Model Checking for Coalition Announcement Logic Rustam GALIMULLIN 1 Natasha ALECHINA 1 Hans VAN DITMARSCH 2 1 University of Nottingham, Nottingham, UK 2 CNRS, LORIA, University of Lorraine, France & ReLaX, Chennai, India
What this talk is about What agents know and don’t know (through the lens of epistemic logic 1 ) The effect of public announcements 2 on agent’s knowledge How agents can achieve certain goals by teaming up in coalitions and making joint announcements 3 Model checking for such a framework 1 Hans van Ditmarsch et al., eds. Handbook of Epistemic Logic . College Publications, 2015. 2 Hans van Ditmarsch, Wiebe van der Hoek, and Barteld Kooi. Dynamic Epistemic Logic . Vol. 337. Synthese Library. Springer, 2008. 3 Thomas ˚ Agotnes and Hans van Ditmarsch. “Coalitions and Announcements”. In: Proceedings of AAMAS 2008 . Ed. by Lin Padgham et al. IFAAMAS, 2008, pp. 673–680. KI 2018 Model Checking for CAL 2 / 53
What this talk is about What agents know and don’t know (through the lens of epistemic logic 1 ) The effect of public announcements 2 on agent’s knowledge How agents can achieve certain goals by teaming up in coalitions and making joint announcements 3 Model checking for such a framework 1 Hans van Ditmarsch et al., eds. Handbook of Epistemic Logic . College Publications, 2015. 2 Hans van Ditmarsch, Wiebe van der Hoek, and Barteld Kooi. Dynamic Epistemic Logic . Vol. 337. Synthese Library. Springer, 2008. 3 Thomas ˚ Agotnes and Hans van Ditmarsch. “Coalitions and Announcements”. In: Proceedings of AAMAS 2008 . Ed. by Lin Padgham et al. IFAAMAS, 2008, pp. 673–680. KI 2018 Model Checking for CAL 3 / 53
Example There are two households, a and b , and an electricity substation c that requires information about how many households consume power. Moreover, it is imperative that individual household’s consumption remains unknown. KI 2018 Model Checking for CAL 4 / 53
Example There are two households, a and b , and an electricity substation c that requires information about how many households consume power. Moreover, it is imperative that individual household’s consumption remains unknown. KI 2018 Model Checking for CAL 5 / 53
Example w 1 w 2 a , b , c a , b , c c p a , ¬ p b p a , p b c c c a , b , c a , b , c c ¬ p a , p b ¬ p a , ¬ p b w 3 w 4 ( M , w 1 ) | = p a ∧ ¬ p b , ( M , w 1 ) | = K a p a , ( M , w 1 ) | = K a p b , ( M , w 1 ) | = ¬ K c p a . KI 2018 Model Checking for CAL 6 / 53
Example w 1 w 2 a , b , c a , b , c c p a , ¬ p b p a , p b c c c a , b , c a , b , c c ¬ p a , p b ¬ p a , ¬ p b w 3 w 4 ( M , w 1 ) | = p a ∧ ¬ p b , ( M , w 1 ) | = K a p a , ( M , w 1 ) | = K a p b , ( M , w 1 ) | = ¬ K c p a . KI 2018 Model Checking for CAL 7 / 53
Example w 1 w 2 a , b , c a , b , c c p a , ¬ p b p a , p b c c c a , b , c a , b , c c ¬ p a , p b ¬ p a , ¬ p b w 3 w 4 ( M , w 1 ) | = p a ∧ ¬ p b , ( M , w 1 ) | = K a p a , ( M , w 1 ) | = K a p b , ( M , w 1 ) | = ¬ K c p a . KI 2018 Model Checking for CAL 8 / 53
Public Announcements In the continuation of the example, suppose that a announces that Exactly one of us, a and b , uses electricity, i.e. ( p a ∧ ¬ p b ) ∨ ( ¬ p a ∧ p b ). KI 2018 Model Checking for CAL 9 / 53
Public Announcements In the continuation of the example, suppose that a announces that Exactly one of us, a and b , uses electricity, i.e. ( p a ∧ ¬ p b ) ∨ ( ¬ p a ∧ p b ). KI 2018 Model Checking for CAL 10 / 53
Example Updated w 1 w 2 a , b , c a , b , c c p a , ¬ p b p a , p b c c c a , b , c a , b , c c ¬ p a , p b ¬ p a , ¬ p b w 3 w 4 KI 2018 Model Checking for CAL 11 / 53
Example Updated w 1 w 2 a , b , c a , b , c c p a , ¬ p b p a , p b c c c a , b , c a , b , c c ¬ p a , p b ¬ p a , ¬ p b w 3 w 4 KI 2018 Model Checking for CAL 12 / 53
Example Updated w 1 a , b , c p a , ¬ p b c a , b , c ¬ p a , p b w 3 ( M , w 1 ) ann | = p a ∧ ¬ p b , ( M , w 1 ) ann | = K a p a , ( M , w 1 ) ann | = K a p b , ( M , w 1 ) ann | = K c (( p a ∧ ¬ p b ) ∨ ( ¬ p a ∧ p b )). KI 2018 Model Checking for CAL 13 / 53
Example Updated w 1 a , b , c p a , ¬ p b c a , b , c ¬ p a , p b w 3 ( M , w 1 ) ann | = p a ∧ ¬ p b , ( M , w 1 ) ann | = K a p a , ( M , w 1 ) ann | = K a p b , ( M , w 1 ) ann | = K c (( p a ∧ ¬ p b ) ∨ ( ¬ p a ∧ p b )). KI 2018 Model Checking for CAL 14 / 53
