Lecture 11: Topics in Formal Epistemology

Transcription

1 Lecture 11: Topics in Formal Epistemology Eric Pacuit ILLC, University of Amsterdam staff.science.uva.nl/ epacuit Lecture Date: May 4, 2006 Caput Logic, Language and Information: Social Software staff.science.uva.nl/ epacuit/caputlli.html

2 Introduction What are we trying to model? Formal Models Epistemic Logic; Aumann Structures (Type Spaces, Knowledge Structures, Bayesian Structures) Common Knowledge Levels of Knowledge A Model of Knowledge for Social Software Communication Graphs

3 What are we trying to model? Greco & Sosa. The Blackwell Guide to Epistemology J. van Benthem. Epistemic Logic and Epistemology

4 What are we trying to model? Greco & Sosa. The Blackwell Guide to Epistemology J. van Benthem. Epistemic Logic and Epistemology Plato: true justied belief

5 What are we trying to model? Greco & Sosa. The Blackwell Guide to Epistemology J. van Benthem. Epistemic Logic and Epistemology Plato: true justied belief Hintikka: truth throughout a set of possibilities the agent considers possible J. Hintikka. Knowledge and Belief. 1962, recently republished.

6 What are we trying to model? Greco & Sosa. The Blackwell Guide to Epistemology J. van Benthem. Epistemic Logic and Epistemology Plato: true justied belief Hintikka: truth throughout a set of possibilities the agent considers possible J. Hintikka. Knowledge and Belief. 1962, recently republished. Dretske: belief supported by reliable correlations F. Dretske. Knowledge and the Flow of Information. CSLI, 1999.

7 What are we trying to model? Greco & Sosa. The Blackwell Guide to Epistemology J. van Benthem. Epistemic Logic and Epistemology Plato: true justied belief Hintikka: truth throughout a set of possibilities the agent considers possible J. Hintikka. Knowledge and Belief. 1962, recently republished. Dretske: belief supported by reliable correlations F. Dretske. Knowledge and the Flow of Information. CSLI, Nozik: true belief plus if φ was false then φ would be believed R. Nozik. Philosophical Explanations

8 What are we trying to model? Greco & Sosa. The Blackwell Guide to Epistemology J. van Benthem. Epistemic Logic and Epistemology Plato: true justied belief Hintikka: truth throughout a set of possibilities the agent considers possible J. Hintikka. Knowledge and Belief. 1962, recently republished. Dretske: belief supported by reliable correlations F. Dretske. Knowledge and the Flow of Information. CSLI, Nozik: true belief plus if φ was false then φ would be believed R. Nozik. Philosophical Explanations

9 Gettier Cases Suppose that Smith and Jones have applied for a certain job.

10 Gettier Cases Suppose that Smith and Jones have applied for a certain job. Smith as strong evidence for the conjunction (a) Jones is the man who will get the job, and Jones has ten coins in his pocket.

11 Gettier Cases Suppose that Smith and Jones have applied for a certain job. Smith as strong evidence for the conjunction (a) Jones is the man who will get the job, and Jones has ten coins in his pocket. Hence, Smith is justied in believing (b) The man who will get the job has ten coins in his pocket

12 Gettier Cases Suppose that Smith and Jones have applied for a certain job. Smith as strong evidence for the conjunction (a) Jones is the man who will get the job, and Jones has ten coins in his pocket. Hence, Smith is justied in believing (b) The man who will get the job has ten coins in his pocket Imagine, further, that unknown to Smith, he himself, not Jones will get the job. And, also, unknown to Smith, he himself has ten coins in his pocket.

13 Gettier Cases Let P := The man who will get the job has ten coins in his pocket Unknown to Smith, he himself, not Jones will get the job. And, also, unknown to Smith, he himself has ten coins in his pocket. It is true that P Smith believes that P Smith is justied in believing that P But we do not say that Smith knows that P E. Gettier. Is Justied True Belief Knowledge?

14 Digression: What does Amazon.com know? Does Amazon.com know my address? X knows p i 1. X has ready access to p 2. p is true 3. X can make use of the informational content of p (i.e., X can exercise certain capacities dependent on its knowing p). S. Chopra and L. White. Attribution of Knowledge to Articial Agents and their Principals

15 Formal Models of Knowledge There is a huge literature! FHMV. Reasoning about Knowledge Epistemic Logic Aumann Structures History Based Structures (Knowledge Structures) (Type Spaces) (Probability Spaces)

16 Possible Worlds A state of the worlds is a complete description of the relevant information including all ground facts, what each agent knows, what every agent knows the other agent knows, and so on...

