a generic constructive solution for concurrent games with expressive constraints on strategies
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
A generic constructive solution for concurrent games with expressive constraints on
strategiesSophie 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
3 Players P2P1 P3
Q Q Q Q
s |= P1 Q
Predicate Q is a move from s for player P1
s
Q’ Q’ Q’
Q’’ Q’’Q’’ Q’’Q’’
:-)
:-(
:-(
Q1 Q1 Q1 Q1
s |= P1 Q1 P2 Q2 P3 Q3
Q2 Q2 Q2 Q2
Q3 Q3 Q3 Q3
Ro ItFr
s
AX(Q1 Q2 Q3 Ro)
Decision modalities PQ
Q{1,3}.
Ro ItFr
s
s |=
Q1 Q1 Q1 Q1
^
There exist moves of P1 and P3such that …
Q1. Q3. P1 Q1 P3 Q3 AX((Q1 Q3) (Ro Fr))
Q3 Q3 Q3 Q3
Infinitary Setting
Strategies: Q. …
^ Q. AG(P Q) …
P Q holds everywhere
(AX(Ro Fr)| Q1 Q3)
Ro ItFr
s
s |= .
Q1,Q3 Q1,Q3
^
Property AX(Ro Fr) holds inside Q1 and Q3
RELATIVIZATION of wrt Q (|Q)
« The subtree designated by Q satisfies »
AX((Q1 Q3) (Ro Fr))Q{1,3}.
Inside Q
(EX |Q) = EX(Q(|Q))
RELATIVIZATION (|Q)
• (EX |Q) EX(Q(|Q))• (R|Q) R• (|Q) (|Q)• ( ’|Q) (|Q) (’|Q)
Q is a set (conjunction) of propositions
• (Z|Q) Z
• (Z. (Z)|Q) Z. ((Z)|Q)
• (Z. (Z)|Q) Z. ((Z)|Q)
(E U |Q) E ((Q(|Q)) U ((Q(|Q))
If CTL -calculus
• (Q.|Q) Q. (|Q)• (PQ|Q) P(QQ)
For example Q.( EFQ’.(’|Q’)|Q) Q.(|Q) E Q U [Q’.(’|Q’Q)]
+
Inside Q
Inside Q’ (inside Q)
’
Q.( EFQ’.(’|Q’)|Q)
The meaning ofRelativization
Q.(|Q) E Q U [Q’.(’|Q’Q)]
Q. (EX Q’. (|Q’) Q)Q. EX (Q Q’. (|Q’))
Variants ofRelativization
Specifying Strategies
^QC. (|QC)
Let C be a coalition of players
and
Dominated Strategies « Q is a strictly dominated strategy »
^ Q’. (Q’ Q) (|Q’R)
^Q’.R. (|QR)(|Q’R) R. (|Q’R)(|QR)
R’. (R’ R) (|QR’)
(|QR) ^
^
^
« Coalition C has a strategy to enforce »
Nash Equilibrium
Theoretical Properties• Bisimulation invariant fragments of MSOwhere 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)
’
No relationshipbetween C and C’
GL
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’)’
^