Reasoning About the Knowledge of Multiple Agents

Reasoning About the Knowledge of Multiple Agents Ashker Ibne Mujib Andrew Reinders

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 factφif φis true in all the worlds he thinks possible.

Drawbacks • Many 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.

“Dean doesn’t know whether Nixon knows that Dean knows that Nixon knows that McCord burgled O’Brien’s office at Watergate”

Formalizing a Language involving Multiple Agents • n: Number of agents • Φ: Set of primitive propositions (usually denoted by letters p, q, r) • K1,…,Kn: Modal operators • If φ, Ψ formulas, so are ¬φ, φ^Ψ and Kiφi = 1, 2, … , n • Kiφ is read as “agent i knows φ”

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.

“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 2 Also let p be the statement – “McCord burgled O’Brien’s office at Watergate” ¬K1 ¬ (K2K1K2p) ^ ¬K1¬(¬K2K1K2p)

CommonKnowledge • The 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.

EG: Everyone in the group G knows. • CG: It is common knowledge among the agents in G

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.

Kripke Structure (M) (M,s) |= φis read “φ is true, or satisfied, in state s of structure M”.

Graph Representation of Kripke Structure • Labeled vertices connected by directed, labeled edges • Vertices are the states of S • Each vertex is labeled by the primitive propositions true and false there • There is an edge from s to tlabeled iexactly if (s,t) Є Ki

Example • Φ = {p} • n = 2 p 1,2 s 2 1 p ¬ p 1,2 u 1,2 t

p • 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 1,2 s 2 1 p ¬ p 1,2 u 1,2 t

p • Agent 1 cannot tell s and t apart • Agent 2 cannot tell s and u apart K1 = {(s,s),(s,t),(t,s),(t,t),(u,u)} K2 = {(s,s),(s,u),(u,s),(t,t),(u,u)} 1,2 s 2 1 p ¬ p 1,2 u 1,2 t

Coordinated Attack Problem • Two 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.

Coordinated Attack Problem • KB p • KA KB p • KB KA KB p Only depth of knowledge is increasing. Common knowledge is never attained!

Modeling Multi-agent Systems • Global 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

Some definitions • Run: 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).

Unbounded Message Delays • A 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 ireceives no messages at time m, and no process receives a message between times m and m´.

Unbounded Message Delays Theorem: 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.

Interpreted System • An interpreted system I consists of a pair (R, π), where R is a system and πis an interpretation for the propositions in Φ which assigns truth values to the primitive propositions at the global states.

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.

Є-Common Knowledge • Within Єunits everyone knows that within Єtime units everyone knows that… • Just as common knowledge corresponds to simultaneous coordination, Єcommon knowledge corresponds to coordinating to within Єtime 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

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

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

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.

Thank You

Reasoning About the Knowledge of Multiple Agents