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Reasoning About the Knowledge of Multiple Agents
Ashker Ibne MujibAndrew Reinders
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The Classical Model(Also called possible-worlds model)There are a number of possible worlds (states of
affairs)Some of these possible worlds may be
indistinguishable to an agent from the true world.
An agent is said to know a fact φ if φ is true in all the worlds he thinks possible.
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DrawbacksMany applications of interest involve
multiple agents.It’s also important to consider what an
agent knows about what the other agents know and don’t know.
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“Dean doesn’t know whether Nixon knows that Dean knows that Nixon knows that McCord burgled O’Brien’s office at Watergate”
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Formalizing a Language involving Multiple Agents
n: Number of agentsΦ: Set of primitive propositions (usually
denoted by letters p, q, r)K1,…,Kn: Modal operatorsIf φ, Ψ formulas, so are ¬φ, φ^Ψ and Kiφ i
= 1, 2, … , nKiφ is read as “agent i knows φ”
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Example
K1K2p ^ ¬K2K1K2p
Agent 1 knows that agent 2 knows p, but agent 2 doesn’t know that agent 1 knows that agent 2 knows p.
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“Dean doesn’t know whether Nixon knows that Dean knows that Nixon knows that McCord burgled O’Brien’s office at Watergate”
Let Dean be agent 1 and Nixon be agent 2Also let p be the statement – “McCord burgled
O’Brien’s office at Watergate”
¬K1 ¬ (K2K1K2p) ^ ¬K1¬(¬K2K1K2p)
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Common KnowledgeThe infinite conjunction of the
statements “everyone knows, and everyone knows that everyone knows, and everyone knows that everyone knows that everyone knows,…”
In order for something to be a convention, it must be common knowledge among the members of the group.
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EG: Everyone in the group G knows.CG: It is common knowledge among the
agents in G
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Kripke Structure (M) M is a tuple (S, π, Κ1 ,…, Κn), where
S: set of states or possible worlds π: an interpretation which associates with each state in S a truth assignment to the primitive propositions (i.e., π(s)(p) Є {true, false} for each state s Є S and each primitive proposition p)Κi : an equivalence relation on S, which is basically agent i’s possibility relation.(s,t) Є Ki , if agent i cannot distinguish state s from state t.
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Kripke Structure (M)
(M,s) |= φ is read “φ is true, or satisfied, in state s of structure M”.
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Properties of (M,s) |= φ (M,s) |= p for a primitive proposition p if
π(s)(p) = true(M,s) |= ¬ φ if (M,s) |≠ φ (M,s) |= φ^Ψ if (M,s) |= φ and (M,s) |= Ψ(M,s) |= Kiφ if (M,s) |= φ for all t such that
(s, t) Є Ki
(M,s) |= EGφ if (M,s) |= Kiφ for all i Є G(M,s) |= CGφ if (M,s) |= Ek
Gφ for k = 1,2,…, where E1
Gφ = EGφ and Ek+1Gφ = EG Ek
Gφ
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Graph Representation of Kripke StructureLabeled vertices connected by directed,
labeled edgesVertices are the states of SEach vertex is labeled by the primitive
propositions true and false thereThere is an edge from s to t labeled i
exactly if (s,t) Є Ki
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Example
Φ = {p}n = 2
s
t u
1 2
¬ p p
1,2
1,2 1,2
p
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M = (S, π, Κ1 ,…, Κn), where
S = {s, t, u}p is true at states s and u, but false at t
π(s)(p) = π(u)(p) = true, π(t)(p) = false
s
t u
1 2
¬ p p
1,2
1,2 1,2
p
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Agent 1 cannot tell s and t apartAgent 2 cannot tell s and u apartK1 = {(s,s),(s,t),(t,s),(t,t),(u,u)}
K2 = {(s,s),(s,u),(u,s),(t,t),(u,u)}
s
t u
1 2
¬ p p
1,2
1,2 1,2
p
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Coordinated Attack ProblemTwo divisions of army are camped on two
hilltops overlooking a common valley where enemy resides.
They will win only if both divisions attack simultaneously.
There is a messenger to exchange news.
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Coordinated Attack ProblemKB pKA KB pKB KA KB p
Only depth of knowledge is increasing.Common knowledge is never attained!
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Modeling Multi-agent SystemsGlobal State: A tuple consisting of each process’
local state, together with the state of the environment.
Environment: Consists of everything that is relevant to the system that is not contained in the state of the processes.
A global state has the form (se,s1,…,sn), where se is the state of the environment and si is agent i’s state, for i = 1,…,n
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Some definitionsRun: A complete description of what
happens over time in one possible execution of the system.
Point: A pair (r,m) consisting of a run r and a time m. At a point (r,m) the system is in some global state r(m).
If r(m) = (se,s1,…,sn), then we take ri(m) to be si, agent i’s local state at point (r,m).
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Unbounded Message DelaysA system R displays unbounded message delays
if, whenever there is a run r Є R such that process i receives a message at time m in r, then for all m´ > m, there is another run r´ that is identical to r up to time m except that process i receives no messages at time m, and no process receives a message between times m and m´.
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Unbounded Message DelaysTheorem: In any run of a system that displays
unbounded message delays, it can never be common knowledge that a message has been delivered.
Corollary: In any run of a system that displays unbounded message delays, it can never be common knowledge among the generals that they are attacking; i.e., if G consists of the two generals, then CG(attack) never holds.
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Interpreted SystemAn interpreted system I consists of a pair
(R, π), where R is a system and π is an interpretation for the propositions in Φ which assigns truth values to the primitive propositions at the global states.
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Theorem: In any system for coordinated attack, when the generals attack, it is common knowledge among the generals that they are attacking. Thus, if I is an interpreted system for coordinated attack, and G consists of the two generals, then at every point (r,m) of I, we have(I,r,m) |= attack => CG(attack).
Corollary: In any system for coordinated attack that displays unbounded message delays, the generals never attack.
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Є-Common KnowledgeWithin Є units everyone knows that
within Є time units everyone knows that…Just as common knowledge corresponds
to simultaneous coordination, Є common knowledge corresponds to coordinating to within Є time units.
Imperfect knowledge
Perfect: what agents can't know is clear Perfect knowledge breaks every cryptosystem Computationally infeasible
Perfect knowledge too aggressive for world Perfect knowledge overestimates adversaries Perfect knowledge overestimates other
agents Not really a model of knowledge
Montague-Scott structures
Agent believes sets of worlds possible, not formulas
Knowledge describes sets of worlds Agent i knows p if {w | p is true in w} is a
possible set of states of worlds
Discarding this gives incomplete reasoning to agents
Doesn't really model knowability
sCT'T'TsCT ii
NPL, other reasoning-weakenings
Instead of simply arbitrarily breaking inference
(p ^ ¬ p) => q can fail p true in s if ¬p not true in adjunct world, s* Reasoning loses power, is now poly-time
computable Adversaries no longer infinitely able to
compute
Information
Information-passing is nontrivial Telling agent i Time is inherently necessary in message
passing to maintain consistency
p¬Kp i
Probability and knowledge
Information not known may have some probability
q's probability may be the same even if outcome is changed by unknown p
Reasoning captures this how? Partition possibilities Simple partitioning may not capture
probabilities based on knowledge of an agent
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References
Reasoning About Knowledge: A Survey. Joseph Y. Halpern in D. Gabbay, C. J. Hogger, and J. A. Robinson, Eds.,Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 4, Oxford University Press, 1995.
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Thank You