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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

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
Games
  • Economy
  • Biology
  • Synthesis and Control of Reactive Systems
  • Checking and Realizability of Specifications
  • Compatibilty of Interfaces
  • Simulation Relations
  • Test Cases Generation
games cont
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
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
Outline
  • Concurrent Games
  • Strategies
  • Relativization
  • Strategies Specifications
  • Theoretical Properties
  • Related Work
slide6

P1

P2

P3

3 Players

slide7

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’’

slide8

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

slide9

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
Infinitary Setting

Strategies:

P Q holds everywhere

^

Q. …

Q.AG(P Q) …

slide11

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  »

relativization 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(QQ)

+

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.(  EFQ’.(’|Q’)|Q)

Q.(|Q)  E Q U [Q’.(’|Q’Q)]

slide14

Q.(|Q)  E Q U [Q’.(’|Q’Q)]

Q.(  EFQ’.(’|Q’)|Q)

The meaning of

Relativization

Inside Q

Inside Q’ (inside Q)

’

slide15

Variants of

Relativization

Q. (EX Q’. (|Q’)Q)

Q. EX (Q Q’. (|Q’))

specifying strategies
Specifying Strategies

Let C be a coalition of players

^

QC. (|QC)

« Coalition C has a strategy to enforce  »

(|QR)

and

Nash Equilibrium

^

^

Q’. (Q’  Q) (|Q’R)

R’. (R’  R) (|QR’)

Dominated Strategies « Q is a strictly dominated strategy »

^

^

^

Q’.R.(|QR)(|Q’R) R. (|Q’R)(|QR)

theoretical properties
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
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
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’)’