ATL I : Classical Results Model Checking wrt m Overview 1 ATL I : Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATL i : Models CGES Model Checking ATL i and ATL I The case ATL iR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATL C Models: CCGS Model Checking Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 12/60
ATL I : Classical Results Model Checking wrt m Model Checking ATL I : 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 : ATL Ir = ATL IR . Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 13/60
ATL I : Classical Results Model Checking wrt m Model Checking ATL I : 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 : ATL Ir = ATL IR . Theorem (Alur, Kupferman & Henzinger 1998) ATL IR (resp. ATL Ir ) model checking is P -complete , and can be done in time linear in the size of the model and the length of the formula . Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 13/60
Complexity wrt n Complexity wrt n Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 14/60
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 15/60
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 ( nd k ) Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 15/60
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 ( nd k ) m is not polynomially bounded in n when agents are present. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 15/60
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 ( nd k ) m is not polynomially bounded in n when agents are present. Agents make models explode! Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 15/60
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 ( nd k ) 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 15/60
Complexity wrt n First Result i = NP Σ P Σ P i − 1 : problems solvable in pol. time by a non-deterministic TM making queries to a Σ P i − 1 oracle i = P Σ P ∆ P i − 1 : problems solvable in pol. time by a deterministic TM making adaptive queries to a Σ P i − 1 oracle Σ P 2 = NP NP 3 = P [ NP NP ] ∆ P Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 16/60
Complexity wrt n First Result i = NP Σ P Σ P i − 1 : problems solvable in pol. time by a non-deterministic TM making queries to a Σ P i − 1 oracle i = P Σ P ∆ P i − 1 : problems solvable in pol. time by a deterministic TM making adaptive queries to a Σ P i − 1 oracle Σ P 2 = NP NP 3 = P [ NP NP ] ∆ P Proposition Model checking ATL IR 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 ATL IR , model checking is Σ P 2 -complete . Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 16/60
Complexity wrt n ATL i : Models CGES Overview 1 ATL I : Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATL i : Models CGES Model Checking ATL i and ATL I The case ATL iR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATL C Models: CCGS Model Checking Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 17/60
Complexity wrt n ATL i : Models CGES Example: Robots and Carriage 1 2 pos 0 1 2 2 pos 1 pos 2 2 1 1 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 18/60
Complexity wrt n ATL i : Models CGES The CGS model wait,wait push,push 1 2 q 0 pos 0 pos 0 push,wait push,wait wait,push wait,push 1 2 2 pos 1 wait,wait pos 2 wait,wait push,push push,push 2 1 1 q 2 q 1 wait,push pos 2 push,wait pos 1 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 19/60
Complexity wrt n ATL i : Models CGES ATL with perfect Information 1 2 pos 0 1 1 2 pos 1 pos 2 2 2 1 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 20/60
Complexity wrt n ATL i : Models CGES ATL with imperfect Information 1 2 1 2 pos 0 pos 0 1 1 1 1 2 2 pos 1 pos 1 pos 2 pos 2 2 2 2 2 1 1 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 20/60
Complexity wrt n ATL i : Models CGES ATL ir : ATL with Imperfect Information wait,wait push,push q 0 1 2 pos 0 pos 0 push,wait push,wait wait,push wait,push 2 1 1 2 2 pos 1 wait,wait pos 2 wait,wait push,push 2 push,push 1 1 q 2 q 1 wait,push pos 2 push,wait pos 1 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 21/60