17 Possible Worlds A state of the worlds is a complete description of the relevant information including all ground facts, what each agent knows, what every agent knows the other agent knows, and so on... A state is A set of ground facts For each agent a possibility function

18 Possible Worlds A state of the worlds is a complete description of the relevant information including all ground facts, what each agent knows, what every agent knows the other agent knows, and so on... A state is A set of ground facts For each agent a possibility function A Possibility function is a function from states to sets of states

19 Possible Worlds A state of the worlds is a complete description of the relevant information including all ground facts, what each agent knows, what every agent knows the other agent knows, and so on... A state is A set of ground facts For each agent a possibility function A Possibility function is a function from states to sets of states Is there a vicious circle? J. Barwise and L. Moss. Vicious Circles

20 Epistemic Logic Let A be a set of agents and At a set of atomic propositions. Modal Language the smallest set generated by (p At) φ := p φ φ ψ iφ Kripke Frame W, {Ri}i A where Ri W W Kripke Model W, {Ri}i A, V where V : At 2 W Truth in a Model 1. M, s = p if p V (s) 2. M, s = φ ψ if M, s = φ and M, s = ψ 3. M, s = φ if M, s = φ 4. M, s = iφ if for each v W, if wrv, then M, v = φ

21 Some Epistemic Axioms Axiom Property Frame Property (φ ψ) ( φ ψ) (valid in all frames) φ φ Truth Axiom Reexivity φ φ Positive Introspection Transitivity φ φ Negative Introspection Euclidean Consistency Serial FHMV. Reasoning about Knowledge

22 Aumann Structures R. Aumann. Interactive Epistemology I: Knowledge Knowledge function: κ with domain W and range an abstract set. for w W, κ(w) is the signal the agent receives from the outside world when the true state of the world is w. Information function: I(w) := {w W κ(w) = κ(w )} w I(w) I(w) and I(w ) are either disjoint or identical So I forms a partition of W.

23 Aumann Structures Let Σ be the family of all events (= 2 W ). Dene K : Σ Σ as follows w KE i I(w) E K has the following properties: 1. KE E 2. E F implies KE KF 3. KE KKE We can take any of the κ, I, K as primitive. Eg. Take K : Σ Σ as primitive. I(w) := K{w}

24 Epistemic Logic with Dierent Terminology Given an arbitrary set operator P : 2 W 2 W, the following properties can be assumed of P. Let E, F be any two subsets W P1 P(E) P(F ) = P(E F ) P2 j JP(Ej) = P( j JEj), for any index set J a P3 P(E) E P4 P(E) P(P(E)) P5 P(E) P(P(E)) P6 P(E) P(E) a When J =, we get K(Ω) = Ω

25 Let P : W 2 W be any function. We dene the following properties of P : Reexive w W, w P (w) Transitive w, v W, v P (w) P (v) P (w) Euclidean w, v W, v P (w) P (w) P (v) Serial w W, P (w) J. Halpern. Set Theoretic Completeness of Epistemic and Conditional Logic

26 Common Knowledge Given two agents i and j, how do we formalize the statement φ is common knowledge between i and j in terms of i, j, φ and (private knowledge)? Let γ be the statement it is common knowledge between i and j that φ. For a good exposition see, M. Chwe. Rational Ritual

27 Three Views of Common Knowledge 1. γ := i knows that φ, j knows that φ, i knows that j knows that φ, j knows that i knows that φ, i knows that j knows that i knows that φ,... D. Lewis. Convention, A Philosophical Study

28 Three Views of Common Knowledge 1. γ := i knows that φ, j knows that φ, i knows that j knows that φ, j knows that i knows that φ, i knows that j knows that i knows that φ,... D. Lewis. Convention, A Philosophical Study γ := i and j know that (φ and γ) G. Harman.???.???.

