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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