Distributed games with causal memory are decidable for series-parallel systems

Transcription

1 Distributed games with causal memory are decidable for series-parallel systems Paul Gastin, enjamin Lerman, Marc Zeitoun LIF, Université Paris 7, UMR CNRS 7089 FSTTCS, dec. 0 Gastin, Lerman, Zeitoun Causal distributed games /

2 Outline Motivation toy example 3 rchitectures, games, memory and strategies Main result Gastin, Lerman, Zeitoun Causal distributed games /

3 Outline Motivation toy example 3 rchitectures, games, memory and strategies Main result Gastin, Lerman, Zeitoun Causal distributed games 3/

4 Context: control and synthesis nvironment acts on Open system S Specification ϕ Two problems (Ramadge, Wonham) Decide whether there exists a finite memory controller st. (S C) = ϕ. Synthesis: if so, compute such a controller. Gastin, Lerman, Zeitoun Causal distributed games /

5 Context: control and synthesis nvironment acts on Controller C enables/disables actions provides info Open system S Specification ϕ Two problems (Ramadge, Wonham) Decide whether there exists a finite memory controller st. (S C) = ϕ. Synthesis: if so, compute such a controller. Gastin, Lerman, Zeitoun Causal distributed games /

6 Control as a game Classical framework Games well suited to solve controller synthesis problems. System game graph. Controller a player. nvironment the opponent. Correct behavior winning condition. Finding a controller finding a winning strategy. In general For reasonable sequential systems and specifications, both problems are decidable. Gastin, Lerman, Zeitoun Causal distributed games 5/

7 Distributed controller synthesis nvironment acts on Open distributed system S S S S 3 S Specification ϕ Two problems, again Decide whether there exists a distributed controller with finite memory, st. (S C ) (S n C n ) = ϕ, and if so, synthesize it. Distributed game system against environment. Winning distributed strategy correct distributed controller. In general, undecidable (Peterson-Reif 979, Pnueli-Rosner 990). Gastin, Lerman, Zeitoun Causal distributed games 6/

8 Distributed controller synthesis nvironment acts on Controlled Open open distributed distributed system system S S C S S C C 3 S 3 S C Specification ϕ Two problems, again Decide whether there exists a distributed controller with finite memory, st. (S C ) (S n C n ) = ϕ, and if so, synthesize it. Distributed game system against environment. Winning distributed strategy correct distributed controller. In general, undecidable (Peterson-Reif 979, Pnueli-Rosner 990). Gastin, Lerman, Zeitoun Causal distributed games 6/

9 Distributed controller synthesis Input Communication architecture. Interactions between system and environment. Output Finite memory distributed controller, which can inhibit some controllable actions, and realizes the specification. s in the sequential games, can use memory. Controllers only observe locally, on their own site. No global view. eg, the choice for an action cannot depend on a concurrent action. One requires the memory to be computable in the distributed architecture. one allows: extra data in communications (piggy-backing), one rules out: additional communications. Gastin, Lerman, Zeitoun Causal distributed games 7/

10 Outline Motivation toy example 3 rchitectures, games, memory and strategies Main result Gastin, Lerman, Zeitoun Causal distributed games 8/

11 Game toy example: lice and ob Want to communicate via the same communication line. t any time, exactly one communication line broken by the environment. nvironment: looks where & are connected, and, atomically, chooses to break a (possibly different) line. / looks its own position and the broken line and, atomically, reconnects to a (possibly different) line. States of process i: Qi = {,, 3, }. Q: lice s line. Q: Number of the broken line. Q3: ob s line. Dependence between moves. Q roken line 3 3 Communication architecture Q Q 3 read-write ability read-only ability Gastin, Lerman, Zeitoun Causal distributed games 9/

12 xample: plays Moves xample of move: reads, on processes, 3 and writes on process 3. Play: sequence of moves. Winning condition: finite play where & are on the same non-broken line at the end. In this example, one can add some fairness: loses if it monopolizes the game Q Q Q Distributed plays are labeled partial orders Gastin, Lerman, Zeitoun Causal distributed games 0/

13 Outline Motivation toy example 3 rchitectures, games, memory and strategies Main result Gastin, Lerman, Zeitoun Causal distributed games /

14 rchitectures (Σ, P, R, W ) P: set of processes ach player a of Σ, atomically 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, W (a) R(a) a, b Σ, R(a) W (b) R(b) W (a) Dependence: a D b R(a) W (b) R(b) W (a) Legal architectures a Forbidden (non symmetric) b OK (cellular) a d b c e OK (purely asynchronous) Gastin, Lerman, Zeitoun Causal distributed games /

15 Games and plays Game on (Σ, P, R, W ) G = (Σ0, Σ, (Q i ) i P, (δ a ) a Σ, q 0, W). Σ = Σ0 Σ : partition of the set of players (actions) in teams 0 and. Players of team 0 cooperate against team. Rules of the game for player a are given by δa Q R(a) Q W (a). Finite play Mazurkiewicz trace on Σ 3 Σ = {(a, p) a Σ, p Q W (a) } {(, q 0 )}, (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, D). Team 0 wins the plays of W and loses other plays. Gastin, Lerman, Zeitoun Causal distributed games 3/

16 Gastin, Lerman, Zeitoun Causal distributed games / Memory Memory ach 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. 3 3 Causal memory (intuitively, maximal history a player can observe) Players gather and forward as much information as possible. y extension: finite abstraction of full causal memory. Thanks to the symmetry condition, causal memory can be implemented.

