Games and distributed synthesis Igor Walukiewicz, CNRS Bordeaux - PowerPoint PPT Presentation

Games and distributed synthesis Igor Walukiewicz, CNRS Bordeaux University

Input C Church, ``Applications of recursive arithmetics to the problem of circuit synthesis'', 1957 Output Controller Plant b a Ramadge & Wonham, ``The control of a discrete event systems’', 1987-89 b

Here we revisit this setting with a very simple model of distributed systems Zielonka ``Note on finite asynchronous automata'', 1987 Emerson & Clarke, ``Using branching time temporal logic to synthesize synchronization skeletons’', 1982 This is not about Pnueli & Rosner model We get multi-player partial information games that are (or may be) decidable.

Church synthesis problem (1957) Input At every cycle the device reads In and outputs Out. C C : A + in → A out Output Behaviour is an infinite sequence: w 0 C ( w 0 ) w 1 C ( w 0 w 1 ) w 2 C ( w 0 w 1 w 2 ) . . . Church synthesis problem C C C given α find C such that for all w ∈ A ω in : C w C ( w ) w C ( w w ) w C ( w w w ) . . . ✏ α . ∈ w 0 C ( w 0 ) w 1 C ( w 0 w 1 ) w 2 C ( w 0 w 1 w 2 ) . . . ✏ α .

Church synthesis problem C C C given α find C such that for all w ∈ A ω in : C w C ( w ) w C ( w w ) w C ( w w w ) . . . ✏ α . ∈ w 0 C ( w 0 ) w 1 C ( w 0 w 1 ) w 2 C ( w 0 w 1 w 2 ) . . . ✏ α . Game between Environment (In) and Player (Out) 1 0 Every infinite play should satisfy α 0 0 1 1 C 0 1 0 1 ∃ Z. strategy ( Z ) ∧ ∀ P. play ( P, Z ) ⇒ α ( P ) 0 0 1 0 0 1 1 1 Thm [Rabin’69, Büchi and Landweber’69] There is an algorithm that given an MSOL α decides if such a C exist is. If there is one then there is a C implementable by a finite automaton.

Game between Environment (In) and Player (Out) 1 0 Every infinite play should satisfy α 0 0 1 1 C 0 1 0 1 ∃ Z. strategy ( Z ) ∧ ∀ P. play ( P, Z ) ⇒ α ( P ) 0 0 1 0 0 1 1 1 Extensions: • Quality of strategies: 
 promptness of response, 
 permissive strategies, 
 memory optimal strategies. • Context free game graphs and specifications. • Quantitative conditions. • Real-time. • Randomisation

Ramadge and Wonham setting (1989) | I | O ∃ ∧ ∀ ⇒ a x Plant: a finite deterministic automaton P over an alphabet A. y A is divided into A ctr and A uctr . b a x Controller: a finite deterministic automaton C over A s.t y b control ∀ s ∈ S C ∀ a ∈ A uctr . δ ( s, a )defined. x,y « No b before y, after y only b’s » Controlled plant: P × C P × C R&W control problem: given P and α find C such the P × C ✏ α .

Church setting is a special case of Ramadge and Wonham a k a 1 . . . . . . And vice versa.

Simple specifications P ? K ⊆ x P C P × C Given P and another finite automaton K find C such that: L ( P × C ) ⊆ L ( K ) P × C ⊆ non-blocking every reachable state in P × C has an outgoing edge. Take A = P × K . 1. Mark red all states ( p, k ) having no transitions from it. 2. Mark red all states having a path on A ∗ uctr to a red state. 3. Mark red all states having transitions only to red states; go back to step 2. Automaton C is the obtained by removing all red states in A . if loc ( a ) ∩ loc ( b ) = ∅ and wabv ∈ L ( A ) then wbav ∈ L ( A ).

Ramadge and Wonham for distributed systems

Ramadge and Wonham for distributed systems We do the same as before but instead of finite automata we take their distributed version.

Zielonka automata (asynchrnous automata) p c c a a q b b b r d d Local states sets S p , S q , S r . Local transitions δ c : S p → S p δ b : S q × S r → S q × S r , . . . Process p executes local action c ,

Zielonka automata (asynchronous automata) 0 1 p c c a a 0 q b b b 0 r d d Local states sets S p , S q , S r . Local transitions δ c : S p → S p δ b : S q × S r → S q × S r , . . . Process p executes local action c , Processes q , r synchronize on action b (and update states). . . .