Complexity wrt n ATL i : 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 ′ ⇒ s a ( q ) = s a ( q ′ ) Uniform strategies for group of agents: q ∼ A q ′ ⇒ s a ( q ) = s a ( q ′ ) , where q ∼ A q ′ is defined by there is an agent a ∈ A with q ∼ a q ′ Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 22/60
Complexity wrt n ATL i : 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 ′ ⇒ s a ( q ) = s a ( q ′ ) Uniform strategies for group of agents: q ∼ A q ′ ⇒ s a ( q ) = s a ( 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 22/60
Complexity wrt n ATL i : Models CGES ATL ir and ATL iR � ir ✐ M, q | = � � A � ϕ iff there exists uniform S A such that, for every path Λ ∈ � q ′ ∼ A q out ( q ′ , S A ) , we have M, Λ[1] | = ϕ ; M, q | = � � A � � ir � ϕ iff there exists uniform S A such that, for every Λ ∈ � q ′ ∼ A q out ( q ′ , S A ) , we have M, Λ[ i ] | = ϕ for every i ≥ 0 ; M, q | = � � A � � ir ϕ U ψ iff there exists uniform S A such that, for every Λ ∈ � q ′ ∼ A q out ( q ′ , S A ) , 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 ( ATL iR )? Instead of equivalences q ′ ∼ A q , one has to consider sequences q ′ ∼ A q . Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 23/60
Complexity wrt n ATL i : Models CGES Example: Robots and Carriage 1 2 pos 0 � ir � ¬ pos 1 ? � � 1 � 1 2 2 pos 1 pos 2 2 1 1 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 24/60
Complexity wrt n ATL i : Models CGES Example: Robots and Carriage 1 2 pos 0 ¬� � 1 � � ir � ¬ pos 1 1 2 2 pos 1 pos 2 2 1 1 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 24/60
Complexity wrt n ATL i : Models CGES Example: Robots and Carriage 1 2 pos 0 ¬� � 1 � � ir � ¬ pos 1 � ir � ¬ pos 1 ? � � 2 � 1 2 2 pos 1 pos 2 2 1 1 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 24/60
Complexity wrt n ATL i : Models CGES Example: Robots and Carriage 1 2 pos 0 ¬� � 1 � � ir � ¬ pos 1 1 2 2 pos 1 ¬� � 2 � � ir � ¬ pos 1 pos 2 2 1 1 Why not � � 2 � � ir � ¬ pos 1 by using the following strategy for agent 2: “push” when in q 0 and “wait” when in q 2 ? This not a feasible strategy, because it does not work in q 1 . Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 24/60
Complexity wrt n Model Checking ATL i and ATL I Overview 1 ATL I : Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATL i : Models CGES Model Checking ATL i and ATL I The case ATL iR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATL C Models: CCGS Model Checking Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 25/60
Complexity wrt n Model Checking ATL i and ATL I Complexity Results for Strategic Logics m , l n , k , l C TL P-complete [1] P-complete [1] A TL Ir P-complete [3] A TL ir Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 26/60
Complexity wrt n Model Checking ATL i and ATL I Complexity Results for Strategic Logics m , l n , k , l C TL P-complete [1] P-complete [1] A TL Ir P-complete [3] A TL ir Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems ... . ACM Prog. Lang. Syst [1] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic . J. ACM [3] Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 26/60
Complexity wrt n Model Checking ATL i and ATL I Complexity Results for Strategic Logics m , l n , k , l C TL P-complete [1] P-complete [1] A TL Ir 3 -complete [5,6] ∆ P P-complete [3] A TL ir Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems ... . ACM Prog. Lang. Syst [1] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic . J. ACM [3] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [5] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [6] Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 26/60