29 Three Views of Common Knowledge 1. γ := i knows that φ, j knows that φ, i knows that j knows that φ, j knows that i knows that φ, i knows that j knows that i knows that φ,... D. Lewis. Convention, A Philosophical Study γ := i and j know that (φ and γ) G. Harman.???.???. 3. There is a shared situation s such that s entails φ s entails i knows φ s entails j knows φ H. Clark and C. Marshall. Denite Reference and Mutual Knowledge

30 Three Views of Common Knowledge 1. γ := i knows that φ, j knows that φ, i knows that j knows that φ, j knows that i knows that φ, i knows that j knows that i knows that φ,... D. Lewis. Convention, A Philosophical Study γ := i and j know that (φ and γ) G. Harman.???.???. 3. There is a shared situation s such that s entails φ s entails i knows φ s entails j knows φ H. Clark and C. Marshall. Denite Reference and Mutual Knowledge

31 Common Knowledge: Iterated View Dene K m : 2 W 2 W for m 1 by K 1 E := i AKiE K m+1 E := K 1 (K m (E)) K 1 E means everyone knows E K 2 E means everyone knows that everyone knows E Dene K : 2 W 2 W K E := K 1 E K 2 E K m E It is easy to verify that KiK E = K E for all i

32 Three Views of Common Knowledge In Relational Models (including Aumann Structures), the rst two views are simply equivalent and it is not clear how to represent the third view.

33 Three Views of Common Knowledge In Relational Models (including Aumann Structures), the rst two views are simply equivalent and it is not clear how to represent the third view. C 0 1,2E := E C 1,2 κ+1 E := E K 1(C 1,2E) κ K2(C 1,2E) κ C λ 1,2E := κ<λc λ 1,2 Let C1,2E = C 1,2E κ where κ is the least ordinal where C 1,2E κ = C κ+1 E Fact In every Relational Model, the above procedure stabilizing at κ ω.

34 Three Views of Common Knowledge In topological models the rst two views can be distinguished. J. van Benthem and D. Sarenac. The Geometry of Knowledge All three can be separated in the situation calculus. J. Barwise. Three Views of Common Knowledge. TARK, 1987.

35 Models of Knowledge for Social Software Let P be a plan the agent is carrying out. π(p ) = {w w C and w enables P } where C is the context of the plan P (decided by an observer). An agent believes φ, provided π(p ) (φ) M. An agent knows φ provide the agent believes φ and φ is true.

36 Models of Knowledge for Social Software If an agent comes to a point in her plan where her appropriate action is If φ do α else do β and she does α, then we will say that she i-believes φ. R. Parikh. WHAT do we know and what do WE know. TARK, 2005.

37 Recall the following 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.

38 Recall the following 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.

39 Recall the following 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. Taking a cue from computer science, we can ask is this procedure correct?

40 Recall the following example Yes, if 1. Ann knows about the talk.

41 Recall the following example Yes, if 1. Ann knows about the talk. 2. Bob knows about the talk.

42 Recall the following example Yes, if 1. Ann knows about the talk. 2. Bob knows about the talk. 3. Ann knows that Bob knows about the talk.

43 Recall the following example 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.

44 Recall the following example 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.

45 Basic Denitions R. Parikh and R. Ramanujam. A History Based Semantics of Messages Let E be a xed set of events. Elements of E E ω will be called histories. Three types: 1. local histories, 2. nite global histories and 3. innite global histories. Given two histories H and H, write H H to mean H is a nite prex of H. Dene FinPre(H) = {H H E, H H, & H H}. Given a history H, Ht is the nite prex of H of length t.

46 Basic Denitions R. Parikh and R. Ramanujam. A History Based Semantics of Messages Let E be a xed set of events. Elements of E E ω will be called histories. Three types: 1. local histories, 2. nite global histories and 3. innite global histories. Given two histories H and H, write H H to mean H is a nite prex of H. Dene FinPre(H) = {H H E, H H, & H H}. Given a history H, Ht is the nite prex of H of length t.

47 Basic Denitions R. Parikh and R. Ramanujam. A History Based Semantics of Messages Let E be a xed set of events. Elements of E E ω will be called histories. Three types: 1. local histories, 2. nite global histories and 3. innite global histories. Given two histories H and H, write H H to mean H is a nite prex of H. Dene FinPre(H) = {H H E, H H, & H H}. Given a history H, Ht is the nite prex of H of length t.

