1. Communicating Timed Automata Pavel Krčál Wang Yi Uppsala University [CAV’06]

2. 2006/07/06 Pavel Krčál, Communicating Timed Automata Goal ABCD CommandsHigh-level inst Precise moves requests mission Real time tasks A, B, C, D Read inputs from channels and write output to channels Channel under/overflow is an issue Channel machines (Communicating finite state machines) Computing in the common (real) time Verification – reachability, boundedness

3. 2006/07/06 Pavel Krčál, Communicating Timed Automata Outline Communicating Finite State Machines (Channel Systems)  Known results Communicating Timed Automata  Definition, Subclasses  Main results One Channel  Reordering technique  How to handle the dense time Two Channels  Reordering technique  Eager reading – Turing power

4. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m ) a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2

5. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines Finite automata connected by unbounded (FIFO) unidirectional channels – a model for protocols Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m ) a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 Asynchronous!

6. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m )

7. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 a Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m )

8. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m )

9. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 b Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m )

10. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 bb Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m )

11. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 bbb Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m )

12. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 bb Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m )

13. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 bb d Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m )

14. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Finite State Machines a!,c1 b?,c2 d!,c1 a!,c1 a?,c1 d?,c1 b!,c2 a?,c1 … … c1 c2 bb da Finite automata connected by unbounded (FIFO) unidirectional channels Labels on transitions: a letter, read/write, channel State: (s 1, …, s n, w 1, …, w m )

15. 2006/07/06 Pavel Krčál, Communicating Timed Automata Some Results (Old) Turing power Equivalent to finite automata people: Brand, Zafiropulo, Pachl, Purush Iyer, Finkel, Abdulla, Jonsson, Schnoebelen, … A B A A B ABC AB Half duplex

16. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Timed Automata (CTA) Replace Finite Automata by Timed Automata  Communication via unbounded FIFO channels  Time is global (time passes globally and for all automata in the same pace)  A, B, C – Timed Automata ACB Negative results carry over Positive results – do not carry over (previous proofs do not work in the timed setting)

17. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Timed Automata – Semantics State: (s A, s B, A, B, w)  s A, s B – locations of A,B  A, B – clock valuations  w – channel content (a word from Σ * ) Transitions:  Time pass: A +t, B +t  Discrete transition: s s’, A produces (w a∙w), B consumes (w∙a w); timed automata guards  Lazy vs. eager reading Language: accepting states, words produced by A We show that both dense & discrete time give the same expressivity. AB

18. 2006/07/06 Pavel Krčál, Communicating Timed Automata Communicating Timed Automata – Results  Accepts non-regular context free languages, e.g., a n ba n  Only regular languages in the untimed case!  Equivalent to Petri nets with one unbounded place (eager reading: One-counter machines) ACB  Non-context free context sensitive languages, e.g., (a n ba n b) *  Petri nets with two unbounded places (eager reading: Turing machines) AB

19. 2006/07/06 Pavel Krčál, Communicating Timed Automata Main Proof Ingredients One channel:  Desynchronization of timed automata (we are able to desynchronize two timed automata and resynchronize them correctly later)  We need to remember clock difference relations and a counter Two channels:  The same desynchronization of timed automata  Ability to check some context sensitive properties (two channels check context free properties in an alternating manner)  With the eager reading, we can check that the word encodes a computation of a two counter machine

20. 2006/07/06 Pavel Krčál, Communicating Timed Automata Untimed Case – Reordering Technique Equivalent to finite automata AB … Reordering of the computation  1 st phase: there is at most one letter in each channel  2 nd phase: letters are not read When A produces a letter then it stops. B runs until it reads the letter from the channel. Then A continues again…

21. 2006/07/06 Pavel Krčál, Communicating Timed Automata Reordering Technique a!,c1 b!,c1 d!,c1  a?,c1  d?,c1 a?,c1 … c1 b?,c1  Untimed Case – Reordering Technique

22. 2006/07/06 Pavel Krčál, Communicating Timed Automata Reordering Technique a!,c1 b!,c1 d!,c1  a?,c1  d?,c1 a?,c1 … c1 b?,c1  a Untimed Case – Reordering Technique

23. 2006/07/06 Pavel Krčál, Communicating Timed Automata Reordering Technique a!,c1 b!,c1 d!,c1  a?,c1  d?,c1 a?,c1 … c1 b?,c1  Untimed Case – Reordering Technique

