ATL, Game Theory and Argumentation Jrgen Dix (joint work with N. - - PowerPoint PPT Presentation

ATL, Game Theory and Argumentation

Jürgen Dix (joint work with N. Bulling, C. Chesnevar, W. Jamroga)

Department of Computer Science Clausthal University of Technology

23rd April, Luxembourg

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results

ATL: What Agents Can Achieve

ATL: Agent Temporal Logic [Alur et al. 1997] Extension of CTL: Temporal logic meets game theory

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results

ATL: What Agents Can Achieve

ATL: Agent Temporal Logic [Alur et al. 1997] Extension of CTL: Temporal logic meets game theory Main idea: usual temporal operators:

❥

, , U plus cooperation modalities A

• A

Φ stands for coalition A has a collective strategy to enforce Φ

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results

ATL: What Agents Can Achieve

ATL: Agent Temporal Logic [Alur et al. 1997] Extension of CTL: Temporal logic meets game theory Main idea: usual temporal operators:

❥

, , U plus cooperation modalities A

• A

Φ stands for coalition A has a collective strategy to enforce Φ What exactly is a strategy?

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results

Vanilla ATL

Full ATL (denoted by ATL∗) is too complex:

Strategies with memory: 2EXPTIME Strategies without memory: PSPACE

“Vanilla” ATL: temporal operators are preceded by exactly one cooperation modality:

• A

❥ Φ, B Φ, B U Φ. Vanilla ATL suffices for most purposes.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results

Example: Rocket and Cargo

Rocket: moves between London (roL) and Paris (roP), Cargo: in London (caL), in Paris (caP), or in rocket (caR). Rocket can be moved only if it has its fuel tank full (fuelOK), When it moves, it consumes fuel, and nofuel holds after each flight. 3 agents, x can load the cargo, unload it, and move the rocket, y can unload the cargo and move the rocket, z can load the cargo and tank the rocket (action fuel), Each agent can also decide to do nothing (nop: “no-operation”);

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ATLI: Classical Results Models: CGS

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results Models: CGS

Rocket Example: Adding Agents and Actions

nofuel roL caR fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK

1 5 6 2 3 4 8 7 9 10 12 11

roL roP roL roL roL roL roP roP roP roP roP caL caL caL caL caR caR caR caP caP caP caP

< > load ,nop ,fuel

1 2

< > unload ,unload ,fuel

1 2

< > nop ,nop ,nop

1 2 3

< > load ,unload ,nop

1 2 3

< > nop ,unload ,load

1 2 3

< > unload ,unload ,nop

1 2 3

< > unload ,nop ,nop

1 2 3

< > unload ,nop ,fuel

1 2

< > load ,unload ,fuel

1 2

< > nop ,nop ,fuel

1 2

< > nop ,unload ,fuel

1 2

< > nop ,nop ,load

1 2 3

< > load ,nop ,load

1 2 3

< > load ,unload ,load

1 2 3

< > load ,nop ,nop

1 2 3

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ATLI: Classical Results Models: CGS

Models: Concurrent Game Structures (CGS)

M = Agt, Q, Π, π, Act, d, o, Agt: a finite set of all agents, |Agt| = k, Q: a set of states, |Q| = n, Π: a set of atomic propositions, π : Q → P(Π): a valuation of propositions, Act: a finite set of (atomic) actions, d : Agt × Q → P(Act) defines actions available to an agent,

• : a det. transition function that assigns outcome states

q′ = o(q, α1, . . . , αk) to states and tuples of actions. What is the size of a CGS? # transitions= m vs. # states = n.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results Models: CGS

Models: Concurrent Game Structures (CGS)

M = Agt, Q, Π, π, Act, d, o, Agt: a finite set of all agents, |Agt| = k, Q: a set of states, |Q| = n, Π: a set of atomic propositions, π : Q → P(Π): a valuation of propositions, Act: a finite set of (atomic) actions, d : Agt × Q → P(Act) defines actions available to an agent,

• : a det. transition function that assigns outcome states

q′ = o(q, α1, . . . , αk) to states and tuples of actions. What is the size of a CGS? # transitions= m vs. # states = n.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results Models: CGS

Strategies and Paths

A strategy is a conditional plan

sa : Q → Act (imperfect recall: ATLIr) sa : Q∗ → Act (perfect recall: ATLIR)

For vanilla ATL: ATLIr = ATLIR. For full ATL: ATLIr = ATLIR.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results Models: CGS

Strategies and Paths

A strategy is a conditional plan

sa : Q → Act (imperfect recall: ATLIr) sa : Q∗ → Act (perfect recall: ATLIR)

For vanilla ATL: ATLIr = ATLIR. For full ATL: ATLIr = ATLIR. A path is an infinite sequence of states that can be affected by subsequent transitions. Paths refer to possible courses of action. SA = sa1, sa2, . . . , sar is a collective strategy for A = {a1, a2, . . . , ar}. Function out(q, SA) returns the set of all paths that may result from agents A executing strategy SA from state q

• nward.

