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
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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]
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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
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• Our approach (Second-Order Kind) DECIDABLE
Outline
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Concurrent Games
Strategies
Relativization
Strategies Specifications
Theoretical Properties
Related Work
P1
3 Players
P2
P3
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
Q1
Q1
Q2
Q2
Q3
It
Fr
Q1
Q3
Q1
Q2
Q3
Q2
Q3
There exist moves of P1 and P3
such that …
^
s |=  Q1.  Q3.
 Q
Q1.  P3 Q3  AX((Q1  Q3)  (Ro  Fr))
P1{1,3}
s
Ro
Q1
Q3
It
Fr
Q1
Q1
Q3
Q1
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((Q1
(AX(Ro Q3)
Fr)|Q1
(Ro
 Q3)
 Fr))
s
Ro
Q1,Q3
It
Fr
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
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(EX |Q)  EX(Q(|Q))
(R|Q)  R
(|Q)  (|Q)
(  ’|Q)  (|Q)  (’|Q)
If -calculus
CTL
+
• (Q.|Q)  Q. (|Q)
• (PQ|Q)  P(QQ)
(EFor
U
|Q)
•(Z|Q)
 Z E ((Q(|Q)) U ((Q(|Q))
example
• (Z.(Z)|Q)
 Z. ((Z)|Q)
Q.(
EFQ’.(’|Q’)|Q)

Q.(|Q)
 E QUZ.
[Q’.(’|Q’Q)]
• (Z. (Z)|Q)
( (Z)|Q)
Q.(|Q)
Q.(  
EFQ’.(’|Q’)|Q)
E Q U [Q’.(’|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  »
Nash Equilibrium (|QR) and
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 Q’. (Q’  Q)  (|Q’R)   R’. (R’  R)   (|QR’)
Dominated Strategies « Q is a strictly dominated strategy »
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Q’.R. (|QR)(|Q’R)  R. (|Q’R)(|QR)
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
– Regular solutions
G |= ^ Q . (|Q )
C
C
Related Works
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Alternating Time Logic [AHK02]
ATL, ATL*, AMC, GL are subsumed
   uses the variant of relativization 
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lC.   EF( lC’.’)  QC. (  EF(QC’.(’QC’)) QC)
 GL

^
^
No relationship
QC.   E
QCU (QC’.(’QC

between C and C’
Quantification under
the scope of a fixpoint
’
Related Works (cont.)
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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)
Eq(Q,Q’)  AG(Q  Q’)
^
Uniq(Q)  (|Q) Q’. (|Q’)  Eq(Q,Q’)’
– Express sets of strategies