48 Basic Denitions R. Parikh and R. Ramanujam. A History Based Semantics of Messages Let E be a xed set of events. Elements of E E ω will be called histories. Three types: 1. local histories, 2. nite global histories and 3. innite global histories. Given two histories H and H, write H H to mean H is a nite prex of H. Dene FinPre(H) = {H H E, H H, & H H}. Given a history H, Ht is the nite prex of H of length t.

49 Basic Denitions R. Parikh and R. Ramanujam. A History Based Semantics of Messages Let E be a xed set of events. Elements of E E ω will be called histories. Three types: 1. local histories, 2. nite global histories and 3. innite global histories. Given two histories H and H, write H H to mean H is a nite prex of H. Dene FinPre(H) = {H H E, H H, & H H}. Given a history H, Ht is the nite prex of H of length t.

50 Protocols A set H E E ω is called a protocol if H is closed under FinPre.

51 Protocols A set H E E ω is called a protocol if H is closed under FinPre. Intuitively, a protocol represents the various ways a social interactive situation may evolve.

52 Protocols A set H E E ω is called a protocol if H is closed under FinPre. Intuitively, a protocol represents the various ways a social interactive situation may evolve. Note that any (social) procedure will generate a protocol, but not all protocols are generated by some procedure.

53 History Based Structures A history based structure based on a set of events E is a tuple H, E1,..., En where H is a protocol and Ei E for each i A.

54 History Based Structures A history based structure based on a set of events E is a tuple H, E1,..., En where H is a protocol and Ei E for each i A. If e Ei then i is aware of event e.

55 History Based Structures A history based structure based on a set of events E is a tuple H, E1,..., En where H is a protocol and Ei E for each i A. If e Ei then i is aware of event e. History based structures are "low-level descriptions" of a social situation.

56 History Based Frames For each i A dene λi : FinPre(H) E i to be the local view function of agent i.

57 History Based Frames For each i A dene λi : FinPre(H) E i to be the local view function of agent i. We can dene for each nite H H, H i H i λi(h) = λi(h )

58 History Based Frames For each i A dene λi : FinPre(H) E i to be the local view function of agent i. We can dene for each nite H H, H i H i λi(h) = λi(h ) A history based frame is a tuple H, E1,..., En, λ1,..., λn.

59 History Based Frames: Local View Functions Note that for each i A, λi is any function on FinPreH. Possible Assumptions: The agents' local clock is consistent with the global clock:

60 History Based Frames: Local View Functions Note that for each i A, λi is any function on FinPreH. Possible Assumptions: The agents' local clock is consistent with the global clock: For all H H if t m, then λi(ht) λi(hm)

61 History Based Frames: Local View Functions Note that for each i A, λi is any function on FinPreH. Possible Assumptions: The agents' local clock is consistent with the global clock: For all H H if t m, then λi(ht) λi(hm) Agents are not wrong about the events that they witness:

62 History Based Frames: Local View Functions Note that for each i A, λi is any function on FinPreH. Possible Assumptions: The agents' local clock is consistent with the global clock: For all H H if t m, then λi(ht) λi(hm) Agents are not wrong about the events that they witness: λi(h) is embeddable in H.

63 History Based Frames: Local View Functions Note that for each i A, λi is any function on FinPreH. Possible Assumptions: The agents' local clock is consistent with the global clock: For all H H if t m, then λi(ht) λi(hm) Agents are not wrong about the events that they witness: λi(h) is embeddable in H. Assumptions about the agents reasoning capabilities, i.e., perfect recall :

64 History Based Frames: Local View Functions Note that for each i A, λi is any function on FinPreH. Possible Assumptions: The agents' local clock is consistent with the global clock: For all H H if t m, then λi(ht) λi(hm) Agents are not wrong about the events that they witness: λi(h) is embeddable in H. Assumptions about the agents reasoning capabilities, i.e., perfect recall : λi(h) is obtained by mapping each event "seen" by i into itself and each event not seen by i to the empty string.