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ATLI: Classical Results Models: CGS

Rocket Agents

nofuel roL caR fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK

1 5 6 2 3 4 8 7 9 10 12 11

roL roP roL roL roL roL roP roP roP roP roP caL caL caL caL caR caR caR caP caP caP caP

< > load ,nop ,fuel

1 2

< > unload ,unload ,fuel

1 2

< > nop ,nop ,nop

1 2 3

< > load ,unload ,nop

1 2 3

< > nop ,unload ,load

1 2 3

< > unload ,unload ,nop

1 2 3

< > unload ,nop ,nop

1 2 3

< > unload ,nop ,fuel

1 2

< > load ,unload ,fuel

1 2

< > nop ,nop ,fuel

1 2

< > nop ,unload ,fuel

1 2

< > nop ,nop ,load

1 2 3

< > load ,nop ,load

1 2 3

< > load ,unload ,load

1 2 3

< > load ,nop ,nop

1 2 3

nofuel → 3 nofuel

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results Models: CGS

nofuel roL caR fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK

1 5 6 2 3 4 8 7 9 10 12 11

roL roP roL roL roL roL roP roP roP roP roP caL caL caL caL caR caR caR caP caP caP caP

< > load ,nop ,fuel

1 2

< > unload ,unload ,fuel

1 2

< > nop ,nop ,

1 2 nop3

< > load ,unload ,

1 2 nop3

< > nop ,unload ,load

1 2 3

< > unload ,unload ,

1 2 nop3

< > unload ,nop ,

1 2 nop3

< > unload ,nop ,fuel

1 2

< > load ,unload ,fuel

1 2

< > nop ,nop ,fuel

1 2

< > nop ,unload ,fuel

1 2

< > nop ,nop ,load

1 2 3

< > load ,nop ,load

1 2 3

< > load ,unload ,load

1 2 3

< > load ,nop ,

1 2 nop3

nofuel → 3 nofuel

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results Models: CGS

nofuel roL caR fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK

1 5 6 2 3 4 8 7 9 10 12 11

roL roP roL roL roL roL roP roP roP roP roP caL caL caL caL caR caR caR caP caP caP caP

< > load ,nop ,fuel

1 2

< > unload ,unload ,fuel

1 2

< > nop ,nop ,nop

1 2 3

< > load ,unload ,nop

1 2 3

< > nop ,unload ,load

1 2 3

< > unload ,unload ,nop

1 2 3

< > unload ,nop ,nop

1 2 3

< > unload ,nop ,fuel

1 2

< > load ,unload ,fuel

1 2

< > nop ,nop ,fuel

1 2

< > nop ,unload ,fuel

1 2

< > nop ,nop ,load

1 2 3

< > load ,nop ,load

1 2 3

< > load ,unload ,load

1 2 3

< > load ,nop ,nop

1 2 3

caL → 1, 3 ♦caP caL → ¬ 1 ♦caP

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results Model Checking wrt m

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results Model Checking wrt m

Model Checking ATLI: wrt m=# of transitions.

Model checking: Does ϕ hold in model M (CGS) and state q? Nice results: model checking ATL is tractable! Perfect = imperfect recall: ATLIr = ATLIR.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATLI: Classical Results Model Checking wrt m

Model Checking ATLI: wrt m=# of transitions.

Model checking: Does ϕ hold in model M (CGS) and state q? Nice results: model checking ATL is tractable! Perfect = imperfect recall: ATLIr = ATLIR.

Theorem (Alur, Kupferman & Henzinger 1998)

ATLIR (resp. ATLIr) model checking is P-complete, and can be done in time linear in the size of the model and the length of the formula.

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Complexity wrt n

Complexity wrt n

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Complexity wrt n

Model Checking ATL (wrt n= # states) m: transitions, n: states, d: actions, k: agents. How does m depend on n and k?

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n

Model Checking ATL (wrt n= # states) m: transitions, n: states, d: actions, k: agents. How does m depend on n and k? m = O(ndk)

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n

Model Checking ATL (wrt n= # states) m: transitions, n: states, d: actions, k: agents. How does m depend on n and k? m = O(ndk) m is not polynomially bounded in n when agents are present.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n

Model Checking ATL (wrt n= # states) m: transitions, n: states, d: actions, k: agents. How does m depend on n and k? m = O(ndk) m is not polynomially bounded in n when agents are present. Agents make models explode!

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n

Model Checking ATL (wrt n= # states) m: transitions, n: states, d: actions, k: agents. How does m depend on n and k? m = O(ndk) m is not polynomially bounded in n when agents are present. Agents make models explode! Do agents make model checking explode?

