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
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