Complexity wrt n Model Checking ATL i and ATL I Complexity Results for Strategic Logics m , l n , k , l n local , k , l C TL P-complete [1] P-complete [1] A TL Ir 3 -complete [5,6] ∆ P P-complete [3] A TL ir 2 -complete [4,7] 3 -complete [7] ∆ P ∆ P Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems ... . ACM Prog. Lang. Syst [1] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic . J. ACM [3] Schobbens (2004). Alternating-time logic with imperfect recall . ENTCS [4] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [5] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [6] Jamroga & Dix (2008). Model checking abilities of agents . Theory of Computing systems [7] Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 26/60
Complexity wrt n Model Checking ATL i and ATL I Complexity Results for Strategic Logics m , l n , k , l n local , k , l C TL P-complete [1] P-complete [1] A TL Ir 3 -complete [5,6] ∆ P P-complete [3] A TL ir 2 -complete [4,7] 3 -complete [7] ∆ P ∆ P Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems ... . ACM Prog. Lang. Syst [1] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic . J. ACM [3] Schobbens (2004). Alternating-time logic with imperfect recall . ENTCS [4] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [5] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [6] Jamroga & Dix (2008). Model checking abilities of agents . Theory of Computing systems [7] Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 26/60
Complexity wrt n Model Checking ATL i and ATL I Complexity Results for Strategic Logics m , l n , k , l n local , k , l C TL P-complete [1] P-complete [1] PSPACE-complete [2] A TL Ir 3 -complete [5,6] ∆ P P-complete [3] A TL ir 2 -complete [4,7] 3 -complete [7] ∆ P ∆ P Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems ... . ACM Prog. Lang. Syst [1] Kupferman, Vardi & Wolper (2000). An automata-theoretic approach to .... J. ACM [2] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic . J. ACM [3] Schobbens (2004). Alternating-time logic with imperfect recall . ENTCS [4] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [5] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [6] Jamroga & Dix (2008). Model checking abilities of agents . Theory of Computing systems [7] Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 26/60
Complexity wrt n Model Checking ATL i and ATL I Complexity Results for Strategic Logics m , l n , k , l n local , k , l C TL P-complete [1] P-complete [1] PSPACE-complete [2] A TL Ir 3 -complete [5,6] EXPTIME-complete [8] ∆ P P-complete [3] A TL ir 2 -complete [4,7] 3 -complete [7] ∆ P ∆ P Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems ... . ACM Prog. Lang. Syst [1] Kupferman, Vardi & Wolper (2000). An automata-theoretic approach to .... J. ACM [2] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic . J. ACM [3] Schobbens (2004). Alternating-time logic with imperfect recall . ENTCS [4] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [5] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [6] Jamroga & Dix (2008). Model checking abilities of agents . Theory of Computing systems [7] Hoek, Lomuscio & Wooldridge (AAMAS 2006). On the complexity of practical ATL model checking. [8] Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 26/60
Complexity wrt n Model Checking ATL i and ATL I Complexity Results for Strategic Logics m , l n , k , l n local , k , l C TL P-complete [1] P-complete [1] PSPACE-complete [2] A TL Ir 3 -complete [5,6] EXPTIME-complete [8] ∆ P P-complete [3] A TL ir 2 -complete [4,7] 3 -complete [7] PSPACE-complete [9] ∆ P ∆ P Clarke, Emerson & Sistla (1986). Automatic verification of finite-state concurrent systems ... . ACM Prog. Lang. Syst [1] Kupferman, Vardi & Wolper (2000). An automata-theoretic approach to .... J. ACM [2] Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic . J. ACM [3] Schobbens (2004). Alternating-time logic with imperfect recall . ENTCS [4] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? [5] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. [6] Jamroga & Dix (2008). Model checking abilities of agents . Theory of Computing systems [7] Hoek, Lomuscio & Wooldridge (AAMAS 2006). On the complexity of practical ATL model checking. [8] Jamroga & Ågotnes (AAMAS 2007). Modular Interpreted Systems [9] Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 26/60