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n

First Result

ΣP

i = NPΣP

i−1: problems solvable in pol. time by a

non-deterministic TM making queries to a ΣP

i−1 oracle

∆P

i = PΣP

i−1: problems solvable in pol. time by a

deterministic TM making adaptive queries to a ΣP

i−1 oracle

ΣP

2 = NPNP

∆P

3 = P[NPNP]

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n

First Result

ΣP

i = NPΣP

i−1: problems solvable in pol. time by a

non-deterministic TM making queries to a ΣP

i−1 oracle

∆P

i = PΣP

i−1: problems solvable in pol. time by a

deterministic TM making adaptive queries to a ΣP

i−1 oracle

ΣP

2 = NPNP

∆P

3 = P[NPNP]

Proposition

Model checking ATLIR is ∆P

3 -complete wrt the number of states

(n), decisions (d) and agents (k) in the model, and the length of the formula (l). For positive ATLIR, model checking is ΣP

2 -complete.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n ATLi: Models CGES

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n ATLi: Models CGES

Example: Robots and Carriage

1 2 1 2 1 1 2 2

pos0 pos1 pos2

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n ATLi: Models CGES

The CGS model

1 2 1 2 1 1 2 2

pos0 pos1 pos2

q0 q2 q1

pos0 pos1

wait,wait wait,wait wait,wait push,push push,push push,push push,wait push,wait wait,push push,wait wait,push wait,push

pos2

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n ATLi: Models CGES

ATL with perfect Information

1 2 1 1 2 2 1 2

pos0 pos1 pos2 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n ATLi: Models CGES

ATL with imperfect Information

1 2 1 1 2 2 1 2

pos0 pos1 pos2

1 2 1 1 2 2 1 2

pos0 pos1 pos2 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n ATLi: Models CGES

ATLir: ATL with Imperfect Information

1 2 1 2 1 1 2 2

pos0 pos1 pos2

q0 q2 q1

pos0 pos1

wait,wait wait,wait wait,wait push,push push,push push,push push,wait push,wait push,wait wait,push

pos2

wait,push wait,push

1 2

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Complexity wrt n ATLi: Models CGES

From CGS to CEGS

Memoryless strategies: We extend CGS by epistemic relations ∼a, one per agent: we obtain CEGS. Uniform strategies per agent: q ∼a q′ ⇒ sa(q) = sa(q′) Uniform strategies for group of agents: q ∼A q′ ⇒ sa(q) = sa(q′), where q ∼A q′ is defined by there is an agent a ∈ A with q ∼a q′

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Complexity wrt n ATLi: Models CGES

From CGS to CEGS

Memoryless strategies: We extend CGS by epistemic relations ∼a, one per agent: we obtain CEGS. Uniform strategies per agent: q ∼a q′ ⇒ sa(q) = sa(q′) Uniform strategies for group of agents: q ∼A q′ ⇒ sa(q) = sa(q′), where q ∼A q′ is defined by there is an agent a ∈ A with q ∼a q′

Strategies with memory

For Q∗ → Act, definitions above are appropriately modified.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n ATLi: Models CGES

ATLir and ATLiR

M, q | = A ir ✐ ϕ iff there exists uniform SA such that, for every path Λ ∈

q′∼Aq out(q′, SA), we have M, Λ[1] |

= ϕ; M, q | = A irϕ iff there exists uniform SA such that, for every Λ ∈

q′∼Aq out(q′, SA), we have M, Λ[i] |

= ϕ for every i ≥ 0; M, q | = A irϕ U ψ iff there exists uniform SA such that, for every Λ ∈

q′∼Aq out(q′, SA), we have M, Λ[i] |

= ψ for some i ≥ 0, and M, Λ[j] | = ϕ for every 0 ≤ j < i.

What about strategies with memory (ATLiR)?

Instead of equivalences q′ ∼A q, one has to consider sequences q′ ∼A q.

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Complexity wrt n ATLi: Models CGES

Example: Robots and Carriage

1 2 1 2 1 1 2 2

pos0 pos1 pos2

• 1

ir¬pos1 ?

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n ATLi: Models CGES

Example: Robots and Carriage

1 2 1 2 1 1 2 2

pos0 pos1 pos2

¬ 1 ir¬pos1

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Complexity wrt n ATLi: Models CGES

Example: Robots and Carriage

1 2 1 2 1 1 2 2

pos0 pos1 pos2

¬ 1 ir¬pos1

• 2

ir¬pos1 ?

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n ATLi: Models CGES

Example: Robots and Carriage

1 2 1 2 1 1 2 2

pos0 pos1 pos2

¬ 1 ir¬pos1 ¬ 2 ir¬pos1 Why not 2 ir¬pos1 by using the following strategy for agent 2: “push” when in q0 and “wait” when in q2? This not a feasible strategy, because it does not work in q1.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Complexity Results for Strategic Logics

m, l n, k, l CTL P-complete [1] P-complete [1] ATLIr P-complete [3] ATLir

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Complexity Results for Strategic Logics

m, l n, k, l CTL P-complete [1] P-complete [1] ATLIr P-complete [3] ATLir

[1] Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems .... ACM Prog. Lang. Syst [3] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic. J. ACM Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Complexity Results for Strategic Logics

m, l n, k, l CTL P-complete [1] P-complete [1] ATLIr P-complete [3] ∆P

3 -complete [5,6]

