91 Agent Oriented Properties: No Miracles: For all finite histories H, H H and events e Σ such that He H and H e H, if H i H then He i H e. Perfect Recall: For all finite histories H, H H and events e Σ such that He H and H e H, if He i H e then H i H. Synchronous: For all finite histories H, H H, if H i H then len(h) = len(h ). Eric Pacuit: LORI, Lecture 3 38

92 Perfect Recall t = 0 e 2 e 4 e 2 e 1 e 2 t = 1 e 1 e 5 e 2 e 3 e 4 e 2 e 7 t = 2 e 1 e 3 e 7 e 6 i e 1 e 3 t = 3 Eric Pacuit: LORI, Lecture 3 39

93 Perfect Recall t = 0 e 2 e 4 e 2 e 1 e 2 t = 1 e 1 e 5 e 2 e 3 i e 4 e 2 e 7 t = 2 e 1 e 3 e 7 e 6 i e 1 e 3 t = 3 Eric Pacuit: LORI, Lecture 3 40

94 Perfect Recall t = 0 i e 2 e 4 i e 2 e 1 e 2 t = 1 e 1 e 5 e 2 e 3 i e 4 e 2 e 7 t = 2 e 1 e 3 e 7 e 6 i e 1 e 3 t = 3 Eric Pacuit: LORI, Lecture 3 41

95 No Miracles t = 0 e 2 e 4 e 2 e 1 e 2 t = 1 e 1 e 5 e 2 e 3 i e 4 e 2 e 7 t = 2 e 1 e 3 e 7 e 6 e 1 e 3 t = 3 Eric Pacuit: LORI, Lecture 3 42

96 No Miracles t = 0 e 2 e 4 e 2 e 1 e 2 t = 1 e 1 e 5 e 2 e 3 i e 4 e 2 e 7 t = 2 e 1 e 3 e 7 e 6 i e 1 e 3 t = 3 Eric Pacuit: LORI, Lecture 3 43

97 No Miracles t = 0 e 2 e 4 e 2 e 1 e 2 t = 1 e 1 e 5 e 2 e 1 i e 4 e 2 e 7 t = 2 i e 1 e 3 e 7 e 6 i e 1 e 5 t = 3 Eric Pacuit: LORI, Lecture 3 44

98 Ideal Agents Assume there are two agents Theorem The logic of ideal agents with respect to a language with common knowledge and future is highly undecidable (for example, by assuming perfect recall). J. Halpern and M. Vardi.. The Complexity of Reasoning abut Knowledge and Time. J. Computer and Systems Sciences, 38, J. van Benthem and EP. The Tree of Knowledge in Action. Proceedings of AiML, Eric Pacuit: LORI, Lecture 3 45

99 End of lecture 3. Eric Pacuit: LORI, Lecture 3 46