Complexity wrt n Model Checking ATL i and ATL I Last Column: meaning of n local MIS A modular interpreted system (MIS) is of the form � A gt , Act, In � where each agent a i has the following internal structure a i = � St i , d i , out i , in i , o i , Π i , π i � . Set of global states is the cartesian product of the St i : n = n 1 · . . . · n k . Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 27/60
Complexity wrt n Model Checking ATL i and ATL I Last Column: meaning of n local MIS A modular interpreted system (MIS) is of the form � A gt , Act, In � where each agent a i has the following internal structure a i = � St i , d i , out i , in i , o i , Π i , π i � . Set of global states is the cartesian product of the St i : n = n 1 · . . . · n k . A MIS viewed as a CGS (for ATL) is very succinct. For ATL ir , we have in addition to CGS all the local epistemic relations ∼ 1 , . . . , ∼ k ( CEGS ). A MIS viewed as a CEGS (for ATL ir ) does not compress that much. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 27/60
Complexity wrt n The case ATL iR Overview 1 ATL I : Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATL i : Models CGES Model Checking ATL i and ATL I The case ATL iR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATL C Models: CCGS Model Checking Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 28/60
Complexity wrt n The case ATL iR What about ATL iR ? ATL iR : 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 ATL iR . Why is it undecidable? q 0 β α p r q 1 q 2 M , q 0 | = � � 1 � � IR ( �♦ p ∧ �♦ r ) M , q 0 �| = � � 1 � � Ir ( �♦ p ∧ �♦ r ) Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 29/60
Complexity wrt n The case ATL iR 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. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 30/60
ATL + Plausibility ATL + Plausibility Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 31/60
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. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 32/60
ATL + Plausibility Plausibility concept ATL: Reasoning about all possible behaviors. ATLP: Reasoning about plausible behaviors. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 33/60
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. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 33/60
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. � ϕ : Agents A have a plausible collective strategy to enforce Pl � � A � ϕ against any plausible response of their opponents. Example: Playing undominated strategies is often plausible,... Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 33/60
ATL + Plausibility Base Logic Overview 1 ATL I : Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATL i : Models CGES Model Checking ATL i and ATL I The case ATL iR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATL C Models: CCGS Model Checking Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 34/60
ATL + Plausibility Base Logic The Base Logic: L base ATLP Definition ( L base ATLP ) The language L base ATLP is defined over nonempty sets: Π of propositions, p ∈ Π , A gt of agents, a ∈ A gt , A ⊆ A gt , and Ω of basic plausibility terms , ω ∈ Ω . � ✐ ϕ ::= p | ¬ ϕ | ϕ ∧ ϕ | � � A � ϕ | � � A � � � ϕ | � � A � � ϕ U ϕ | Pl A ϕ | ( set-pl ω ) ϕ Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 35/60
ATL + Plausibility Base Logic The Base Logic: L base ATLP Definition ( L base ATLP ) The language L base ATLP is defined over nonempty sets: Π of propositions, p ∈ Π , A gt of agents, a ∈ A gt , A ⊆ A gt , and Ω of basic plausibility terms , ω ∈ Ω . � ✐ ϕ ::= p | ¬ ϕ | ϕ ∧ ϕ | � � A � ϕ | � � A � � � ϕ | � � A � � ϕ U ϕ | Pl A ϕ | ( set-pl ω ) ϕ M , q | = Pl B � � A � � γ if A can enforce γ , when agents in B play only plausible strategies Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 35/60