ATLir

[1] Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems .... ACM Prog. Lang. Syst [3] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic. J. ACM [5] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [6] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Complexity Results for Strategic Logics

m, l n, k, l nlocal, k, l CTL P-complete [1] P-complete [1] ATLIr P-complete [3] ∆P

3 -complete [5,6]

ATLir ∆P

2 -complete [4,7]

∆P

3 -complete [7]

[1] Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems .... ACM Prog. Lang. Syst [3] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic. J. ACM [4] Schobbens (2004). Alternating-time logic with imperfect recall. ENTCS [5] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [6] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [7] Jamroga & Dix (2008). Model checking abilities of agents. Theory of Computing systems Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Complexity Results for Strategic Logics

m, l n, k, l nlocal, k, l CTL P-complete [1] P-complete [1] ATLIr P-complete [3] ∆P

3 -complete [5,6]

ATLir ∆P

2 -complete [4,7]

∆P

3 -complete [7]

[1] Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems .... ACM Prog. Lang. Syst [3] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic. J. ACM [4] Schobbens (2004). Alternating-time logic with imperfect recall. ENTCS [5] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [6] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [7] Jamroga & Dix (2008). Model checking abilities of agents. Theory of Computing systems Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Complexity Results for Strategic Logics

m, l n, k, l nlocal, k, l CTL P-complete [1] P-complete [1] PSPACE-complete [2] ATLIr P-complete [3] ∆P

3 -complete [5,6]

ATLir ∆P

2 -complete [4,7]

∆P

3 -complete [7]

[1] Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems .... ACM Prog. Lang. Syst [2] Kupferman, Vardi & Wolper (2000). An automata-theoretic approach to .... J. ACM [3] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic. J. ACM [4] Schobbens (2004). Alternating-time logic with imperfect recall. ENTCS [5] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [6] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [7] Jamroga & Dix (2008). Model checking abilities of agents. Theory of Computing systems Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Complexity Results for Strategic Logics

m, l n, k, l nlocal, k, l CTL P-complete [1] P-complete [1] PSPACE-complete [2] ATLIr P-complete [3] ∆P

3 -complete [5,6] EXPTIME-complete [8]

ATLir ∆P

2 -complete [4,7]

∆P

3 -complete [7]

[1] Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems .... ACM Prog. Lang. Syst [2] Kupferman, Vardi & Wolper (2000). An automata-theoretic approach to .... J. ACM [3] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic. J. ACM [4] Schobbens (2004). Alternating-time logic with imperfect recall. ENTCS [5] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [6] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [7] Jamroga & Dix (2008). Model checking abilities of agents. Theory of Computing systems [8] Hoek, Lomuscio & Wooldridge (AAMAS 2006). On the complexity of practical ATL model checking. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Complexity Results for Strategic Logics

m, l n, k, l nlocal, k, l CTL P-complete [1] P-complete [1] PSPACE-complete [2] ATLIr P-complete [3] ∆P

3 -complete [5,6] EXPTIME-complete [8]

ATLir ∆P

2 -complete [4,7]

∆P

3 -complete [7]

PSPACE-complete [9]

[1] Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems .... ACM Prog. Lang. Syst [2] Kupferman, Vardi & Wolper (2000). An automata-theoretic approach to .... J. ACM [3] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic. J. ACM [4] Schobbens (2004). Alternating-time logic with imperfect recall. ENTCS [5] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [6] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [7] Jamroga & Dix (2008). Model checking abilities of agents. Theory of Computing systems [8] Hoek, Lomuscio & Wooldridge (AAMAS 2006). On the complexity of practical ATL model checking. [9] Jamroga & Ågotnes (AAMAS 2007). Modular Interpreted Systems Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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Complexity wrt n Model Checking ATLi and ATLI

Last Column: meaning of nlocal

MIS A modular interpreted system (MIS) is of the form Agt, Act, In where each agent ai has the following internal structure ai = Sti, di, outi, ini, oi, Πi, πi. Set of global states is the cartesian product of the Sti: n = n1 · . . . · nk.

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Complexity wrt n Model Checking ATLi and ATLI

Last Column: meaning of nlocal

MIS A modular interpreted system (MIS) is of the form Agt, Act, In where each agent ai has the following internal structure ai = Sti, di, outi, ini, oi, Πi, πi. Set of global states is the cartesian product of the Sti: n = n1 · . . . · nk. A MIS viewed as a CGS (for ATL) is very succinct. For ATLir, we have in addition to CGS all the local epistemic relations ∼1, . . . , ∼k (CEGS). A MIS viewed as a CEGS (for ATLir) does not compress that much.

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Complexity wrt n The case ATLiR

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

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Complexity wrt n The case ATLiR

What about ATLiR?

