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A generic constructive solution for concurrent games with expressive constraints on strategies. Sophie Pinchinat IRISA, Université de Rennes 1, France RSISE, Canberra, Australia Marie Curie Fellow, EU FP6. Games. Economy Biology Synthesis and Control of Reactive Systems
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A generic constructive solution for concurrent games with expressive constraints on strategies Sophie Pinchinat IRISA, Université de Rennes 1, France RSISE, Canberra, Australia Marie Curie Fellow, EU FP6
Games • Economy • Biology • Synthesis and Control of Reactive Systems • Checking and Realizability of Specifications • Compatibilty of Interfaces • Simulation Relations • Test Cases Generation • …
Games (Cont.) • Concurrent Game Structures [AHK98] • Generalization of Kripke Structures • Based on Global States • Several Players make Decisions • Effect Transitions • Specifications of Game Objectives • Alternating Time Logic ATL,CTL*, AMC… [AHK98] generalize Temporal Logic CTL, CTL*, -calculus • Strategy Logic [CHP07] • Our approach
Specifications • Existence of strategies to achieve an objective • Alternating Time Logic • Model-Checking Problems • Strategy Logic (First-order Kind) • Synthesis Problems • Non-elementary - Effective Subclasses • Our approach (Second-Order Kind) DECIDABLE
Outline • Concurrent Games • Strategies • Relativization • Strategies Specifications • Theoretical Properties • Related Work
P1 P2 P3 3 Players
Predicate Q is a move from s for player P1 s |= P1 Q s :-) Q Q Q Q :-( Q’ Q’ Q’ Q’’ :-( Q’’ Q’’ Q’’ Q’’
Decision modalities PQ s |= P1 Q1 P2 Q2 P3 Q3 AX(Q1 Q2 Q3 Ro) s Ro It Fr Q1 Q1 Q1 Q1 Q2 Q2 Q2 Q2 Q3 Q3 Q3 Q3
There exist moves of P1 and P3 such that … ^ s |= Q1. Q3.P1 Q1 P3 Q3 Q{1,3}. AX((Q1 Q3) (Ro Fr)) s Ro It Fr Q1 Q1 Q1 Q1 Q3 Q3 Q3 Q3
Infinitary Setting Strategies: P Q holds everywhere ^ Q. … Q.AG(P Q) …
Property AX(Ro Fr) holds inside Q1 and Q3 ^ s |= . Q{1,3}. (AX(Ro Fr)| Q1 Q3) AX((Q1 Q3) (Ro Fr)) s Ro It Fr Q1,Q3 Q1,Q3 RELATIVIZATION of wrt Q(|Q) « The subtree designated by Q satisfies »
(EX |Q) = EX(Q(|Q)) Inside Q
RELATIVIZATION (|Q) Q is a set (conjunction) of propositions • (EX |Q)EX(Q(|Q)) • (R|Q)R • (|Q)(|Q) • (’|Q)(|Q) (’|Q) • (Q.|Q) Q. (|Q) • (PQ|Q)P(QQ) + If CTL -calculus (E U |Q)E ((Q(|Q))U ((Q(|Q)) • (Z|Q)Z • (Z. (Z)|Q)Z. ((Z)|Q) • (Z. (Z)|Q)Z. ((Z)|Q) For example Q.( EFQ’.(’|Q’)|Q) Q.(|Q) E Q U [Q’.(’|Q’Q)]
Q.(|Q) E Q U [Q’.(’|Q’Q)] Q.( EFQ’.(’|Q’)|Q) The meaning of Relativization Inside Q Inside Q’ (inside Q) ’
Variants of Relativization Q. (EX Q’. (|Q’)Q) Q. EX (Q Q’. (|Q’))
Specifying Strategies Let C be a coalition of players ^ QC. (|QC) « Coalition C has a strategy to enforce » (|QR) and Nash Equilibrium ^ ^ Q’. (Q’ Q) (|Q’R) R’. (R’ R) (|QR’) Dominated Strategies « Q is a strictly dominated strategy » ^ ^ ^ Q’.R.(|QR)(|Q’R) R. (|Q’R)(|QR)
Theoretical Properties • Bisimulation invariant fragments of MSO where quantifiers and fixpoints can interleave • Involved automata constructions • Automata with variables [AN01] • Projection [Rab69] • Non-elementary (nEXPTIME/(n+1)EXPTIME) where n is the number of quantifiers alternations • Strategies synthesis • Model-checking G |= • Regular solutions ^ QC. (|QC)
Related Works • Alternating Time Logic [AHK02] ATL, ATL*, AMC, GL are subsumed uses the variant of relativization ^ ^ lC. EF(lC’.’) QC. ( EF(QC’.(’QC’))QC) GL ^ ^ No relationship between C and C’ QC. E QCU (QC’.(’QC’)) Quantification under the scope of a fixpoint ’
Related Works (cont.) • Strategy Logic [CHP07] “x is strictly dominated”: x’[y.(x,y) (x’,y)y (x’,y) (x,y)] First-order Cannot • Compare strategies (equality, uniqueness) • Express sets of strategies Eq(Q,Q’) AG(Q Q’) ^ Uniq(Q) (|Q) Q’. (|Q’) Eq(Q,Q’)’