ATL + Plausibility Base Logic Plausibility Terms Ω : Set of basic plausibility terms, ω ∈ Ω Hard-wired sets of strategies: ω NE : Nash equilibria How to activate? ω PO : Pareto optimal strategies Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 36/60
ATL + Plausibility Base Logic Plausibility Terms Ω : Set of basic plausibility terms, ω ∈ Ω Hard-wired sets of strategies: ω NE : Nash equilibria How to activate? ω PO : Pareto optimal strategies ( set-pl ω ) : Sets plausible strategies to [ [ ω ] ] ⊆ Σ Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 36/60
ATL + Plausibility Base Logic Plausibility Terms Ω : Set of basic plausibility terms, ω ∈ Ω Hard-wired sets of strategies: ω NE : Nash equilibria How to activate? ω PO : Pareto optimal strategies ( set-pl ω ) : Sets plausible strategies to [ [ ω ] ] ⊆ Σ And where do the terms come from? Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 36/60
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 ) Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 37/60
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 37/60
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 37/60
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. ∀ σ 1 ∃ σ 2 . . . ∀ σ n ϕ ( σ, σ 1 , . . . σ n ) � �� � ∈L base ATLP (Ω ∪{ σ,σ 1 ,...,σ n } ) Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 37/60
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. ∀ σ 1 ∃ σ 2 . . . ∀ σ n ϕ ( σ, σ 1 , . . . σ n ) � �� � ∈L base ATLP (Ω ∪{ σ,σ 1 ,...,σ n } ) Example: ω DOM = σ. ∀ σ ′ ( σ better than σ ′ ) Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 37/60
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. ∀ σ 1 ∃ σ 2 . . . ∀ σ n ϕ ( σ, σ 1 , . . . σ n ) � �� � ∈L base ATLP (Ω ∪{ σ,σ 1 ,...,σ n } ) Example: ω DOM = σ. ∀ σ ′ ( σ better than σ ′ ) How to determine whether a strategy is good? Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 37/60
ATL + Plausibility Models: CGSP Overview 1 ATL I : Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATL i : Models CGES Model Checking ATL i and ATL I The case ATL iR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATL C Models: CCGS Model Checking Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 38/60
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 . Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 39/60
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. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 39/60
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. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 39/60
ATL + Plausibility Models: CGSP Concurrent game structures with plausibility M = ( A gt , Q , Π , π, Act, d, δ, Υ , Ω , [ [ · ] ]) Υ ⊆ Σ : set of (plausible) strategy profiles Example: Υ = { ( head , head ) } Ω = { ω 1 , ω 2 , . . . } : set of plausibility terms Example: ω NE stands for all Nash equilibria [ [ · ] ] : Q → (Ω → P (Σ)) : plausibility mapping, it assigns a set of strategy profiles to each state and plausibility term Example: [ [ ω NE ] ] q = { ( head , head ) , ( tail , tail ) } Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 40/60
ATL + Plausibility Models: CGSP Concurrent game structures with plausibility M = ( A gt , Q , Π , π, Act, d, δ, Υ , Ω , [ [ · ] ]) Υ ⊆ Σ : set of (plausible) strategy profiles Example: Υ = { ( head , head ) } Ω = { ω 1 , ω 2 , . . . } : set of plausibility terms Example: ω NE stands for all Nash equilibria [ [ · ] ] : Q → (Ω → P (Σ)) : plausibility mapping, it assigns a set of strategy profiles to each state and plausibility term Example: [ [ ω NE ] ] q = { ( head , head ) , ( tail , tail ) } Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 40/60