ATLiR: Imperfect information, perfect recall. Undecidable: we do not yet have a formal proof! We have not found any result in the literature which directly implies the undecidability of ATLiR. Why is it undecidable?

q0 q1

p

q2

r α β M, q0 | = 1 IR(♦p ∧ ♦r) M, q0 | = 1 Ir(♦p ∧ ♦r)

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Complexity wrt n The case ATLiR

References

Joint work with Wojtek Jamroga. W.Jamroga and J.Dix Model checking abilities of agents. Theory of Computing Systems, 42 (3), 366–410, 2008.

• W. Jamroga and J. Dix.

Do agents make model checking explode? In Proceedings of CEEMAS’05, LNAI 3690, pages 398–407, 2005.

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ATL + Plausibility

ATL + Plausibility

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ATL + Plausibility

Agents usually act rationally!

We would like to extend ATL with a notion of plausibility, reason about rational behavior of agents, have a logic that can express any solution concept, compare different game theoretic solution concepts.

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ATL + Plausibility

Plausibility concept

ATL: Reasoning about all possible behaviors. ATLP: Reasoning about plausible behaviors.

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ATL + Plausibility

Plausibility concept

ATL: Reasoning about all possible behaviors.

• A

ϕ: Agents A have a collective strategy to enforce ϕ against any response of their opponents.

ATLP: Reasoning about plausible behaviors.

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ATL + Plausibility

Plausibility concept

ATL: Reasoning about all possible behaviors.

• A

ϕ: Agents A have a collective strategy to enforce ϕ against any response of their opponents.

ATLP: Reasoning about plausible behaviors.

Pl A ϕ: Agents A have a plausible collective strategy to enforce ϕ against any plausible response of their opponents. Example: Playing undominated strategies is often plausible,...

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ATL + Plausibility Base Logic

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

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ATL + Plausibility Base Logic

The Base Logic: Lbase

ATLP

Definition (Lbase

ATLP)

The language Lbase

ATLP is defined over nonempty sets:

Π of propositions, p ∈ Π, Agt of agents, a ∈ Agt, A ⊆ Agt, and Ω of basic plausibility terms, ω ∈ Ω . ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | A ✐ ϕ | A ϕ | A ϕ U ϕ | Pl Aϕ | (set-pl ω)ϕ

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ATL + Plausibility Base Logic

The Base Logic: Lbase

ATLP

Definition (Lbase

ATLP)

The language Lbase

ATLP is defined over nonempty sets:

Π of propositions, p ∈ Π, Agt of agents, a ∈ Agt, A ⊆ Agt, and Ω of basic plausibility terms, ω ∈ Ω . ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | A ✐ ϕ | A ϕ | A ϕ U ϕ | Pl Aϕ | (set-pl ω)ϕ M, q | = Pl B A γ if A can enforce γ, when agents in B play only plausible strategies

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ATL + Plausibility Base Logic

Plausibility Terms

Ω: Set of basic plausibility terms, ω ∈ Ω Hard-wired sets of strategies: ωNE: Nash equilibria ωPO : Pareto optimal strategies How to activate?

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ATL + Plausibility Base Logic

Plausibility Terms

Ω: Set of basic plausibility terms, ω ∈ Ω Hard-wired sets of strategies: ωNE: Nash equilibria ωPO : Pareto optimal strategies How to activate? (set-pl ω) : Sets plausible strategies to [ [ω] ] ⊆ Σ

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Base Logic

Plausibility Terms

Ω: Set of basic plausibility terms, ω ∈ Ω Hard-wired sets of strategies: ωNE: Nash equilibria ωPO : Pareto optimal strategies How to activate? (set-pl ω) : Sets plausible strategies to [ [ω] ] ⊆ Σ And where do the terms come from?

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ATL + Plausibility Base Logic

How to describe strategies?

Plausibility terms: abstract labels, no structure! Idea: Formulas that describe plausible strategies! Select all s such that s is better than any other strategy s′ Complex plausibility terms: ω = σ. ϕ(σ)

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ATL + Plausibility Base Logic

How to describe strategies?

Plausibility terms: abstract labels, no structure! Idea: Formulas that describe plausible strategies! Complex plausibility terms: ω = σ. ϕ(σ)

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Base Logic

How to describe strategies?

Plausibility terms: abstract labels, no structure! Idea: Formulas that describe plausible strategies! Complex plausibility terms: ω = σ. ϕ(σ)

• Property that σ should fulfill

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Base Logic

How to describe strategies?

Plausibility terms: abstract labels, no structure! Idea: Formulas that describe plausible strategies! Complex plausibility terms: ω = σ.∀σ1∃σ2 . . . ∀σn ϕ(σ, σ1, . . . σn)

• ∈Lbase

ATLP(Ω∪{σ,σ1,...,σn}) Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Base Logic

How to describe strategies?

Plausibility terms: abstract labels, no structure! Idea: Formulas that describe plausible strategies! Complex plausibility terms: ω = σ.∀σ1∃σ2 . . . ∀σn ϕ(σ, σ1, . . . σn)

• ∈Lbase

ATLP(Ω∪{σ,σ1,...,σn})

Example: ωDOM = σ.∀σ′ (σ better than σ′)

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Base Logic

How to describe strategies?

