Multi-Agent Logics

■  Coalition Logic■  Alternating-time Logic (ATL)■  Epistemic Variants (ATEL, ATEL-A)

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Coalition Logic

■   We repeat some slides from MAIR…

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Non-normal modal logic

■  We have seen that in normal modal logic (based on Kripke semantics) it holds that:

■   ² (¤ϕ ∧ ¤(ϕ → ψ)) → ¤ψ

■  If one does not want this property one should use non-normal modal logic (based on neighborhood semantics)

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Neighborhood semantics

■  Accessibility relation between a world and its accessible worlds is generalized to a relation between a world and a collection of sets of worlds

w

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“Minimal” Models

■   M = h S, ¼, N i■  Where

–  S is a set of worlds–   ¼ is a truth assignment function–  N is a function S ! P(P(S))

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Interpretation

■  The sets in the collection are called neighborhoods and represent propositions that are necessary in world w

■  So semantics:■  M,w ² ¤Á $ kÁkM 2 N(w)where kÁkM is the truth set of Á in M:■   kÁkM = {w | M,w ² Á}

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Logic

■  The following system is sound & complete w.r.t. the class of minimal models: PC (incl MP) +

Á $ à _________ ¤Á $ ¤Ã

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Extensions

■  Of course, this is a very weak logic, but it is possible again to put constraints on the models in order to get again properties such as D, T, 4, 5! (Cf. Chellas, Ch. 7)

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Coalition Logic

■  Operator [C]Á–   “coalition C is able to ensure Á”–   Semantics based on neighborhood models, with

extra properties–   M = h S, ¼, E iwhere

•   S is a nonempty set of worlds/states•   ¼ is a truth assignment function•   E : S ! (P(AGT) ! P(P(S))) effectivity function

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(Playable) effectivity function

■  E satisfies–   not ; 2 E(w)(C)–   S 2 E(w)(C)–   (not S\X 2 E(w)(;)) ) X 2 E(w)(AGT)–   X 2 E(w)(C) & X µ Y ) Y 2 E(w)(C)–   X 2 E(w)(C1) & Y 2 E(w)(C2) &

C1 Å C2 = ; ) X Å Y 2 E(w)(C1 [ C2)

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Interpretation

■   M,s ² [C]Á $ kÁkM 2 E(w)(C)

■  Intuition: E(w)(C) is the collection of sets X µ S such that C can force the world to be in some state of X (where X represents a proposition)

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Sound & complete axiomatization ■  PC (incl (MP))■   ¬[C]?■   [C]>■   ¬[;]¬Á ! [AGT]Á■   [C](Á Æ Ã) ! [C]Á Æ [C]Ã■   [C1]Á Æ [C2]à ! [C1 [ C2] (Á Æ Ã)

if C1 Å C2 = ;■   Á $ Ã / [C]Á $ [C]Ã