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’)’

^