Multi-Agent Logics - Utrecht Logics Coalition Logic Alternating-time Logic (ATL) Epistemic Variants

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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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MAIR2013 John-Jules Meyer 261

Non-normal modal logic

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

( ( )) 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, , N i Where

S is a set of worlds 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 $ kkM 2 N(w)where kkM is the truth set of in M: kkM = {w | M,w }

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Logic

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

$ _________ $

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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] coalition C is able to ensure Semantics based on neighborhood models, with

extra properties M = h S, , E iwhere

S is a nonempty set of worlds/states 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 Y ) Y 2 E(w)(C) X 2 E(w)(C1) & Y 2 E(w)(C2) &

C1 C2 = ; ) X Y 2 E(w)(C1 [ C2)

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Interpretation

M,s [C] $ kkM 2 E(w)(C)

Intuition: E(w)(C) is the collection of sets X 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]> [;] ! [AGT] [C]( ) ! [C] [C] [C1] [C2] ! [C1 [ C2] ( )

if C1 C2 = ; $ / [C] $ [C]