24. 2006/07/06 Pavel Krčál, Communicating Timed Automata Reordering Technique a!,c1 b!,c1 d!,c1  a?,c1  d?,c1 a?,c1 … c1 b?,c1  b Untimed Case – Reordering Technique

25. 2006/07/06 Pavel Krčál, Communicating Timed Automata Reordering Technique a!,c1 b!,c1 d!,c1  a?,c1  d?,c1 a?,c1 … c1 b?,c1  b Untimed Case – Reordering Technique

26. 2006/07/06 Pavel Krčál, Communicating Timed Automata Reordering Technique a!,c1 b!,c1 d!,c1  a?,c1  d?,c1 a?,c1 … c1 b?,c1  Untimed Case – Reordering Technique

27. 2006/07/06 Pavel Krčál, Communicating Timed Automata Reordering Technique a!,c1 b!,c1 d!,c1  a?,c1  d?,c1 a?,c1 … c1 b?,c1  d Untimed Case – Reordering Technique

28. 2006/07/06 Pavel Krčál, Communicating Timed Automata Reordering Technique a!,c1 b!,c1 d!,c1  a?,c1  d?,c1 a?,c1 … c1 b?,c1  d Untimed Case – Reordering Technique

29. 2006/07/06 Pavel Krčál, Communicating Timed Automata Reordering Technique a!,c1 b!,c1 d!,c1  a?,c1  d?,c1 a?,c1 … c1 b?,c1  Untimed Case – Reordering Technique

30. 2006/07/06 Pavel Krčál, Communicating Timed Automata CTA with One Channel We try to modify the reordering technique such that it works also for timed automata But for this we need to desynchronize timed automata - desynchronized semantics [CONCUR’98, BJLY] A desired semantics:  language equivalent to the original one  a state with finite control and a counter Reordering of the computation  1 st phase: there is at most one letter in the channel  2 nd phase: letters are not read

31. 2006/07/06 Pavel Krčál, Communicating Timed Automata CTA with One Channel We are able to desynchronize timed automata and resynchronize them correctly later! We need to limit all possible resynchronizations (only some are correct) Clock Difference Relations [FSTTCS’05, PK]  t A – x ◊ t B – y  t A – x ◊ 1 – (t B – y)  x – t A ◊ t B – y  …  x – a clock of A, y – a clock of B, ◊  {,=} Semantics: ( A, B ) satisfies t A – x ◊ t B – y  fr( A (t A ))-fr( A (x)) ◊ fr( B (t B ))-fr( B (y))

32. 2006/07/06 Pavel Krčál, Communicating Timed Automata CTA with One Channel Desynchronization + CDR Now we can encode the state of a CTA (with desynchronized semantics) by finite state control and a counter finite unbounded place/counter state: (s A, s B, D A, D B, t A ◊t B, CDR, w, N)

33. 2006/07/06 Pavel Krčál, Communicating Timed Automata One Counter Machines Counter – number of a’s in the channel Control unit – locations of A, B q 1 : C++; goto q 2 A: s 1 s 2 B: s 1 s 2 q 1 : if C=0 then goto q 2 else C--; goto q 3 a!  b!    a? b? A: s 1 s 2 s 3 B: s 1 s 2 s 3 error

34. 2006/07/06 Pavel Krčál, Communicating Timed Automata CTA with Two Channels Similar desynchronization, needs two unbounded places Eager reading: can simulate Two-Counter Machines  Two channels can check whether the input word is a n ba n ba n ba n b…  Each pair a n ba n is context free (one channel is enough to check this), overlap is checked using ‘alternation’  Counters C,D (valued c,d) are encoded by number of a’s: n = 2c∙3d  C: doubling/halving of the number of a’s (a n ba 2n is context free), D: multiplication/division by three  Test for zero: modulo two/three

35. 2006/07/06 Pavel Krčál, Communicating Timed Automata Conclusions Synchrony makes analysis more difficult One channel:  Some context free languages (contrast with the asynchronous case)  Petri Nets with one unbounded place/One-counter machine  Reachability/boundedness questions decidable Two channels:  Some context sensitive languages  Petri Nets with two unbounded places/Turing Machine  Eager reading – most questions undecidable Further questions?  Abstraction of the channels?  Controllers for CTA?