Plausibility terms: abstract labels, no structure! Idea: Formulas that describe plausible strategies! Complex plausibility terms: ω = σ.∀σ1∃σ2 . . . ∀σn ϕ(σ, σ1, . . . σn)

• ∈Lbase

ATLP(Ω∪{σ,σ1,...,σn})

Example: ωDOM = σ.∀σ′ (σ better than σ′) How to determine whether a strategy is good?

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ATL + Plausibility Models: CGSP

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

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ATL + Plausibility Models: CGSP

Some Game Theory

NF games: Normal Form: Players move simultanously. EF games: Extensive Form: Alternate moves. Moves can depend on the whole history. This is a tree.

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ATL + Plausibility Models: CGSP

Some Game Theory

NF games: Normal Form: Players move simultanously. EF games: Extensive Form: Alternate moves. Moves can depend on the whole history. This is a tree. EF NF: One can easily transform a EF game into a NF game. EF CGS: Each EF game can be modelled as a CGS. CGS EF: CGS can have cycles or simultaneous moves.

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ATL + Plausibility Models: CGSP

Some Game Theory

NF games: Normal Form: Players move simultanously. EF games: Extensive Form: Alternate moves. Moves can depend on the whole history. This is a tree. EF NF: One can easily transform a EF game into a NF game. EF CGS: Each EF game can be modelled as a CGS. CGS EF: CGS can have cycles or simultaneous moves. We want to define CGSP that correspond to NF games.

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ATL + Plausibility Models: CGSP

Concurrent game structures with plausibility

M = (Agt, Q, Π, π, Act, d, δ, Υ, Ω, [ [·] ]) Υ ⊆ Σ: set of (plausible) strategy profiles Example: Υ = {(head, head)} Ω = {ω1, ω2, . . . }: set of plausibility terms Example: ωNE stands for all Nash equilibria [ [·] ] : Q → (Ω → P(Σ)): plausibility mapping, it assigns a set

• f strategy profiles to each state and plausibility term

Example: [ [ωNE] ]q = {(head, head), (tail, tail)}

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Models: CGSP

Concurrent game structures with plausibility

M = (Agt, Q, Π, π, Act, d, δ, Υ, Ω, [ [·] ]) Υ ⊆ Σ: set of (plausible) strategy profiles Example: Υ = {(head, head)} Ω = {ω1, ω2, . . . }: set of plausibility terms Example: ωNE stands for all Nash equilibria [ [·] ] : Q → (Ω → P(Σ)): plausibility mapping, it assigns a set

• f strategy profiles to each state and plausibility term

Example: [ [ωNE] ]q = {(head, head), (tail, tail)}

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Models: CGSP

Concurrent game structures with plausibility

M = (Agt, Q, Π, π, Act, d, δ, Υ, Ω, [ [·] ]) Υ ⊆ Σ: set of (plausible) strategy profiles Example: Υ = {(head, head)} Ω = {ω1, ω2, . . . }: set of plausibility terms Example: ωNE stands for all Nash equilibria [ [·] ] : Q → (Ω → P(Σ)): plausibility mapping, it assigns a set

• f strategy profiles to each state and plausibility term

Example: [ [ωNE] ]q = {(head, head), (tail, tail)}

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Models: CGSP

Concurrent game structures with plausibility

M = (Agt, Q, Π, π, Act, d, δ, Υ, Ω, [ [·] ]) Υ ⊆ Σ: set of (plausible) strategy profiles Example: Υ = {(head, head)} Ω = {ω1, ω2, . . . }: set of plausibility terms Example: ωNE stands for all Nash equilibria [ [·] ] : Q → (Ω → P(Σ)): plausibility mapping, it assigns a set

• f strategy profiles to each state and plausibility term

Example: [ [ωNE] ]q = {(head, head), (tail, tail)}

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ATL + Plausibility Models: CGSP

General Solution Concepts: CGSP

Idea: Agents have preferences: η = η1, . . . , ηk ηi: ATL path formulæ (payoff) Example: η2 = ♦money2

q0

start

q1

money1

q2

money2

q3

money1 money2

q4 q5

money2 hh tt ht th hh ht th tt hh ht th tt

nn nn nn

No payoffs needed as for classical solution concepts!

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ATL + Plausibility Models: CGSP

General Solution Concepts: CGSP

Idea: Agents have preferences: η = η1, . . . , ηk ηi: ATL path formulæ (payoff) Example: η2 = ♦money2

q0

start

q1

money1

q2

money2

q3

money1 money2

q4 q5

money2 hh tt ht th hh ht th tt hh ht th tt

nn nn nn

No payoffs needed as for classical solution concepts!

