Concurrent automata vs. asynchronous systems

Transcription

1 Concurrent automata vs. asynchronous systems Une application d un résultat d ndré rnold avec l aide du théorème de Zielonka Réunion TGD 3 novembre 2005 Rémi MORIN

2 Concurrent automata vs. asynchronous systems Rémi MORIN Laboratoire d Informatique Fondamentale de Marseille 1 Background with 4 examples 2 Technical result with proof on a simple example 3 Extension to dynamic independencies Zielonka s result rnold s result 4 Discussion

3 From finite automata to regular trees 1 Nodes are words but letters are lost!

4 Modeling of concurrency 2 Since the two words and are equivalent:.

5 be an alphabet. n independence relation over Introduction to Mazurkiewicz traces [Mazurkiewicz 1977] 3 Let is a binary, symmetric, and irreflexive relation. The trace equivalence associated with an independence alphabet is the least equivalence relation over such that for some. It is a congruence. trace is the equivalence class of a word. trace is a prefix of a trace if. In that case we put.

6 $ # # " synchronous systems [Bednarczyk 1988] 4 n asynchronous system (S) over is a deterministic transition system! such that, : Example ' * ( ' Each asynchronous system is associated with a set of words, a set of traces ) and a domain (.

7 n example with some questions [Husson 1996] 5 Is it the domain of an asynchronous system? Is it the domain of an asynchronous system with finitely many states? Is it the domain of an asynchronous system with a finite alphabet? Here we focus on finite asynchronous systems

8 First try 6 with We have. States correspond to traces!

9 +,.- +,0/ +,21 +,.3 4,.- 4,.- 4,.- 4,.- +,0/ +,21 4,0/ 4,0/ +,.3 4,0/ +,21 +,.3 4,21 4,21 +,.3 Second try with an infinite alphabet 7 with 65 7 if5 8 7

10 synchronous systems and their domains Forward-stability vs. coherence 7

11 $ # $ $ 9 # Two closure properties Two subclasses 8 n asynchronous system is forward-stable if, : n asynchronous system is coherent if ;:pairwise independent:, < < < < < $ 9 < < Theorem (First result and main technical contribution) For any coherent finite S such that the two domains ( there exists a forward-stable finite S ( and >= >= are isomorphic.

12 First easy step: Forward-stable completion with some fresh states 9 Observe that - is coherent but not forward-stable Can you guess a forward-stable? - at this point? x B B C D C E

13 Z W R T T and W X T X B B X RR M V B T with C D C L S T BR X S;Y BR X S Y BR S FHG IJ K Second step: dding problematic loops 10 We use an extended alphabet if and ; if This involves some exponential state-explosion! is forward-stable, too! BR X S Y T. S TU MON P M Q

14 Third step: Implementation without deadlock as a synchronized product 11 By means of Zielonka s theorem [Zielonka 1987, Ştefănescu 2003]... ` ` a a [H\ ]^ F G IJ F G IJ b c _ b d K K This involves some further extended alphabet and some new (exponential) state-explosion!

15 K C e C f N N [ \ ]^ _ [H\ ] ^ _ N F G IJ K N FHG IJ K The fourth step is easy: Separate occurrences of the same action 12 i.e. F G IJ K N N F G I J N N

16 R B B B D C D _ l B N N N N C fkj P C N = N = N = = [H\ ]^ = N N = The last step removes fresh states and loops: Synchronise because of the S T -loop g h with g i 13

17 synchronous systems and their domains Forward-stability vs. coherence Extension to dynamic independencies 13

18 m " # o ' # w # " " " y z v p s x #q ##t and #r #t " " " " p s s p m ##q and #r #r " " p p " utomata with concurrency relations [Droste 1990] 14 n automaton with concurrency relations over the alphabet structure! mn o such that is a 1. is a non-empty set of states, with an initial state ; 2. is a set of transitions; 3. if then #q ; 4. mn is a family of independence relations over ; 5. if m then there exist. The trace equivalence over such that u associated with, is the least equivalence, { u

19 #q #q : : # # # # #t : : # #r #~ #r #~ : : #} #} # Stably concurrent automata [DrosteKuske 1994] 15 n automaton with concurrency relations automaton (SC) if, ;: is called a stably concurrent (distinct):

20 From dynamic to static independence relations 16 By means of rnold s result on CCI sets of P-traces [rnold 1991] Theorem (Second result) For any finite SC domains ( there exists a finite S ( and >= >= are isomorphic. such that the two This connection preserves coherence. Composing the two results, we get: Corollary (Main result of this paper) For any finite coherent SC such that the two domains ( there exists a finite forward-stable S ( and >= >= are isomorphic. This subsumes two results from [Schmitt 98] and [Thiagarajan 02].

21 synchronous systems and their domains Forward-stability vs. coherence Extension to dynamic independencies Short discussion about the cube axioms 16

22 Pq Pr Pt P ˆ partial order What are the domains of asynchronous systems? 17 Theorem corresponds to some possibly infinite S iff Š ƒ : it has a least element ; : it is finitary, i.e. 2 is finite for all ; : : i.e. is a prime-algebraic partial order.

23 P What are the domains of forward-stable asynchronous systems? 18 Theorem partial order corresponds to some possibly infinite forwardstable S iff it satisfies Pq,,, Pr P Pt, and : i.e. it is a coherent prime-algebraic partial order.

24 Œ Ž Where do these cube axioms come from? 19 Œ Theorem Let be an automaton with concurrency relations. The domain ( concurrent automaton. satisfies Pq,, Pr Pt and P iff it is a stably Theorem Let be an automaton with concurrency relations. The domain ( stably concurrent automaton. satisfies Pq,,, Pr P Pt and P iff it is a coherent