Distributed games with causal memory
are decidable for series-parallel systems
Paul Gastin, Benjamin Lerman, Marc Zeitoun
Firstname.Name@liafa.jussieu.fr
LIAFA, Université Paris 7, UMR CNRS 7089
FSTTCS, dec. 04
Gastin, Lerman, Zeitoun
Causal distributed games
1/21
Outline
1
Motivation
2
A toy example
3
Architectures, games, memory and strategies
4
Main result
Gastin, Lerman, Zeitoun
Causal distributed games
2/21
Outline
1
Motivation
2
A toy example
3
Architectures, games, memory and strategies
4
Main result
Gastin, Lerman, Zeitoun
Causal distributed games
3/21
Context: control and synthesis
Environment E
acts on
Open system
S
Specification
ϕ
Two problems (Ramadge, Wonham)
I
I
Decide whether there exists a finite memory controller st. (S ⊗ C) k E |= ϕ.
Synthesis: if so, compute such a controller.
Gastin, Lerman, Zeitoun
Causal distributed games
4/21
Context: control and synthesis
Environment E
acts on
enables/disables actions
Controller
C
provides info
Open system
S
Specification
ϕ
Two problems (Ramadge, Wonham)
I
I
Decide whether there exists a finite memory controller st. (S ⊗ C) k E |= ϕ.
Synthesis: if so, compute such a controller.
Gastin, Lerman, Zeitoun
Causal distributed games
4/21
Control as a game
Classical framework
Games well suited to solve controller synthesis problems.
System
Controller
I Environment
I Correct behavior
I Finding a controller
I
I
;
;
;
;
;
game graph.
a player.
the opponent.
winning condition.
finding a winning strategy.
In general
For reasonable sequential systems and specifications, both problems are decidable.
Gastin, Lerman, Zeitoun
Causal distributed games
5/21
Distributed controller synthesis
Environment E
acts on
Open distributed system S
S1
S2
S3
S4
Specification
ϕ
Two problems, again
I
I
I
I
Decide whether there exists a distributed controller with finite memory, st.
(S1 ⊗ C1 ) k · · · k (Sn ⊗ Cn ) k E |= ϕ, and if so, synthesize it.
Distributed game system against environment.
Winning distributed strategy ; correct distributed controller.
In general, undecidable (Peterson-Reif 1979, Pnueli-Rosner 1990).
Gastin, Lerman, Zeitoun
Causal distributed games
6/21
Distributed controller synthesis
Environment E
acts on
Controlled
distributed
system
S
Open open
distributed
system
S
C1
S1
S2
C2
C3
S3
S4
C4
Specification
ϕ
Two problems, again
I
I
I
I
Decide whether there exists a distributed controller with finite memory, st.
(S1 ⊗ C1 ) k · · · k (Sn ⊗ Cn ) k E |= ϕ, and if so, synthesize it.
Distributed game system against environment.
Winning distributed strategy ; correct distributed controller.
In general, undecidable (Peterson-Reif 1979, Pnueli-Rosner 1990).
Gastin, Lerman, Zeitoun
Causal distributed games
6/21
Distributed controller synthesis
I
Input
I
I
I
I
I
Communication architecture.
Interactions between system and environment.
Output Finite memory distributed controller, which can inhibit some
controllable actions, and realizes the specification.
As in the sequential games, can use memory.
I Controllers only observe locally, on their own site. No global view.
I eg, the choice for an action cannot depend on a concurrent action.
One requires the memory to be computable in the distributed architecture.
I one allows: extra data in communications (piggy-backing),
I one rules out: additional communications.
Gastin, Lerman, Zeitoun
Causal distributed games
7/21
Outline
1
Motivation
2
A toy example
3
Architectures, games, memory and strategies
4
Main result
Gastin, Lerman, Zeitoun
Causal distributed games
8/21
A toy example: Alice and Bob
Game
I
I
I
I
I
Want to communicate via the same communication line.
At any time, exactly one communication line broken by the environment.
Environment: looks where A&B are connected, and, atomically, chooses to
break a (possibly different) line.
A/B looks its own position and the broken line and, atomically, reconnects to
a (possibly different) line.