65 Temporal Epistemic Logic A formula φ KT n can have the following form φ := p φ φ ψ Kiφ φ φ Uψ

66 Temporal Epistemic Logic A formula φ KT n can have the following form φ := p φ φ ψ Kiφ φ φ Uψ Kiφ: "agent i knows φ" φ: "φ is true at the next moment" φuψ: "φ is true until ψ becomes true" F φ: Uφ: "φ is true sometime in the future" Gφ: F φ: "φ is always true"

67 Temporal Epistemic Logic: Truth If H H is a innite global history, then H, t = φ is intended to mean φ is true at time t in history H: 1. H, t = φ Uψ i there exists m t such that H, m = ψ and for all l such that t l < m, H, l = φ 2. H, t = Kiφ i for all H H such that Ht i H t, H, t = φ 2. H, t = Kiφ i for all m 0, for all H H such that Ht i H m, H, m = φ

68 Temporal Epistemic Logic: Truth If H H is a innite global history, then H, t = φ is intended to mean φ is true at time t in history H: 1. H, t = φ Uψ i there exists m t such that H, m = ψ and for all l such that t l < m, H, l = φ 2. H, t = Kiφ i for all H H such that Ht i H t, H, t = φ 2. H, t = Kiφ i for all m 0, for all H H such that Ht i H m, H, m = φ

69 Returning to the Example Let mac be the event that represents the message from Ann to Carol Let mcb be the event that represents the message from Carol to Bob Let P be the proposition that means Ann's talk is at 2 PM". HmACH mcb = KBKAKBP

70 Temporal Epistemic Logic: Axiomatizations Sound and complete axiomatizations (using a dierent semantics)

71 Temporal Epistemic Logic: Axiomatizations Sound and complete axiomatizations (using a dierent semantics) 1. Linear time: [HvdMV, 2003] 2. Branching time: [vdmw, 2004] 3. Past time operators: [FvdMR, 2004] 4. Group strategies (AT L): [GvD, 2002]

72 Histories or Runs? See [FHMV] and [HvdMV] for more information. Let L be a set of local states. A run r R is a function r : N L n+1 A system for n agents is a set R. r(t) has the form le, l1,..., ln, where le is the state of the environment, li for i = 1,..., n is the local state of each agent. A point, or global state, is an element (r, t) R N. An interpreted system I = (R, π), where R is a system and π : (R N) At {true, false} Agent i cannot distinguish two points if it is in the same state in both: (r, t) i (r, t ) i r(t)i = r (t )i.

73 Histories or Runs? See [FHMV] and [HvdMV] for more information. Let L be a set of local states. A run r R is a function r : N L n+1 A system for n agents is a set R. r(t) has the form le, l1,..., ln, where le is the state of the environment, li for i = 1,..., n is the local state of each agent. A point, or global state, is an element (r, t) R N. An interpreted system I = (R, π), where R is a system and π : (R N) At {true, false} Agent i cannot distinguish two points if it is in the same state in both: (r, t) i (r, t ) i r(t)i = r (t )i.

74 Histories or Runs? See [FHMV] and [HvdMV] for more information. Let L be a set of local states. A run r R is a function r : N L n+1 A system for n agents is a set R. r(t) has the form le, l1,..., ln, where le is the state of the environment, li for i = 1,..., n is the local state of each agent. A point, or global state, is an element (r, t) R N. An interpreted system I = (R, π), where R is a system and π : (R N) At {true, false} Agent i cannot distinguish two points if it is in the same state in both: (r, t) i (r, t ) i r(t)i = r (t )i.

75 Histories or Runs? See [FHMV] and [HvdMV] for more information. Let L be a set of local states. A run r R is a function r : N L n+1 A system for n agents is a set R. r(t) has the form le, l1,..., ln, where le is the state of the environment, li for i = 1,..., n is the local state of each agent. A point, or global state, is an element (r, t) R N. An interpreted system I = (R, π), where R is a system and π : (R N) At {true, false} Agent i cannot distinguish two points if it is in the same state in both: (r, t) i (r, t ) i r(t)i = r (t )i.

76 Histories or Runs? See [FHMV] and [HvdMV] for more information. Let L be a set of local states. A run r R is a function r : N L n+1 A system for n agents is a set R. r(t) has the form le, l1,..., ln, where le is the state of the environment, li for i = 1,..., n is the local state of each agent. A point, or global state, is an element (r, t) R N. An interpreted system I = (R, π), where R is a system and π : (R N) At {true, false} Agent i cannot distinguish two points if it is in the same state in both: (r, t) i (r, t ) i r(t)i = r (t )i.

