Logic and Artificial Intelligence Lecture 12

Transcription

1 Logic and Artificial Intelligence Lecture 12 Eric Pacuit Currently Visiting the Center for Formal Epistemology, CMU Center for Logic and Philosophy of Science Tilburg University ai.stanford.edu/ epacuit October 10, 2011 Logic and Artificial Intelligence 1/33

2 Dynamic Epistemic Logic: Literature A. altag and L. Moss. Logics for Epistemic Programs W. van der Hoek, H. van Ditmarsch and. Kooi. Dynamic Epistemic Logic Logic and Artificial Intelligence 2/33

3 Dynamic Epistemic Logic Logic and Artificial Intelligence 3/33

4 Dynamic Epistemic Logic M E The initial model (Ann and ob are ignorant about P 2PM ). Private announcement to Ann about the talk. Logic and Artificial Intelligence 3/33

5 Dynamic Epistemic Logic M E Initial epistemic model. Abstract description of an epistemic event. Logic and Artificial Intelligence 3/33

6 Abstract Description of the Event Ann looks at the card while ob is looking away. Logic and Artificial Intelligence 4/33

7 Abstract Description of the Event Ann looks at the card while ob is looking away. A A, e 1 P e 2 Ann knows which event took place. Logic and Artificial Intelligence 4/33

8 Product Update Logic and Artificial Intelligence 5/33

9 Product Update A, M A, A E A, P A, P P s t e 1 e 2 Logic and Artificial Intelligence 5/33

10 Product Update A, M A, A E A, P A, P P s t e 1 e 2 (s, e 1 ) P P (s, e 2 ) (t, e 1 ) P P (t, e 2 ) Logic and Artificial Intelligence 5/33

11 Product Update A, M A, A E A, P A, P P s t e 1 e 2 (s, e 1 ) P P (s, e 2 ) (t, e 1 ) P P (t, e 2 ) Logic and Artificial Intelligence 5/33

12 Product Update A, M A, A E A, P A, P P s t e 1 e 2 (s, e 1 ) P P (s, e 2 ) P (t, e 2 ) Logic and Artificial Intelligence 5/33

13 Product Update A, M A, A E A, P A, P P s t e 1 e 2 (s, e 1 ) P P (s, e 2 ) A, P (t, e 2 ) Logic and Artificial Intelligence 5/33

14 Product Update Details Let M = W, R, V be a Kripke model. An event model is a tuple A = A, S, Pre, where S A A and Pre : L (A). Logic and Artificial Intelligence 6/33

15 Product Update Details Let M = W, R, V be a Kripke model. An event model is a tuple A = A, S, Pre, where S A A and Pre : L (A). The update model M A = W, R, V where Logic and Artificial Intelligence 6/33

16 Product Update Details Let M = W, R, V be a Kripke model. An event model is a tuple A = A, S, Pre, where S A A and Pre : L (A). The update model M A = W, R, V where W = {(w, a) w = Pre(a)} Logic and Artificial Intelligence 6/33

17 Product Update Details Let M = W, R, V be a Kripke model. An event model is a tuple A = A, S, Pre, where S A A and Pre : L (A). The update model M A = W, R, V where W = {(w, a) w = Pre(a)} (w, a)r (w, a ) iff wrw and asa Logic and Artificial Intelligence 6/33

18 Product Update Details Let M = W, R, V be a Kripke model. An event model is a tuple A = A, S, Pre, where S A A and Pre : L (A). The update model M A = W, R, V where W = {(w, a) w = Pre(a)} (w, a)r (w, a ) iff wrw and asa (w, a) V (p) iff w V (p) Logic and Artificial Intelligence 6/33

19 Product Update Details Let M = W, R, V be a Kripke model. An event model is a tuple A = A, S, Pre, where S A A and Pre : L (A). The update model M A = W, R, V where W = {(w, a) w = Pre(a)} (w, a)r (w, a ) iff wrw and asa (w, a) V (p) iff w V (p) M, w = [A, a]ϕ iff M, w = Pre(a) implies M A, (w, a) = ϕ. Logic and Artificial Intelligence 6/33

20 Example Ann would like ob to attend her talk; however, she only wants ob 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 ob the time and subject of her talk. Is this procedure correct? Logic and Artificial Intelligence 7/33

21 Example Ann would like ob to attend her talk; however, she only wants ob 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 ob the time and subject of her talk. Is this procedure correct? Yes, if 1. Ann knows about the talk. 2. ob knows about the talk. 3. Ann knows that ob knows about the talk. 4. ob does not know that Ann knows that he knows about the talk. 5. And nothing else. Logic and Artificial Intelligence 7/33

22 Dynamic Epistemic Logic Recall the Ann and ob example: Charles tells ob that the talk is at 2PM. Logic and Artificial Intelligence 8/33

23 Dynamic Epistemic Logic Recall the Ann and ob example: Charles tells ob that the talk is at 2PM. A e 1 P P e 2 e 3 A A, Ann knows which event took place. Logic and Artificial Intelligence 8/33