States of process i: Qi = {1, 2, 3, 4}.
I
I
I
I
Q1 : Alice’s line.
Q2 : Number of the broken line.
Q3 : Bob’s line.
Dependence between moves A – E – B.
1
2
1
Broken line
2
A
B
3
3
4
4
Gastin, Lerman, Zeitoun
Causal distributed games
Communication architecture
Q1
A
Q2
E B
Q3
read-write ability
read-only ability
9/21
Example: plays
Moves
I
I
I
I
Example of move: B reads 1, 2 on processes 2, 3 and writes 4 on process 3.
Play: sequence of moves.
Winning condition: finite play where A & B are on the same non-broken line
at the end.
In this example, one can add some fairness: E loses if it monopolizes the game
Q1
1
Q2 # 1
Q3
1
1
1 #
1
2
1
A
B 2
3
B 4
E
A
2
B
2
A
B
4
4
E
E
3
A
1
B 1
E
4
B
1
Distributed plays are labeled partial orders
Gastin, Lerman, Zeitoun
Causal distributed games
10/21
Outline
1
Motivation
2
A toy example
3
Architectures, games, memory and strategies
4
Main result
Gastin, Lerman, Zeitoun
Causal distributed games
11/21
Architectures
I
I
I
(Σ, P, R, W )
P: set of processes
Each player a of Σ, atomically
I
I
reads the state of processes in a fixed set R(a) ⊆ P.
writes the state on processes of a fixed set W (a) ⊆ P.
∀a ∈ Σ,
∀a, b ∈ Σ,
I
∅ 6= W (a) ⊆ R(a)
R(a) ∩ W (b) 6= ∅ ⇐⇒ R(b) ∩ W (a) 6= ∅
Dependence: a D b ⇐⇒ R(a) ∩ W (b) 6= ∅ ⇐⇒ R(b) ∩ W (a) 6= ∅
Legal architectures
a
A
E B
b
Forbidden (non symmetric)
Gastin, Lerman, Zeitoun
OK (cellular)
Causal distributed games
a
d
c
e
b
OK (purely asynchronous)
12/21
Games and plays
Game on (Σ, P, R, W )
I
I
I
G = (Σ0 , Σ1 , (Qi )i∈P , (δa )a∈Σ , q 0 , W).
Σ = Σ0 ] Σ1 : partition of the set of players (actions) in teams 0 and 1.
Players of team 0 cooperate against team 1.
Rules of the game for player a are given by δa ⊆ QR(a) × QW (a) .
Finite play
1
1 #
1
A
2
B
2
B
4
E
3
A
1
E
4
Σ0 = {(a, p) | a ∈ Σ, p ∈ QW (a) } ∪ {(#, q 0 )},
I
I
I
Mazurkiewicz trace on Σ0
B
1
(a, p) D (b, q) ⇔ a D b
Move: extension of the current trace following the rules.
The game is neither positional nor turned based.
Winning condition: set of Mazurkiewicz traces W ⊆ R(Σ 0 , D).
Team 0 wins the plays of W and loses other plays.
Gastin, Lerman, Zeitoun
Causal distributed games
13/21
Memory
Memory
I
I
I
I
Each player has a partial view of the history of a play.
Strategies use this view in order to choose next move.
Memoryless: a move can depend only on the current state.
Local memory: a player remembers write actions on the processes it writes to.
1
#
2
A
1
B
1
1
A
E 3
B
2
4
E 4B
B
1
A
2
1
1#
1
1
A
1
E
3
B
2
B
4
E
4
B
1
B
2
Causal memory (intuitively, maximal history a player can observe)
I
I
I
Players gather and forward as much information as possible.
By extension: finite abstraction of full causal memory.
Thanks to the symmetry condition, causal memory can be implemented.
Gastin, Lerman, Zeitoun
Causal distributed games
14/21
Memory
Memory
I
I
I
I
Each player has a partial view of the history of a play.
Strategies use this view in order to choose next move.
Memoryless: a move can depend only on the current state.
Local memory: a player remembers write actions on the processes it writes to.
1
#
2
A
1
B
1
1
A
E 3
B
2
??