Zielonka automata (asynchrnous automata) 0 1 p c c a a 1 0 q b b b 0 1 r d d Local states sets S p , S q , S r . Local transitions δ c : S p → S p δ b : S q × S r → S q × S r , . . . Process p executes local action c , Processes q , r synchronize on action b (and update states). . . .

Distributed alphabet P 1 a 1 P : finite set of processes. A : finite set of letters. a 2 P 2 a 2 loc : A æ ( 2 P \ ÿ ) : distribution of d P 3 a 3 letters over processes.

Zielonka’s theorem Distributed alphabet P 1 a 1 P : finite set of processes. A : finite set of letters. a 2 P 2 a 2 loc : A æ ( 2 P \ ÿ ) : distribution of d P 3 a 3 letters over processes. if loc ( a ) ∩ loc ( b ) = ∅ and wabv ∈ L ( AA ) then wbav ∈ L ( AA ). A language is trace closed if it is closed under permutation of independent letters. Thm [Zielonka 87] Every trace closed regular language can be recognised by a Zielonka automaton.

R&W for Zielonka automata ⊆ KK PP ?? x Plant: a Zielonka automaton PP over a distributed alphabet A. A is divided into A ctr and A uctr . Controller: a Zielonka automaton CC over A s.t. CC control ∀ s ∈ S C ∀ a ∈ A uctr . δ ( s, a ) defined. ∀ ∈ ∀ ∈ Controlled plant: PP × CC PP × CC Given PP and another Zielonka automaton KK find CC such that: PP × CC ⊆ K L ( PP × CC ) ⊆ L ( K ) non-blocking every reachable state in PP × CC has an outgoing edge.

A simple example x=g(y) or y=f(x)? Do there exist such f(x) and g(y)? Yes, if x,y range over {0,1} No, if x,y range over {0,1,2}

Causal memory a 1 , b 1 P 1 x 1 , y 1 c 12 � � � � � � a 2 , b 2 P 2 x 2 , y 2 c 12 c 23 � � � � a 3 , b 3 P 3 x 3 , y 3 c 23 � � � « No b before y, after y only b’s » c 1 , 2 c 1 , 2 x 1 a 1 x 1 b 1 c 2 , 3 x 2 x 2 b 2 y 3 b 3

Causal memory a 1 , b 1 x 1 , y 1 c 12 � � � � � � a 2 , b 2 x 2 , y 2 c 12 c 23 � � � � a 3 , b 3 x 3 , y 3 c 23 � � � « No b before y, after y only b’s » c 1 , 2 c 1 , 2 x 1 a 1 x 1 b 1 c 2 , 3 x 2 x 2 b 2 y 3 b 3

What we know Thm [Genest & Gimbert & Muscholl & W.’13, Muscholl & W.’14] The R&W problem for Zielonka automata is decidable for automata with acyclic communication graph. P 1 a 1 P : finite set of processes. A : finite set of letters. a 2 P 2 a 2 loc : A æ ( 2 P \ ÿ ) : distribution of d P 3 a 3 letters over processes. Communication graph: nodes are processes, edges ( p, q ) if p, q ∈ loc ( a ) for some a .

What we know Thm [Genest & Gimbert & Muscholl & W.’13, Muscholl & W.’14] The R&W for Zielonka automata is decidable for automata with acyclic communication graph. P 1 a 1 P : finite set of processes. A : finite set of letters. a 2 P 2 a 2 loc : A æ ( 2 P \ ÿ ) : distribution of d P 3 a 3 letters over processes. Communication graph: nodes are processes, edges ( p, q ) if p, q ∈ loc ( a ) for some a .

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Conclusions Input We have started with the classical Church setting C Output + nonblocking L ( P × C ) ⊆ L ( K ) Took the R&W formulation of it p c c And instantiated with a very simple distributed model a a q of Zielonka automata b b b r d d This gives us a notion of games with partial information (causal memory) that we do not know how to solve. n-players cooperating against the environment, each having causal view of the past: when two players perform a common action, they exchange all information about what they have seen in the past.