24 Dynamic Epistemic Logic Recall the Ann and ob example: Charles tells ob that the talk is at 2PM. A e 1 P P e 2 e 3 A A, Ann knows which event took place. Logic and Artificial Intelligence 8/33

25 Dynamic Epistemic Logic Recall the Ann and ob example: Charles tells ob that the talk is at 2PM. A e 1 P P e 2 e 3 A A, ob thinks a different event took place. Logic and Artificial Intelligence 8/33

26 Dynamic Epistemic Logic Recall the Ann and ob example: Charles tells ob that the talk is at 2PM. A e 1 P P e 2 e 3 A A, That is, ob learns the time of the talk, but Ann learns nothing. Logic and Artificial Intelligence 8/33

27 Dynamic Epistemic Logic Logic and Artificial Intelligence 9/33

28 Dynamic Epistemic Logic (M E 1 ) E 2 The initial model (Ann and ob are ignorant about P 2PM ). Private announcement to Ann about the talk. Logic and Artificial Intelligence 9/33

29 Dynamic Epistemic Logic (M E 1 ) E 2 The initial model (Ann and ob are ignorant about P 2PM ). Private announcement to Ann about the talk. Logic and Artificial Intelligence 9/33

30 Product Update Logic and Artificial Intelligence 10/33

31 Product Update E 2 A P s M E 1 A, A, P t e 1 P P e 2 e 3 A, A Logic and Artificial Intelligence 10/33

32 Product Update A e 1 P P e 2 A, A, A P P e 3 s t A, (s, e 1 ) P P (s, e 2 ) (s, e 3 ) P P (t, e 3 ) Logic and Artificial Intelligence 10/33

33 Product Update A e 1 P P e 2 A, A, A P P e 3 s t A, (s, e 1 ) P P (s, e 2 ) (s, e 3 ) P P (t, e 3 ) Logic and Artificial Intelligence 10/33

34 Product Update A e 1 P P e 2 A, A, A P P e 3 s t A, (s, e 1 ) = K K A K P (s, e 1 ) P P (s, e 2 ) (s, e 3 ) P P (t, e 3 ) Logic and Artificial Intelligence 10/33

35 Product Update A e 1 P P e 2 A, A, A P P e 3 s t A, (s, e 1 ) = K K A K P (s, e 1 ) P P (s, e 2 ) (s, e 3 ) P P (t, e 3 ) Logic and Artificial Intelligence 10/33

36 Product Update A e 1 P P e 2 A, A, A P P e 3 s t A, (s, e 1 ) = K K A K P (s, e 1 ) P P (s, e 2 ) A (s, e 3 ) P P (t, e 3 ) Logic and Artificial Intelligence 10/33

37 Product Update A e 1 P P e 2 A, A P P e 3 s t A, (s, e 1 ) = K K A K P (s, e 1 ) P P (s, e 2 ) (s, e 3 ) P A P (t, e 3 ) Logic and Artificial Intelligence 10/33

38 1. The agents observational powers. 2. The type of change triggered by the event. 3. The underlying protocol specifying which events (observations, messages, actions) are available (or permitted) at any given moment. Logic and Artificial Intelligence 11/33

39 1. The agents observational powers. 2. The type of change triggered by the event. 3. The underlying protocol specifying which events (observations, messages, actions) are available (or permitted) at any given moment. Logic and Artificial Intelligence 11/33

40 A recurring issue in any formal model representing agents (changing) informational attitudes is how to account for the fact that the agents are limited in their access to the available inference steps, possible observations and available messages. This may be because the agents are not logically omniscient and so do not have unlimited reasoning ability. ut it can also be because the agents are following a predefined protocol that explicitly or implicitly limits statements available for observation and/or communication. Logic and Artificial Intelligence 12/33

41 A recurring issue in any formal model representing agents (changing) informational attitudes is how to account for the fact that the agents are limited in their access to the available inference steps, possible observations and available messages. 1. This may be because the agents are not logically omniscient and so do not have unlimited reasoning ability. ut it can also be because the agents are following a predefined protocol that explicitly or implicitly limits statements available for observation and/or communication. Logic and Artificial Intelligence 12/33

42 A recurring issue in any formal model representing agents (changing) informational attitudes is how to account for the fact that the agents are limited in their access to the available inference steps, possible observations and available messages. 1. This may be because the agents are not logically omniscient and so do not have unlimited reasoning ability. 2. ut it can also be because the agents are following a predefined protocol that explicitly or implicitly limits statements available for observation and/or communication. Logic and Artificial Intelligence 12/33

43 A recurring issue in any formal model representing agents (changing) informational attitudes is how to account for the fact that the agents are limited in their access to the available inference steps, possible observations and available messages. 1. This may be because the agents are not logically omniscient and so do not have unlimited reasoning ability. ut it can also be because the agents are following a predefined protocol that explicitly or implicitly limits statements available for observation and/or communication. Logic and Artificial Intelligence 12/33