4
A
E 4B
B
1
1
A
2
1
1#
1
A
1
E
3
B
2
B
4
A
??
E
4
B
1
B
2
Causal memory (intuitively, maximal history a player can observe)
I
I
I
Players gather and forward as much information as possible.
By extension: finite abstraction of full causal memory.
Thanks to the symmetry condition, causal memory can be implemented.
Gastin, Lerman, Zeitoun
Causal distributed games
14/21
Memory
Memory
I
I
I
I
Each player has a partial view of the history of a play.
Strategies use this view in order to choose next move.
Memoryless: a move can depend only on the current state.
Local memory: a player remembers write actions on the processes it writes to.
1
#
2
A
1
B
1
1
A
E 3
B
2
??
4
A
E 4B
B
1
1
A
2
1
1#
1
A
1
E
3
B
2
B
4
A
??
E
4
B
1
B
2
Causal memory (intuitively, maximal history a player can observe)
I
I
I
Players gather and forward as much information as possible.
By extension: finite abstraction of full causal memory.
Thanks to the symmetry condition, causal memory can be implemented.
Gastin, Lerman, Zeitoun
Causal distributed games
14/21
Memory
Memory
I
I
I
I
Each player has a partial view of the history of a play.
Strategies use this view in order to choose next move.
Memoryless: a move can depend only on the current state.
Local memory: a player remembers write actions on the processes it writes to.
1
#
2
A
1
B
1
1
A
E 3
B
2
??
4
A
E 4B
B
1
1
A
2
1
1#
1
A
1
E
3
B
2
B
4
A
??
E
4
B
1
B
2
Causal memory (intuitively, maximal history a player can observe)
I
I
I
Players gather and forward as much information as possible.
By extension: finite abstraction of full causal memory.
Thanks to the symmetry condition, causal memory can be implemented.
Gastin, Lerman, Zeitoun
Causal distributed games
14/21
Memory
Memory
I
I
I
I
Each player has a partial view of the history of a play.
Strategies use this view in order to choose next move.
Memoryless: a move can depend only on the current state.
Local memory: a player remembers write actions on the processes it writes to.
1
#
2
A
1
B
1
1
A
E 3
B
2
??
4
A
E 4B
B
1
1
A
2
1
1#
1
A
1
E
3
B
2
B
4
A
??
E
4
B
1
B
2
Causal memory (intuitively, maximal history a player can observe)
I
I
I
Players gather and forward as much information as possible.
By extension: finite abstraction of full causal memory.
Thanks to the symmetry condition, causal memory can be implemented.
Gastin, Lerman, Zeitoun
Causal distributed games
14/21
Memory
Memory
I
I
I
I
Each player has a partial view of the history of a play.
Strategies use this view in order to choose next move.
Memoryless: a move can depend only on the current state.
Local memory: a player remembers write actions on the processes it writes to.
1
#
2
A
1
B
1
1
A
E 3
B
2
??
4
A
E 4B
B
1
1
A
2
1
1#
1
A
1
E
3
B
2
B
4
A
??
E
4
B
1
B
2
Causal memory (intuitively, maximal history a player can observe)
I
I
I
Players gather and forward as much information as possible.
By extension: finite abstraction of full causal memory.
Thanks to the symmetry condition, causal memory can be implemented.
Gastin, Lerman, Zeitoun
Causal distributed games
14/21
Winning strategies
Tuple (fa )a∈Σ0 where fa indicates to player a ∈ Σ0 how to play.
Memoryless
Full local memory
Full causal memory fa :
fa : QR(a)
∗
fa : QW (a) QR(a)
M(Σ0 , D0 ) × QR(a)
→ QW (a) ∪ Stop
→ QW (a) ∪ Stop
→ QW (a) ∪ Stop
Rem ∃ memoryless strategy ⇒∃ local mem. strategy ⇒ ∃ causal mem. strategy
f -consistent f -maximal plays
For a strategy f = (fa )a∈Σ0 , one looks at all plays t which are
I consistent with f : each a-move in t is played according f .
a
I
maximal: f predicts Stop for all possible a-move of Σ 0 on t.
Winning strategies
A strategy f is winning if all f -consistent f -maximal plays are in W.