17 Memory Memory ach 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. 3?? 3?? Causal memory (intuitively, maximal history a player can observe) Players gather and forward as much information as possible. y extension: finite abstraction of full causal memory. Thanks to the symmetry condition, causal memory can be implemented. Gastin, Lerman, Zeitoun Causal distributed games /

18 Memory Memory ach 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. 3?? 3?? Causal memory (intuitively, maximal history a player can observe) Players gather and forward as much information as possible. y extension: finite abstraction of full causal memory. Thanks to the symmetry condition, causal memory can be implemented. Gastin, Lerman, Zeitoun Causal distributed games /

19 Memory Memory ach 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. 3?? 3?? Causal memory (intuitively, maximal history a player can observe) Players gather and forward as much information as possible. y extension: finite abstraction of full causal memory. Thanks to the symmetry condition, causal memory can be implemented. Gastin, Lerman, Zeitoun Causal distributed games /

20 Memory Memory ach 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. 3?? 3?? Causal memory (intuitively, maximal history a player can observe) Players gather and forward as much information as possible. y extension: finite abstraction of full causal memory. Thanks to the symmetry condition, causal memory can be implemented. Gastin, Lerman, Zeitoun Causal distributed games /

21 Memory Memory ach 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. 3?? 3?? Causal memory (intuitively, maximal history a player can observe) Players gather and forward as much information as possible. y extension: finite abstraction of full causal memory. Thanks to the symmetry condition, causal memory can be implemented. Gastin, Lerman, Zeitoun Causal distributed games /

22 Winning strategies Tuple (f a ) a Σ0 where f a indicates to player a Σ 0 how to play. Memoryless f a : Q R(a) Q W (a) Stop Full local memory f a : Q W (a) Q R(a) Q W (a) Stop Full causal memory f a : M(Σ, D ) Q R(a) Q W (a) Stop Rem memoryless strategy local mem. strategy causal mem. strategy f-consistent f-maximal plays For a strategy f = (f a ) a Σ0, one looks at all plays t which are consistent with f: each a-move in t is played according fa. maximal: f predicts Stop for all possible a-move of Σ0 on t. Winning strategies strategy f is winning if all f-consistent f-maximal plays are in W. Gastin, Lerman, Zeitoun Causal distributed games 5/

23 (Un)decidability of distributed games ven for reachability conditions, distributed games are not determined. ven 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, ernet-janin-walukiewicz It is undecidable whether team 0 has a winning strategy with local memory even: 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 6/

24 Outline Motivation toy example 3 rchitectures, games, memory and strategies Main result Gastin, Lerman, Zeitoun Causal distributed games 7/

25 Causal memory games on cographs Class of dependence alphabets containing singletons and closed by parallel composition (, D D ), sequential composition (, D D ). ehaviors 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 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,M93]. Gastin, Lerman, Zeitoun Causal distributed games 8/

26 Controlled reachability conditions q(t): global state reached on t. P (t): set of global states seen along finite prefixes of t. }{{} q(t) P (t) Controlled reachability condition: F Q P (Q P { }). play t wins if (P, q)(t) F. Particular cases: reachability, safety. xample: Reach(X) corresponds to { } F = (P, q) X (P q) { } (P, ) X P Gastin, Lerman, Zeitoun Causal distributed games 9/

27 Induction on Σ Proof outline Σ 0 has a winning strategy there is a small winning strategy. Difficult case: series product. µ. From a play, extract subplays on, + and induced strategies.. Replace these, -strategies by strategies with bounded memory (induction). 3. Finally glue these bounded memory strategies on, to obtain a strategy on Σ, with bounded memory. Main problem 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 0/

28 Perspectives Generalization to recognizable winning conditions of infinite traces. Generalization to arbitrary dependence alphabets. Generalization to non symmetric architectures. Reasonable complexity bounds for synthesis. Control of systems with asynchronous communications (eg, MSC). May randomized strategies help? Conjectures Distributed games with causal memory are decidable. lternating asynchronous automata accept only than recognizable languages. The previous conjectures are equivalent. Gastin, Lerman, Zeitoun Causal distributed games /

29 Perspectives Generalization to recognizable winning conditions of infinite traces. Generalization to arbitrary dependence alphabets. Generalization to non symmetric architectures. Reasonable complexity bounds for synthesis. Control of systems with asynchronous communications (eg, MSC). May randomized strategies help? Conjectures Distributed games with causal memory are decidable. lternating asynchronous automata accept only than recognizable languages. The previous conjectures are equivalent. n O d f T a l k Thank you! Gastin, Lerman, Zeitoun Causal distributed games /