ATL + Plausibility Models: CGSP Concurrent game structures with plausibility M = ( A gt , Q , Π , π, Act, d, δ, Υ , Ω , [ [ · ] ]) Υ ⊆ Σ : set of (plausible) strategy profiles Example: Υ = { ( head , head ) } Ω = { ω 1 , ω 2 , . . . } : set of plausibility terms Example: ω NE stands for all Nash equilibria [ [ · ] ] : Q → (Ω → P (Σ)) : plausibility mapping, it assigns a set of strategy profiles to each state and plausibility term Example: [ [ ω NE ] ] q = { ( head , head ) , ( tail , tail ) } Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 40/60
ATL + Plausibility Models: CGSP Concurrent game structures with plausibility M = ( A gt , Q , Π , π, Act, d, δ, Υ , Ω , [ [ · ] ]) Υ ⊆ Σ : set of (plausible) strategy profiles Example: Υ = { ( head , head ) } Ω = { ω 1 , ω 2 , . . . } : set of plausibility terms Example: ω NE stands for all Nash equilibria [ [ · ] ] : Q → (Ω → P (Σ)) : plausibility mapping, it assigns a set of strategy profiles to each state and plausibility term Example: [ [ ω NE ] ] q = { ( head , head ) , ( tail , tail ) } Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 40/60
ATL + Plausibility Models: CGSP General Solution Concepts: CGSP Idea : Agents have preferences : � η = � η 1 , . . . , η k � η i : ATL path formulæ ( payoff ) Example: η 2 = ♦ money 2 start q 0 nn nn tt th hh ht money 1 money 2 nn q 1 q 2 hh ht ht tt th th hh tt q 3 q 4 q 5 money 1 money 2 money 2 No payoffs needed as for classical solution concepts! Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 41/60
ATL + Plausibility Models: CGSP General Solution Concepts: CGSP Idea : Agents have preferences : � η = � η 1 , . . . , η k � η i : ATL path formulæ ( payoff ) Example: η 2 = ♦ money 2 start q 0 nn nn tt th hh ht money 1 money 2 nn q 1 q 2 hh ht ht tt th th hh tt q 3 q 4 q 5 money 1 money 2 money 2 No payoffs needed as for classical solution concepts! Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 41/60
ATL + Plausibility Models: CGSP General Solution Concepts: CGSP Idea : Agents have preferences : � η = � η 1 , . . . , η k � η i : ATL path formulæ ( payoff ) Example: η 2 = ♦ money 2 CGSP +preferences � normal form game Each CGSP M with � η corresponds to a normal form game S . q 0 start n n tt th η 1 \ η 2 s hh s ht s th s tt n n hh ht money 1 money 2 0 , 0 0 , 1 0 , 1 s hh 1 , 1 nn q 1 q 2 hh ht ht � s ht 0 , 0 0 , 1 0 , 1 0 , 1 tt th th 0 , 1 0 , 1 0 , 0 s th 1 , 1 hh tt q 3 q 4 q 5 s tt 0 , 1 0 , 1 0 , 0 0 , 1 money 1 money 2 money 2 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 41/60
ATL + Plausibility Models: CGSP General Solution Concepts: CGSP Idea : Agents have preferences : � η = � η 1 , . . . , η k � η i : ATL path formulæ ( payoff ) Example: η 2 = ♦ money 2 CGSP +preferences � normal form game Each CGSP M with � η corresponds to a normal form game S . q 0 start n n tt th η 1 \ η 2 s hh s ht s th s tt n n hh ht money 1 money 2 0 , 0 0 , 1 0 , 1 s hh 1 , 1 nn q 1 q 2 hh ht ht � s ht 0 , 0 0 , 1 0 , 1 0 , 1 tt th th 0 , 1 0 , 1 0 , 0 s th 1 , 1 hh tt q 3 q 4 q 5 s tt 0 , 1 0 , 1 0 , 0 0 , 1 money 1 money 2 money 2 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 41/60
ATL + Plausibility Models: CGSP Characterizing Solution Concepts � a ∈ A gt BR � η NE � η ( σ ): a ( σ ) Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 42/60
ATL + Plausibility Models: CGSP Characterizing Solution Concepts � � BR � η ( set-pl σ [ A gt \{ a } ]) Pl A gt � η a → ( set-pl σ ) � � � a � �∅� � η a a ( σ ): � a ∈ A gt BR � η NE � η ( σ ): a ( σ ) Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 42/60
ATL + Plausibility Models: CGSP Characterizing Solution Concepts � � BR � η ( set-pl σ [ A gt \{ a } ]) Pl A gt � η a → ( set-pl σ ) � � � a � �∅� � η a a ( σ ): � a ∈ A gt BR � η NE � η ( σ ): a ( σ ) SPN � η ( σ ): � � NE � η ( σ ) � �∅� Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 42/60
ATL + Plausibility Models: CGSP Characterizing Solution Concepts � � BR � η ( set-pl σ [ A gt \{ a } ]) Pl A gt � η a → ( set-pl σ ) � � � a � �∅� � η a a ( σ ): � a ∈ A gt BR � η NE � η ( σ ): a ( σ ) SPN � η ( σ ): � � NE � η ( σ ) � �∅� � � ∀ σ ′ Pl A gt a ∈ A gt (( set-pl σ ′ ) � � η a → ( set-pl σ ) � PO � η ( σ ): �∅� �∅� � η a ) ∨ � � a ∈ A gt (( set-pl σ ) � � η a ∧ ¬ ( set-pl σ ′ ) � �∅� �∅� � η a . Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 42/60