Gastin, Lerman, Zeitoun
Causal distributed games
15/21
(Un)decidability of distributed games
I
Even for reachability conditions, distributed games are not determined.
I
Even for reachability conditions, team 0 may need memory to win.
Rational winning conditions (folklore)
Proposition For rational winning conditions. it is undecidable whether team 0 has a
winning distributed strategy (with causal, local memory, or memoryless).
Peterson-Reif, Madhusudan-Thiagarajan, Bernet-Janin-Walukiewicz
It is undecidable whether team 0 has a winning strategy with local memory even:
I
I
for reachability or safety conditions,
with 3 players + environment.
With causal memory, this undecidability result (with 3 players) does not hold.
Gastin, Lerman, Zeitoun
Causal distributed games
16/21
Outline
1
Motivation
2
A toy example
3
Architectures, games, memory and strategies
4
Main result
Gastin, Lerman, Zeitoun
Causal distributed games
17/21
Causal memory games on cographs
Class of dependence alphabets containing singletons and closed by
I
I
parallel composition
sequential composition
(A ∪ B, DA ∪ DB ),
(A ∪ B, DA ∪ DB ∪ A × B ∪ B × A).
Behaviors on cographs are called series-parallel.
Main decidability result
Theorem Distributed games with causal memory are decidable for controlled
reachability conditions and symmetric series-parallel systems.
Corollary Distributed games with causal memory with recognizable winning condition
on finite traces are decidable for symmetric series-parallel systems.
Proof technique
I
I
I
I
Construction of bounded memory strategy from an arbitrary winning strategy.
Induction on Σ.
Glue strategies for small games obtained by induction.
Main problem: compute information in a distributed way [Th90,CMZ93,EM93].
Gastin, Lerman, Zeitoun
Causal distributed games
18/21
Controlled reachability conditions
I
q(t): global state reached on t.
I
P (t): set of global states seen along finite prefixes of t.
#
|
I
I
I
I
{z
P (t)
}
q(t)
Controlled reachability condition: F ⊆ 2QP × (QP ] {∞}).
A play t wins if (P , q)(t) ∈ F.
Particular cases: reachability, safety.
Example: Reach(X) corresponds to
n
o n
o
F = (P , q) | X ∩ (P ∪ q) 6= ∅ ∪ (P , ∞) | X ∩ P 6= ∅
Gastin, Lerman, Zeitoun
Causal distributed games
19/21
Proof outline
Induction on Σ
Σ0 has a winning strategy ⇒ there is a “small winning strategy.
Difficult case: series product.
#
A
B
A
···
µ
1. From a play, extract subplays on A, B + and induced strategies.
2. Replace these A, B-strategies by strategies with bounded memory (induction).
3. Finally glue these bounded memory strategies on A, B to obtain a strategy on
Σ, with bounded memory.
Main problem
I
I
Players of team 0 have to know on which small game they are is playing.
This information must be computed in a distributed way.
Gastin, Lerman, Zeitoun
Causal distributed games
20/21
Perspectives
I
Generalization to recognizable winning conditions of infinite traces.
I
Generalization to arbitrary dependence alphabets.
I
I
I
I
Generalization to non symmetric architectures.
Reasonable complexity bounds for synthesis.
Control of systems with asynchronous communications (eg, MSC).
May randomized strategies help?
Conjectures
I
I
I
Distributed games with causal memory are decidable.
Alternating asynchronous automata accept only than recognizable languages.
The previous conjectures are equivalent.
Gastin, Lerman, Zeitoun
Causal distributed games
21/21
Perspectives
I
Generalization to recognizable winning conditions of infinite traces.
I
Generalization to arbitrary dependence alphabets.
I
I
I
I
Generalization to non symmetric architectures.
Reasonable complexity bounds for synthesis.
Control of systems with asynchronous communications (eg, MSC).
May randomized strategies help?
Conjectures
I
I
I
Distributed games with causal memory are decidable.
Alternating asynchronous automata accept only than recognizable languages.
The previous conjectures are equivalent.
O
T
E
f
n
d
a
k
l
Thank you!
Gastin, Lerman, Zeitoun
Causal distributed games
21/21