Models Definition (Epistemic Model) An epistemic model is a triple M = ( W , ∼ , V ), where W is a non-empty set of states, ∼ : A → P ( W × W ) assigns an equivalence relation to each agent, V : P → P ( W ) is the valuation function. A pair ( M , w ) with w ∈ W is called a pointed model. An announcement in a pointed model ( M , w ) results in an updated pointed model ( M , w ) ϕ with W ϕ = � ϕ � M , ∼ ϕ a = ∼ a ∩ ( � ϕ � M × � ϕ � M ), and V ϕ ( p ) = V ( p ) ∩ � ϕ � M . KI 2018 Model Checking for CAL 15 / 53
Models Definition (Epistemic Model) An epistemic model is a triple M = ( W , ∼ , V ), where W is a non-empty set of states, ∼ : A → P ( W × W ) assigns an equivalence relation to each agent, V : P → P ( W ) is the valuation function. A pair ( M , w ) with w ∈ W is called a pointed model. An announcement in a pointed model ( M , w ) results in an updated pointed model ( M , w ) ϕ with W ϕ = � ϕ � M , ∼ ϕ a = ∼ a ∩ ( � ϕ � M × � ϕ � M ), and V ϕ ( p ) = V ( p ) ∩ � ϕ � M . KI 2018 Model Checking for CAL 16 / 53
Semantics Definition (Semantics) ( M , w ) | = p w ∈ V ( p ) iff ( M , w ) | = ¬ ϕ iff ( M , w ) �| = ϕ ( M , w ) | = ϕ ∧ ψ ( M , w ) | = ϕ and ( M , w ) | = ψ iff ( M , w ) | = K a ϕ iff ∀ v ∈ W : w ∼ a v implies ( M , v ) | = ϕ = ϕ implies ( M , w ) ϕ | ( M , w ) | = [ ϕ ] ψ ( M , w ) | = ψ iff Formula [ ϕ ] ψ is read as after a public announcement of ϕ , ψ holds in the resulting model. Dual of [] = ϕ and ( M , w ) ϕ | ( M , w ) | = � ϕ � ψ iff ( M , w ) | = ψ KI 2018 Model Checking for CAL 17 / 53
Semantics Definition (Semantics) ( M , w ) | = p w ∈ V ( p ) iff ( M , w ) | = ¬ ϕ iff ( M , w ) �| = ϕ ( M , w ) | = ϕ ∧ ψ ( M , w ) | = ϕ and ( M , w ) | = ψ iff ( M , w ) | = K a ϕ iff ∀ v ∈ W : w ∼ a v implies ( M , v ) | = ϕ = ϕ implies ( M , w ) ϕ | ( M , w ) | = [ ϕ ] ψ ( M , w ) | = ψ iff Formula [ ϕ ] ψ is read as after a public announcement of ϕ , ψ holds in the resulting model. Dual of [] = ϕ and ( M , w ) ϕ | ( M , w ) | = � ϕ � ψ iff ( M , w ) | = ψ KI 2018 Model Checking for CAL 18 / 53
Announcements by Coalitions We are interested in the following restrictions on announcements: Announcements are made by agents Agents can only announce what they know Coalitions of agents can announce conjunctions of formulas, where each conjunct is a formula known by an agent in the coalition Agents outside of the coalitions also make an announcement that can preclude coalition to reach its goal KI 2018 Model Checking for CAL 19 / 53
Announcements by Coalitions We are interested in the following restrictions on announcements: Announcements are made by agents Agents can only announce what they know Coalitions of agents can announce conjunctions of formulas, where each conjunct is a formula known by an agent in the coalition Agents outside of the coalitions also make an announcement that can preclude coalition to reach its goal KI 2018 Model Checking for CAL 20 / 53
Announcements by Coalitions Coalition Announcement Logic (CAL) allows us to reason about announcements by coalition of agents. This is Public Announcement Logic (PAL) with the following added operators: � [ G ] � ϕ : ‘there is an announcement by agents from G such that whatever agents A \ G outside of the coalition announce, ϕ holds,’ or [ � G � ] ϕ : ‘whatever agents from G announce, there is an announcement by the agents from the outside of the coalition, such that ϕ holds.’ KI 2018 Model Checking for CAL 21 / 53
Announcements by Coalitions Coalition Announcement Logic (CAL) allows us to reason about announcements by coalition of agents. This is Public Announcement Logic (PAL) with the following added operators: � [ G ] � ϕ : ‘there is an announcement by agents from G such that whatever agents A \ G outside of the coalition announce, ϕ holds,’ or [ � G � ] ϕ : ‘whatever agents from G announce, there is an announcement by the agents from the outside of the coalition, such that ϕ holds.’ KI 2018 Model Checking for CAL 22 / 53
Semantics of CAL Let ψ G be a shorthand for a formula of the type K a ϕ a ∧ . . . ∧ K b ϕ b , where a , . . . , b ∈ G , and ϕ a , . . . , ϕ b are formulas of epistemic logic. For example, ψ G may be K a p ∧ K b ( p → q ) for G = { a , b } . Definition (Semantics) ( M , w ) | = [ � G � ] ϕ iff ∀ ψ G ∃ χ A \ G : ( M , w ) | = ψ G → � ψ G ∧ χ A \ G � ϕ ( M , w ) | = � [ G ] � ϕ iff ∃ ψ G ∀ χ A \ G : ( M , w ) | = ψ G ∧ [ ψ G ∧ χ A \ G ] ϕ KI 2018 Model Checking for CAL 23 / 53
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