77 Histories or Runs? See [FHMV] and [HvdMV] for more information. Let L be a set of local states. A run r R is a function r : N L n+1 A system for n agents is a set R. r(t) has the form le, l1,..., ln, where le is the state of the environment, li for i = 1,..., n is the local state of each agent. A point, or global state, is an element (r, t) R N. An interpreted system I = (R, π), where R is a system and π : (R N) At {true, false} Agent i cannot distinguish two points if it is in the same state in both: (r, t) i (r, t ) i r(t)i = r (t )i.

78 Histories or Runs? See [FHMV] and [HvdMV] for more information. Let L be a set of local states. A run r R is a function r : N L n+1 A system for n agents is a set R. r(t) has the form le, l1,..., ln, where le is the state of the environment, li for i = 1,..., n is the local state of each agent. A point, or global state, is an element (r, t) R N. An interpreted system I = (R, π), where R is a system and π : (R N) At {true, false} Agent i cannot distinguish two points if it is in the same state in both: (r, t) i (r, t ) i r(t)i = r (t )i.

79 (Some!) Relevant Papers Histories or Runs? EP. Comments on History Based Structures. JAL, Bringing in common knowledge creates a lot of complexity. Halpern, van der Meyden and Vardi. Reasoning about Knowledge and Common Knowledge in a Distributed Environment J. van Benthem and EP. The Tree of Knowledge: Towards a Common Perspective Extending the language stit operators J. Horty. Agency and Deontic Logic EP, R. Parikh and E. Cogan. The Logic of Knowledge Based Obligation. KRA 2006.

80 Knowledge, Time and Communication The occurrence of an event, such as communication, changes the agents' states of knowledge. How should we model this change of information? There have been a number of recent papers concerned with adding a notion of update to otherwise static Kripke structures (Dynamic Epistemic Semantics).

81 Knowledge, Time and Communication The occurrence of an event, such as communication, changes the agents' states of knowledge. How should we model this change of information? There have been a number of recent papers concerned with adding a notion of update to otherwise static Kripke structures (Dynamic Epistemic Semantics). Assume: Agents are assumed to communicate (only) according to a communication graph The starting point is not Kripke structures, but rather subset models (Topologic models). EP and R. Parikh. The Logic of Communication Graphs

82 Background: Topologic Topologic is a bimodal logic with a knowledge modality (K) and an eort modality ( ) rst studied by [MP]

83 Background: Topologic Topologic is a bimodal logic with a knowledge modality (K) and an eort modality ( ) rst studied by [MP] Subset Frame: W, W is a set of states 2 W is a set of subsets of W, i.e., a set of observations Neighborhood Situation: Given a subset frame W,, (w, U) is called a neighborhood situation, where w U and U. Model: W,, V, where V : Φ0 2 W is a valuation function.

84 Topologic: Truth Truth: Formulas are interpreted at neighborhood situations. w, U = p i w V (p) w, U = φ i w, U = φ w, U = φ ψ i w, U = φ and w, U = ψ w, U = Kφ i for all v U, v, U = φ w, U = φ i there is a V such that w V U and w, V = φ

85 From Topologic to Communication Graphs Topologic: a bimodal logic with a knowledge modality (K) and an eort modality ( ). φ Kφ if φ is true then after some `eort' φ is known

86 From Topologic to Communication Graphs Topologic: a bimodal logic with a knowledge modality (K) and an eort modality ( ). φ Kφ if φ is true then after some `eort' φ is known Extend to a multi-agent epistemic logic where eort = communication

87 From Topologic to Communication Graphs Topologic: a bimodal logic with a knowledge modality (K) and an eort modality ( ). φ Kφ if φ is true then after some `eort' φ is known Extend to a multi-agent epistemic logic where eort = communication Kiφ Kjφ

88 From Topologic to Communication Graphs Topologic: a bimodal logic with a knowledge modality (K) and an eort modality ( ). φ Kφ if φ is true then after some `eort' φ is known Extend to a multi-agent epistemic logic where eort = communication Kiφ Kjφ if i knows φ then after some communication j knows φ

89 An Example If Bush wants some information from a particular CIA operative, say Bob, he must get this information through Goss. Suppose that φ is the exact whereabouts of Bin Laden and KBobφ KBushφ

90 An Example If Bush wants some information from a particular CIA operative, say Bob, he must get this information through Goss. Suppose that φ is the exact whereabouts of Bin Laden and KBobφ KBushφ Bush can nd out the exact whereabouts of Bin Laden