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ATL + Plausibility Models: CGSP

General Solution Concepts: CGSP

Idea: Agents have preferences: η = η1, . . . , ηk ηi: ATL path formulæ (payoff) Example: η2 = ♦money2

CGSP +preferences normal form game

Each CGSP M with η corresponds to a normal form game S.

q0 start q1

money1

q2

money2

q3

money1 money2

q4 q5

money2 hh tt ht th hh ht th tt hh ht th tt

nn n n n n

• η1\η2

shh sht sth stt shh 1, 1 0, 0 0, 1 0, 1 sht 0, 0 0, 1 0, 1 0, 1 sth 0, 1 0, 1 1, 1 0, 0 stt 0, 1 0, 1 0, 0 0, 1

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ATL + Plausibility Models: CGSP

General Solution Concepts: CGSP

Idea: Agents have preferences: η = η1, . . . , ηk ηi: ATL path formulæ (payoff) Example: η2 = ♦money2

CGSP +preferences normal form game

Each CGSP M with η corresponds to a normal form game S.

q0 start q1

money1

q2

money2

q3

money1 money2

q4 q5

money2 hh tt ht th hh ht th tt hh ht th tt

nn n n n n

• η1\η2

shh sht sth stt shh 1, 1 0, 0 0, 1 0, 1 sht 0, 0 0, 1 0, 1 0, 1 sth 0, 1 0, 1 1, 1 0, 0 stt 0, 1 0, 1 0, 0 0, 1

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ATL + Plausibility Models: CGSP

Characterizing Solution Concepts

NE

η(σ):

• a∈Agt BR

η a(σ)

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ATL + Plausibility Models: CGSP

Characterizing Solution Concepts

BR

η a(σ):

(set-pl σ[Agt\{a}])Pl Agt

• a

ηa → (set-pl σ) ∅ ηa

• NE

η(σ):

• a∈Agt BR

η a(σ)

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ATL + Plausibility Models: CGSP

Characterizing Solution Concepts

BR

η a(σ):

(set-pl σ[Agt\{a}])Pl Agt

• a

ηa → (set-pl σ) ∅ ηa

• NE

η(σ):

• a∈Agt BR

η a(σ)

SPN

η(σ):

• ∅

NE

η(σ)

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ATL + Plausibility Models: CGSP

Characterizing Solution Concepts

BR

η a(σ):

(set-pl σ[Agt\{a}])Pl Agt

• a

ηa → (set-pl σ) ∅ ηa

• NE

η(σ):

• a∈Agt BR

η a(σ)

SPN

η(σ):

• ∅

NE

η(σ)

PO

η(σ):

∀σ′ Pl Agt

a∈Agt((set-pl σ′)

∅ ηa → (set-pl σ) ∅ ηa)∨

• a∈Agt((set-pl σ)

∅ ηa ∧ ¬(set-pl σ′) ∅ ηa

• .

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ATL + Plausibility Models: CGSP

Characterizing Solution Concepts

BR

η a(σ):

(set-pl σ[Agt\{a}])Pl Agt

• a

ηa → (set-pl σ) ∅ ηa

• NE

η(σ):

• a∈Agt BR

η a(σ)

SPN

η(σ):

• ∅

NE

η(σ)

PO

η(σ):

∀σ′ Pl Agt

a∈Agt((set-pl σ′)

∅ ηa → (set-pl σ) ∅ ηa)∨

• a∈Agt((set-pl σ)

∅ ηa ∧ ¬(set-pl σ′) ∅ ηa

• .

Theorem

These notions correspond to those in game theory: [ [σ.NE

η(σ)]

]q

M = NE strategies in S(M,

η, q). Similarly for SPN and PO and UNDOM.

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ATL + Plausibility Models: CGSP

Example

Both agents play without restrictions: M, q0 | = ¬ a2 ♦money2 Both agents play a Nash equilibrium strategy: M, q0 | = (set-pl σ.NEη(σ))Pl Agt a2 ♦money2

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ATL + Plausibility Models: CGSP

The Full Language: LATLP

Plausibility terms: σ.∀σ1∃σ2 . . . ∀σn ϕ where ϕ ∈ Lbase

ATLP

What about nesting (set-pl ·) operators? (set-pl . . . (set-pl . . . (set-pl . . .) . . .) . . .)

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ATL + Plausibility Models: CGSP

The Full Language: LATLP

Plausibility terms: σ.∀σ1∃σ2 . . . ∀σn ϕ where ϕ ∈ Lbase

ATLP

What about nesting (set-pl ·) operators? (set-pl . . . (set-pl . . . (set-pl . . .) . . .) . . .) We get a hierarchy of logics: Lk

ATLP: k nestings

LATLP := limk→∞ Lk

ATLP

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Model Checking

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Plausibility Model Checking

Model Checking Complexity

Theorem Model checking Lbase

ATLP is ∆P 3 -complete.

This is in the line with game theoretical results!