44 Epistemic Temporal Logic R. Parikh and R. Ramanujam. A Knowledge ased Semantics of Messages. Journal of Logic, Language and Information, 12: , 1985, FHMV. Reasoning about Knowledge. MIT Press, Logic and Artificial Intelligence 13/33

45 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 Logic and Artificial Intelligence 14/33

46 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 Logic and Artificial Intelligence 14/33

47 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 Logic and Artificial Intelligence 14/33

48 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 ϕ (the group A commonly knows ϕ). Logic and Artificial Intelligence 15/33

49 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 Logic and Artificial Intelligence 16/33

50 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. Logic and Artificial Intelligence 17/33

51 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. Logic and Artificial Intelligence 17/33

52 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 Logic and Artificial Intelligence 18/33

53 Returning to the Example Ann would like ob to attend her talk; however, she only wants ob to attend if he is interested in the subject of her talk, not because he is just being polite. Logic and Artificial Intelligence 19/33

54 Returning to the Example Ann would like ob to attend her talk; however, she only wants ob 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 ob the time and subject of her talk. Logic and Artificial Intelligence 19/33

55 Returning to the Example Ann would like ob to attend her talk; however, she only wants ob 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 ob the time and subject of her talk. Is this procedure correct? Logic and Artificial Intelligence 19/33

56 t = 0 m 2PM m 3PM t = 1 m A C t m A C t t = 2 t = 3 m C m C t m C t m C t t

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

58 t = 0 m 2PM m 3PM t = 1 m A C t m A C t t = 2 t = 3 m C m C t m C t m C t t ob s uncertainty: H, 3 = K P 2PM

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

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

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

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

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

64 Parameters of the Logical Framework Logic and Artificial Intelligence 21/33

65 Parameters of the Logical Framework 1. Expressivity of the formal language. Does the language include a common knowledge operator? A future operator? oth? Logic and Artificial Intelligence 21/33

66 Parameters of the Logical Framework 1. Expressivity of the formal language. Does the language include a common knowledge operator? A future operator? oth? 2. Structural conditions on the underlying event structure. Do we restrict to protocol frames (finitely branching trees)? Finitely branching forests? Or, arbitrary ETL frames? Logic and Artificial Intelligence 21/33

67 Parameters of the Logical Framework 1. Expressivity of the formal language. Does the language include a common knowledge operator? A future operator? oth? 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? Logic and Artificial Intelligence 21/33

68 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 ). Logic and Artificial Intelligence 22/33

69 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 Logic and Artificial Intelligence 23/33

70 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 Logic and Artificial Intelligence 24/33

71 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 Logic and Artificial Intelligence 25/33

72 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 Logic and Artificial Intelligence 26/33

73 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 Logic and Artificial Intelligence 27/33

74 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 Logic and Artificial Intelligence 28/33

75 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 enthem and EP. The Tree of Knowledge in Action. Proceedings of AiML, Logic and Artificial Intelligence 29/33

76 Constrained Public Announcement 1. A A vs. A A Logic and Artificial Intelligence 30/33

77 Constrained Public Announcement 1. A A vs. A A 2. A K i P A K i A P Logic and Artificial Intelligence 30/33

78 Constrained Public Announcement 1. A A vs. A A 2. A K i P A K i A P 3. A K i P A K i (A A P) Logic and Artificial Intelligence 30/33

79 Constrained Public Announcement 1. A A vs. A A 2. A K i P A K i A P 3. A K i P A K i (A A P) 4. A K i P A K i ( A A P) Logic and Artificial Intelligence 30/33

80 1. The agents observational powers. 2. The type of change triggered by the event. 3. The underlying protocol specifying which events (observations, messages, actions) are available (or permitted) at any given moment. Logic and Artificial Intelligence 31/33

81 1. The agents observational powers. 2. The type of change triggered by the event. 3. The underlying protocol specifying which events (observations, messages, actions) are available (or permitted) at any given moment. Logic and Artificial Intelligence 31/33

82 Agents may differ in precisely how they incorporate new information into their epistemic states. These differences are based, in part, on the agents perception of the source of the information. For example, an agent may consider a particular source of information infallible (not allowing for the possibility that the source is mistaken) or merely trustworthy (accepting the information as reliable, though allowing for the possibility of a mistake). Logic and Artificial Intelligence 32/33

83 Informative Actions E D A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i i C i D i E Logic and Artificial Intelligence 33/33

84 Informative Actions E D A ϕ C Incorporate the new information ϕ(!ϕ): A i Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i i C i D i E Logic and Artificial Intelligence 33/33

85 Informative Actions E D A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i i C i D i E Logic and Artificial Intelligence 33/33

86 Informative Actions E D A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i i C i D i E Logic and Artificial Intelligence 33/33

87 Informative Actions E D A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i i C i D i E Logic and Artificial Intelligence 33/33