ATL + Plausibility Models: CGSP Characterizing Solution Concepts � � BR � η ( set-pl σ [ A gt \{ a } ]) Pl A gt � η a → ( set-pl σ ) � � � a � �∅� � η a a ( σ ): � a ∈ A gt BR � η NE � η ( σ ): a ( σ ) SPN � η ( σ ): � � NE � η ( σ ) � �∅� � � ∀ σ ′ Pl A gt a ∈ A gt (( set-pl σ ′ ) � � η a → ( set-pl σ ) � PO � η ( σ ): �∅� �∅� � η a ) ∨ � � a ∈ A gt (( set-pl σ ) � � η a ∧ ¬ ( set-pl σ ′ ) � �∅� �∅� � η a . Theorem These notions correspond to those in game theory: ] q [ σ.NE � η ( σ )] [ M = NE strategies in S ( M, � η, q ) . Similarly for SPN and PO and UNDOM. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 42/60
ATL + Plausibility Models: CGSP Example Both agents play without restrictions : M , q 0 | = ¬� � a 2 � � ♦ money 2 Both agents play a Nash equilibrium strategy: = ( set-pl σ. NE η ( σ )) Pl A gt � M, q 0 | � a 2 � � ♦ money 2 Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 43/60
ATL + Plausibility Models: CGSP The Full Language: L ATLP Plausibility terms : σ. ∀ σ 1 ∃ σ 2 . . . ∀ σ n ϕ where ϕ ∈ L base ATLP What about nesting ( set-pl · ) operators? ( set-pl . . . ( set-pl . . . ( set-pl . . . ) . . . ) . . . ) Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 44/60
ATL + Plausibility Models: CGSP The Full Language: L ATLP Plausibility terms : σ. ∀ σ 1 ∃ σ 2 . . . ∀ σ n ϕ where ϕ ∈ L base ATLP What about nesting ( set-pl · ) operators? ( set-pl . . . ( set-pl . . . ( set-pl . . . ) . . . ) . . . ) We get a hierarchy of logics: L k ATLP : k nestings L ATLP := lim k →∞ L k ATLP Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 44/60
ATL + Plausibility Model Checking Overview 1 ATL I : Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATL i : Models CGES Model Checking ATL i and ATL I The case ATL iR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATL C Models: CCGS Model Checking Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 45/60
ATL + Plausibility Model Checking Model Checking Complexity Theorem Model checking L base ATLP is ∆ P 3 -complete . This is in the line with game theoretical results! Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 46/60
ATL + Plausibility Model Checking Model Checking Complexity Expressivity vs. Complexity 0 1 2 . . . i . . . unbounded L 1 ∆ P ∆ P ∆ P ∆ P . . . . . . PSPACE ATLP 3 4 5 i + 3 L 2 ∆ P ∆ P ∆ P ∆ P . . . . . . PSPACE ATLP 4 6 7 5 + i − max { 0 , 1 − i } . . . . . . . . . L k ∆ P ∆ P ∆ P ∆ P ATLP . . . . . . PSPACE k + 2 k + 4 k + 6 i + 2k + 1 − max { 0 , k − i − 1 } i > k + 1 For particular well-behaved models, model checking is even polynomial . Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 47/60
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. Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 48/60
ATL + Coalition Formation ATL + Coalition Formation Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 49/60
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 ATL c : � | 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 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 ATL c : � | 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 ATL C Overview 1 ATL I : Classical Results Models: CGS Model Checking wrt m 2 Complexity wrt n ATL i : Models CGES Model Checking ATL i and ATL I The case ATL iR 3 ATL + Plausibility Base Logic Models: CGSP Model Checking 4 ATL + Coalition Formation ATL C Models: CCGS Model Checking Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 51/60
ATL + Coalition Formation ATL C 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 a 1 a 2 a 3 CF ( A gt ): coalitional frameworks over A gt Jürgen Dix · Clausthal University of Technology 23rd April, Luxembourg 52/60
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