91 An Example If Bush wants some information from a particular CIA operative, say Bob, he must get this information through Goss. Suppose that φ is the exact whereabouts of Bin Laden and KBobφ KBushφ Bush can nd out the exact whereabouts of Bin Laden KBushφ

92 An Example If Bush wants some information from a particular CIA operative, say Bob, he must get this information through Goss. Suppose that φ is the exact whereabouts of Bin Laden and KBobφ KBushφ Bush can nd out the exact whereabouts of Bin Laden KBushφ but of course, we cannot nd out such information

93 An Example If Bush wants some information from a particular CIA operative, say Bob, he must get this information through Goss. Suppose that φ is the exact whereabouts of Bin Laden and KBobφ KBushφ Bush can nd out the exact whereabouts of Bin Laden KBushφ but of course, we cannot nd out such information i RoomD.028 Kiφ

94 A pre-requisite for Bush learning φ, Goss will also have come to know φ.

95 A pre-requisite for Bush learning φ, Goss will also have come to know φ. KBushφ (KBushφ KGossφ)

96 A pre-requisite for Bush learning φ, Goss will also have come to know φ. KBushφ (KBushφ KGossφ) So, there is a connection between the fact that the CIA restricts the ow of information and the truth of certain formulas in our language, such as Kiφ and KBushφ (KBushφ KGossφ) i RoomD.028

97 Communication Graphs Let A be a set of agents. A communication graph is a directed graph GA = (A, E) such that for all i A, (i, i) E (i, j) E means that i can directly receive information from agent j, without j knowing this fact. Assume that: 1. all the agents share a common language 2. the agents make available all possible pieces of information

98 The Logic of Communication Graphs: Introduction Suppose that G = (A, EG) is a xed communication graph. Assume that agents are initially given some private information and communicate according to the communication graph G.

99 The Logic of Communication Graphs: Introduction Suppose that G = (A, EG) is a xed communication graph. Assume that agents are initially given some private information and communicate according to the communication graph G. More formally, initially, each agent i knows or is informed (say by nature) of the truth values of a certain subset Ati of propositional variables, and the Ati as well as this fact are common knowledge. Thus the other agents know that i knows the truth values of elements of Ati, but, typically, not what these values actually are.

100 The Logic of Communication Graphs: Introduction Let W be the set of boolean valuations on At. An element v W is called a state. Call any vector of partial boolean valuations v = (v1,..., vn) consistent if for each p dom(vi) dom(vj), vi(p) = vj(p) for all i, j = 1,..., n.

101 The Logic of Communication Graphs: Introduction Let W be the set of boolean valuations on At. An element v W is called a state. Call any vector of partial boolean valuations v = (v1,..., vn) consistent if for each p dom(vi) dom(vj), vi(p) = vj(p) for all i, j = 1,..., n. All this information (including the structure of the communication graph) is common knowledge and only the precise values of the vi are private.

102 A Logic for Communication Graphs: Semantics Fix a communication graph G = (A, E) and a set of states W.

103 A Logic for Communication Graphs: Semantics Fix a communication graph G = (A, E) and a set of states W. A communication event is a tuple (i, j, φ) intended to mean that i learns information φ from j, where φ is a ground (propositional) formula there is an edge between i and j in G Let ΣG be the set of all such events (based on communication graph G).

104 A Logic for Communication Graphs: Semantics Fix a communication graph G = (A, E) and a set of states W. A communication event is a tuple (i, j, φ) intended to mean that i learns information φ from j, where φ is a ground (propositional) formula there is an edge between i and j in G Let ΣG be the set of all such events (based on communication graph G). A history is a nite sequence of events, λi(h) be i's local history. Assume that λi satises the perfect recall assumption.

105 A Logic for Communication Graphs: Semantics A communication graph frame is a pair G, At where G is a communication graph, and At = (At1,..., Atn) is an assignment of sub-languages to the agents. A communication graph model based on a frame G, At is a triple G, At, v, where v is a consistent vector of partial boolean valuations for At.

