Discussions on Pacuit and Parikhs Reasoning about Communication Graphs

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Discussions on Pacuit and Parikhs Reasoning
about Communication Graphs

• Jiahong Guo
• School of Philosophy and Sociology
• Beijing Normal University
• Beijing, 100875

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Introduction to Reasoning about Communication
Graphs (CGs)

• Main literature The logic of communication
  graphs (EP RP, DALT 2004, LNAI 3476, pp.
  256-269, 2005, Springer-Verlag Berlin Heidelberg
  2005), Reasoning about Communication Graphs (EP
  RP 2006)
• Background
• Main ideas and methods
• Interesting issues involved
• Limitations and their suggestions

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Background (list not complete)

• Logic Dynamics (van Benthem 1996)
• DEL (Baltag 1998, van Ditmarsch, van der Hoek
  Kooi 2007, van Benthem 2007)
• Belief Revision (AGM85, Hansson 99, Liu 2008)
• Communication (Plaza 1989, van Benthem 2002)
  One is a lonely number
• Social Software (Parikh 2001, Pacuit 2006?)
• The issue of constructing and verifying social
  procedures, formal analysis tools are applied
• Back

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Main Ideas and Methods of CGS

• Knowledge and communication modalities
  introduced Ki?, ?? mean that agent i knows that
  ? and after some communication ? is true
  respectively
• One sided connected CG for multi-agent situation
  GA(A, E), where A is a set of agents, E ? A?A
  (i, i)i?A

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Main Ideas and Methods of CGS-2

• Communication event ?G(i, j, ?)??LDNF, (i,
  j)?EG
• Communication history H
• Given the set of events ?G, a history is a
  finite sequence of events is local history
  corresponding to H, ?i(H) is defined a binary
  relation ? over histories defined the notion of
  legal history defined

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Main Ideas and Methods of CGS-3

• Partial propositional varibles Ati and partial
  valuations vi for each agent i
• Fix n agents, At(At1, , Atn) is an
  assignment of sub-languages to the agents. A
  communication graph model MltG, At, vgt, where
  v(v1, , vn) such that for each agent i,
  dom(vi)Ati.

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Main Ideas and Methods of CGS-4

• Legal pairs and formulas satisfiability in a pair
  (w, H) of a model M defined by mutual recursion
• w, ? ?M L L is satisfied only by legal pairs
• w, H (i, j, ?) ?M L iff w, H ?M L, (i, j)?E and
  w, H ?M Kj?
• w, H ?M p iff w(p)1, where p?At
• w, H ?M ?? iff ?H, H?H, L(w, H), and w, H ?M
  ?
• w, H ?M Ki? iff ?(v,H), if (w, H)?i (v, H), and
  L(v, H), then v, H ?M ?
• Back

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Interesting Issues Involved (parts)

• To know a proposition trough communication
• If ? is a ground formula, Kj? ? ?Ki? is
  satisfiable in a graph model M if only legal
  pairs are considered, it is valid the meaning is
  that i learns ? from j
• What is true may come to be known (van
  Benthem 2003, a kind of learnability ? ? ?Ki?
  is satisfiable in a graph model provided ? is a
  ground formula and it is true)

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Interesting Issues Involved (parts)2

• Compressed histories c(H) and their equivalences
  if a formula ? is satisfiable in some graph model
  then it is satisfiable in a history in which no
  communication (i, j, ?) occurs twice (no repeated
  events)
• Theories of axiomatic system valid in
  communication graph frames (such as Ki????Ki? is
  valid)

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Interesting Issues Involved (parts)3

• Communications can occur partially and secretly
  Let G(A, E) be a communication graph. Then for
  each (i, j)?E, for all l?A such that l?i and l?j
  and all ground formulas ?, the scheme
• Kj? ? ?Kl? ? ?(Ki? ? ?Kl?) is G-valid
• Non-symmetric communications
• Learning information from other agents without
  their awareness
• Back

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Limitations and their suggestions

• Belief cases to be extended
• Subjective knowledge and subjective belief
• Learnability (through communication) seems not
  always valid while concerning the subjective
  knowledge or belief
• Each agent has a partial valuation
• They may change as agents learn new ground
  information it is possible that new valuation
  cannot be described in terms of the original
  valuation

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Limitations and their suggestions-2

• Complete axiomatization not finished as showed in
  the literature need a completeness proof
• More general extensions communications between
  group of agents (Hoshi 2006, Roelofsen 2005)
  where DEL methods are combined

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Knowledge and Belief

• Pure Knowledge is an idealization
• Its difficult to separate pure knowledge from
  belief especially in communication activities
• A system for knowledge and belief (Kraus
  Lehmann 1986)
• For single agent S5 (for knowledge) KD (for
  belief) Interrelation Axioms (Ki??Bi?,
  Bi??KiBi?), formulas 4 5 for belief can be
  deduced from the above system.
• For many agents plus axioms of common
  knowledge and distributed knowledge, and common
  knowledge implies distributed knowledge (C??D?)
  

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Communication with Moorean Type Information
(belief cases)

• It seems that p??Bip cannot be learned by agent i
  through communication. If Kj(p??Bip) is true,
  Kj(p??Bip)? ?Bi(p??Bip) is always false since
  after communication, i becomes to believe p. But
  such kind of communications do occur.
• What happens when received a MTF
• Informal analysis four possible updates

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Four possible updates with a MTF

• j informs i that p??Bip for belief cases and i,
  it is actually the information equivalent to
  Bj(p??Bip)
• Case1 p was actually true and the agent i did
  believe it (and even knew it). Then i thinks that
  her friend has made a mistake, or at least j did
  not understand what her original belief
  (epistemic) state is. Hence i will not accept
  what her friend said and the original belief
  state will remain untouched.

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Case 2

• p was not the case but i did believe it. Then the
  effect of js information for is belief revision
  is not direct. Although i did not know that p was
  false, i believed that it was true. Then js
  assertion p is the case is attractive
  (meaningful) to i. But the other part of js
  assertion is clearly rejected by i, at least j is
  wrong in judging her friends belief state
  (actually it is wrong to judge p as well). Even
  if j is supposed sincere, it is common for people
  to make mistakes.

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Case 3

• p was actually true and i did not believe it.
  Then if the agent i is aware that she did not
  believe p (we can get it from negative
  introspection), then for a normal (rational)
  human being, i will accept her friends
  suggestion and to believe p.

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Case 4

• p was not the case and i did not believe it. We
  may need to consider two possibilities in this
  situation. One is i knew that p was not the case
  (then she did not believe p), then she will think
  that j has made a mistake, at least in judging p
  although she has correctly asserted is belief
  state. In this case, i will not change her belief
  state after getting js information p??Bip. The
  other is that i did not know p was not the case
  (whether p). If i trusts her friend j very much,
  she will accept js information and to believe p.

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What is learned from A MTF

• Case 3 is a significant case for is revision,
  genuine information learned there
• In that case, actually we can say that
  Kj(p??Bip)
• We expect to have the following results (for case
  3) in augmented communication graph models
• w, H ?M Kj(p??Bip) ? (p??Bip)
• w, H (i, j, p??Bip) ?M Bip
• Then the agent i knows that in the pair (w,
  H), p??Bip it is what the agent i learned after
  communication that we are interested in it may
  need more augmented language to represent
  learning those higher-order information

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Further interesting issues

• Analysis for knowledge cases in communicating
  with Moorean formulas
• Restrictions on E
• To Formalize the process of communication with
  unlimited knowledge or belief (such as
  distributed knowledge, and even arbitrary DEL
  formulas)

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• THANK YOU!