a generic constructive solution for concurrent games with expressive constraints on strategies

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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

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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 - PowerPoint PPT Presentation

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Page 1: A generic constructive solution for concurrent games with   expressive constraints on strategies

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

Page 2: A generic constructive solution for concurrent games with   expressive constraints on strategies

Games

• Economy• Biology• Synthesis and Control of Reactive Systems• Checking and Realizability of Specifications• Compatibilty of Interfaces• Simulation Relations• Test Cases Generation• …

Page 3: A generic constructive solution for concurrent games with   expressive constraints on strategies

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

Page 4: A generic constructive solution for concurrent games with   expressive constraints on strategies

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

Page 5: A generic constructive solution for concurrent games with   expressive constraints on strategies

Outline

• Concurrent Games• Strategies• Relativization• Strategies Specifications• Theoretical Properties• Related Work

Page 6: A generic constructive solution for concurrent games with   expressive constraints on strategies

3 Players P2P1 P3

Page 7: A generic constructive solution for concurrent games with   expressive constraints on strategies

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’’

:-)

:-(

:-(

Page 8: A generic constructive solution for concurrent games with   expressive constraints on strategies

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

Page 9: A generic constructive solution for concurrent games with   expressive constraints on strategies

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

Page 10: A generic constructive solution for concurrent games with   expressive constraints on strategies

Infinitary Setting

Strategies: Q. …

^ Q. AG(P Q) …

P Q holds everywhere

Page 11: A generic constructive solution for concurrent games with   expressive constraints on strategies

(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}.

Page 12: A generic constructive solution for concurrent games with   expressive constraints on strategies

Inside Q

(EX |Q) = EX(Q(|Q))

Page 13: A generic constructive solution for concurrent games with   expressive constraints on strategies

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)]

+

Page 14: A generic constructive solution for concurrent games with   expressive constraints on strategies

Inside Q

Inside Q’ (inside Q)

Q.( EFQ’.(’|Q’)|Q)

The meaning ofRelativization

Q.(|Q) E Q U [Q’.(’|Q’Q)]

Page 15: A generic constructive solution for concurrent games with   expressive constraints on strategies

Q. (EX Q’. (|Q’) Q)Q. EX (Q Q’. (|Q’))

Variants ofRelativization

Page 16: A generic constructive solution for concurrent games with   expressive constraints on strategies

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

Page 17: A generic constructive solution for concurrent games with   expressive constraints on strategies

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)

Page 18: A generic constructive solution for concurrent games with   expressive constraints on strategies

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

Page 19: A generic constructive solution for concurrent games with   expressive constraints on strategies

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’)’

^