106 Uncertainty of the Agents 1. Agents are uncertain about the actual state of the world. 2. Agents are uncertain about the exact sequence of communications that has taken place

107 Uncertainty of the Agents 1. Agents are uncertain about the actual state of the world. 2. Agents are uncertain about the exact sequence of communications that has taken place Let w be a state and H a nite history. Dene the relation i as follows: (w, H) i (v, H ) i w At i = v At i and λi(h) = λi(h ).

108 Two issues are important: Denition of Truth I

109 Denition of Truth I Two issues are important: 1. Not all histories are "legal":

110 Denition of Truth I Two issues are important: 1. Not all histories are "legal": For an event (i, j, φ) to take place after a history H, it must be the case that after H, j knows φ.

111 Denition of Truth I Two issues are important: 1. Not all histories are "legal": For an event (i, j, φ) to take place after a history H, it must be the case that after H, j knows φ. 2. The information which an agent learns by reading" a formula φ may be more than just the fact that φ is true.

112 Denition of Truth I Two issues are important: 1. Not all histories are "legal": For an event (i, j, φ) to take place after a history H, it must be the case that after H, j knows φ. 2. The information which an agent learns by reading" a formula φ may be more than just the fact that φ is true. Suppose that i learns p q from j, but j is not connected, directly or indirectly, to anyone who might know the initial truth value of q. In this case i has learned more than p q, i has learned p as well.

113 Denition of Truth II Introduce a propositional symbol L which is satised only by legal pairs (w, H), denoted L(w, H).

114 Denition of Truth II Introduce a propositional symbol L which is satised only by legal pairs (w, H), denoted L(w, H). w, ɛ =M L w, H; (i, j, φ) =M L i w, H =M L and w, H =M Kjφ w, H =M p i w(p) = 1, where p At w, H =M φ i w, H =M φ w, H =M φ ψ i w, H =M φ and w, H =M ψ w, H =M φ i H, H H, L(w, H ), and w, H =M φ w, H =M Kiφ i (v, H ) if (w, H) i (v, H ), and L(v, H ), then v, H =M φ

115 Denition of Truth II Introduce a propositional symbol L which is satised only by legal pairs (w, H), denoted L(w, H). w, ɛ =M L w, H; (i, j, φ) =M L i w, H =M L and w, H =M Kjφ w, H =M p i w(p) = 1, where p At w, H =M φ i w, H =M φ w, H =M φ ψ i w, H =M φ and w, H =M ψ w, H =M φ i H, H H, L(w, H ), and w, H =M φ w, H =M Kiφ i (v, H ) if (w, H) i (v, H ), and L(v, H ), then v, H =M φ

116 Results I The agents learn at least as much as what is implied by the sequence of communications.

117 Results I The agents learn at least as much as what is implied by the sequence of communications. Dene the sets Xi(w, H) as follows: 1. Xi(w, ɛ) = {v v At i = w At i } 2. Xi(w, H; (i, j, φ)) = Xi(w, H) ˆφ 3. i m then Xi(w, H; (m, j, φ)) = Xi(w, H)

118 Results I The agents learn at least as much as what is implied by the sequence of communications. Dene the sets Xi(w, H) as follows: 1. Xi(w, ɛ) = {v v At i = w At i } 2. Xi(w, H; (i, j, φ)) = Xi(w, H) ˆφ 3. i m then Xi(w, H; (m, j, φ)) = Xi(w, H) Theorem Let M = G, At, v be any communication graph model and φ a ground formula. If Xi(w, H) φ, then (w, H) =M Ki(φ).

119 Results I The agents learn at least as much as what is implied by the sequence of communications. Dene the sets Xi(w, H) as follows: 1. Xi(w, ɛ) = {v v At i = w At i } 2. Xi(w, H; (i, j, φ)) = Xi(w, H) ˆφ 3. i m then Xi(w, H; (m, j, φ)) = Xi(w, H) Theorem Let M = G, At, v be any communication graph model and φ a ground formula. If Xi(w, H) φ, then (w, H) =M Ki(φ). The converse is not true

120 Results II Theorem The satisability problem is decidable.

121 Results II Theorem The satisability problem is decidable. The structure of the communication graph is reected in the set of valid formulas Theorem Let G = (A, E) be a communication graph. Then (i, j) E if and only if, for all l A such that l i and l j and all ground formulas φ, the scheme Kjφ Klφ (Kiφ Klφ) is valid in all communication graph models based on G.

122 Next Week: (Last Class) Topics in Voting Theory