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
Jan 06, 2016
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’)’
^