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*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)

259

Coalition Logic

We repeat some slides from MAIR

260

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)

MAIR2013 John-Jules Meyer 262

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

MAIR2013 John-Jules Meyer 263

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))

MAIR2013 John-Jules Meyer 264

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 }

MAIR2013 John-Jules Meyer 265

Logic

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

$ _________ $

MAIR2013 John-Jules Meyer 266

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)

MAIR2013 John-Jules Meyer 268

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

MAIR2013 John-Jules Meyer 269

(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)

MAIR2013 John-Jules Meyer 270

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)

MAIR2013 John-Jules Meyer 271

Sound & complete axiomatization PC (incl (MP)) [C]? [C]> [;] ! [AGT] [C]( ) ! [C] [C] [C1] [C2] ! [C1 [ C2] ( )

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