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ATL + Plausibility Model Checking

Model Checking Complexity

Expressivity vs. Complexity 1 2 . . . i . . . unbounded L1

ATLP

∆P

3

∆P

4

∆P

5

. . . ∆P

i+3

. . . PSPACE L2

ATLP

∆P

4

∆P

6

∆P

7

. . . ∆P

5+i−max{0,1−i}

. . . PSPACE . . . . . . . . . Lk

ATLP

i > k + 1

∆P

k+2

∆P

k+4

∆P

k+6

. . . ∆P

i+2k+1−max{0,k−i−1}

. . . PSPACE

For particular well-behaved models, model checking is even polynomial.

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ATL + Plausibility Model Checking

References

Joint work with Wojtek Jamroga and Nils Bulling

• N. Bulling and W.Jamroga and J.Dix

Reasoning about Temporal Properties of Rational Play Annals of Mathematics and AI, 53 (1), 2009.

• W. Jamroga and N. Bulling.

A framework for reasoning about rational agents. In Proceedings of AAMAS’07, pages 592–594, 2007.

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ATL + Coalition Formation

ATL + Coalition Formation

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ATL + Coalition Formation

Motivation

ATL: A γ Group A of agents can enforce property γ. Where does A come from? Is it reasonable to assume that these agents work together? Idea: Focus on reasonable coalitions ATLc: |A| γ A is able to form a reasonable coalition which enforces γ.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Coalition Formation

Motivation

ATL: A γ Group A of agents can enforce property γ. Where does A come from? Is it reasonable to assume that these agents work together? Idea: Focus on reasonable coalitions ATLc: |A| γ A is able to form a reasonable coalition which enforces γ.

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

50/60

ATL + Coalition Formation ATLC

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

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ATL + Coalition Formation ATLC

How to model conflicts?

Based on Amgoud (2005):

Definition (Coalitional framework)

A coalitional framework is a tuple cf = (C, A) where C: non-empty set of elements A ⊆ C × C: attack or defeat relation CF(Agt): coalitional frameworks over Agt a1 a2 a3

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ATL + Coalition Formation ATLC

How to determine coalitions?

Definition (Coalitional framework semantics)

A semantics is a mapping sem : CF(C) → P(P(C))

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ATL + Coalition Formation ATLC

How to determine coalitions?

Definition (Coalitional framework semantics)

A semantics is a mapping sem : CF(C) → P(P(C)) a1 a2 a3 sem : {{a1, a2}, {a1}} Is {a1, a2} sensible?

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Coalition Formation ATLC

How to determine coalitions?

Definition (Coalitional framework semantics)

A semantics is a mapping sem : CF(C) → P(P(C)) Consider for example Stable Coalitions: Let C ⊆ C. C is called stable coalition iff C is conflict-free and it defeats all elements not in C. a1 a2 . . . an a′

1

a′

2

. . . a′

k

C

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ATL + Coalition Formation ATLC

Coalitional ATL: Syntax

Definition (ATLc)

The language LATLc is defined over nonempty sets: Π of propositions Agt of agents ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | A ✐ ϕ | A ϕ | A ϕ U ϕ |

• |A|

❥ ϕ | |A| ϕ | |A| ϕ U ϕ

Semantics

M, q | = |A| γ iff there is a reasonable coalition B (wrt A) which can enforce γ

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ATL + Coalition Formation ATLC

Coalitional ATL: Syntax

Definition (ATLc)

The language LATLc is defined over nonempty sets: Π of propositions Agt of agents ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | A ✐ ϕ | A ϕ | A ϕ U ϕ |

• |A|

❥ ϕ | |A| ϕ | |A| ϕ U ϕ

Semantics

M, q | = |A| γ iff there is a reasonable coalition B (wrt A) which can enforce γ

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Coalition Formation Models: CCGS

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

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ATL + Coalition Formation Models: CCGS

Coalitional concurrent game structures

M = Agt, Q, Π, π, Act, d, o, ζ, sem ζ : Q → CF(Agt) sem: (argumentation) semantics

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ATL + Coalition Formation Models: CCGS

Coalitional concurrent game structures

M = Agt, Q, Π, π, Act, d, o, ζ, sem ζ : Q → CF(Agt) sem: (argumentation) semantics

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Coalition Formation Model Checking

Overview

1 ATLI: Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATLi: Models CGES Model Checking ATLi and ATLI The case ATLiR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATLC Models: CCGS Model Checking

Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg

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ATL + Coalition Formation Model Checking

Model Checking Complexity

Theorem Model checking ATLc is in ∆P

2 = PNP for standard

argumentation semantics (stable, preferred, admissible, . . .).

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ATL + Coalition Formation Model Checking

References

Joint work with Nils Bulling and Carlos Chesñevar.

• N. Bulling, C. Chesñevar, and J. Dix.

Modelling coalitions: ATL+argumentation. In Proceedings of AAMAS’08, Estoril, Portugal, May 2008. ACM Press. Revised Version in LNAI 5384, pp.190-211, 2009.

• N. Bulling and J. Dix.

A Finer grained Modelling of Rational Coalitions using GOALS. In Proceedings of the 14th Argentinean Conference on Computer Science CACIC’08, September 2008.

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ATL + Coalition Formation Model Checking

Thank you.

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