Communicating Timed Automata

1
Communicating Timed Automata

• Pavel Krcál
• Wang Yi
• Uppsala University
• CAV06

2
Goal
Precise moves
mission
A
B
C
D
Commands
High-level inst
requests

• 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
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
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1

c1
b?,c2
d?,c1
d!,c1
a?,c1

a!,c1
c2
5
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 (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1

c1
b?,c2
d?,c1
d!,c1
a?,c1

a!,c1
c2
Asynchronous!
6
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1

c1
b?,c2
d?,c1
d!,c1
a?,c1

a!,c1
c2
7
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1
a

c1
b?,c2
d?,c1
d!,c1
a?,c1

a!,c1
c2
8
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1

c1
b?,c2
d?,c1
d!,c1
a?,c1

a!,c1
c2
9
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1

c1
b?,c2
d?,c1
b
d!,c1
a?,c1

a!,c1
c2
10
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1

c1
b?,c2
d?,c1
b
b
d!,c1
a?,c1

a!,c1
c2
11
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1

c1
b?,c2
d?,c1
b
b
b
d!,c1
a?,c1

a!,c1
c2
12
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1

c1
b?,c2
d?,c1
b
b
d!,c1
a?,c1

a!,c1
c2
13
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1
d

c1
b?,c2
d?,c1
b
b
d!,c1
a?,c1

a!,c1
c2
14
Communicating Finite State Machines

• Finite automata connected by unbounded (FIFO)
  unidirectional channels
• Labels on transitions a letter, read/write,
  channel
• State (s1, , sn, w1, , wm)

a!,c1
b!,c2
a?,c1
d
a

c1
b?,c2
d?,c1
b
b
d!,c1
a?,c1

a!,c1
c2
15
Some Results (Old)

• Turing power
• Equivalent to finite automata
• people Brand, Zafiropulo, Pachl, Purush Iyer,
  Finkel, Abdulla, Jonsson, Schnoebelen,

A
B
A
B
A
A
B
A
B
C
Half duplex
16
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

A
C
B

• Negative results carry over
• Positive results do not carry over (previous
  proofs do not work in the timed setting)

17
Communicating Timed Automata Semantics
A
B

• State (sA, sB, ?A, ?B, w)
• sA, sB locations of A,B
• ?A, ?B clock valuations
• w channel content (a word from S)
• Transitions
• Time pass ?At, ?Bt
• Discrete transition s s, A produces (w
  aw),
• B consumes (wa 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.

18
Communicating Timed Automata Results
A
B

• Accepts non-regular context free languages, e.g.,
  anban
• Only regular languages in the untimed case!
• Equivalent to Petri nets with one unbounded place
  (eager reading One-counter machines)

A
C
B

• Non-context free context sensitive languages,
  e.g., (anbanb)
• Petri nets with two unbounded places (eager
  reading Turing machines)

19
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
Untimed Case Reordering Technique
A
B

• Equivalent to finite automata

• Reordering of the computation
• 1st phase there is at most one letter in each
  channel
• 2nd 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
Reordering Technique
Untimed Case Reordering Technique
a!,c1
d?,c1
a?,c1
b!,c1

c1
?
?
a?,c1
?
d!,c1
b?,c1
22
Reordering Technique
Untimed Case Reordering Technique
a!,c1
d?,c1
a?,c1
b!,c1
a

c1
?
?
a?,c1
?
d!,c1
b?,c1
23
Reordering Technique
Untimed Case Reordering Technique
a!,c1
d?,c1
a?,c1
b!,c1

c1
?
?
a?,c1
?
d!,c1
b?,c1
24
Reordering Technique
Untimed Case Reordering Technique
a!,c1
d?,c1
a?,c1
b!,c1
b

c1
?
?
a?,c1
?
d!,c1
b?,c1
25
Reordering Technique
Untimed Case Reordering Technique
a!,c1
d?,c1
a?,c1
b!,c1
b

c1
?
?
a?,c1
?
d!,c1
b?,c1
26
Reordering Technique
Untimed Case Reordering Technique
a!,c1
d?,c1
a?,c1
b!,c1

c1
?
?
a?,c1
?
d!,c1
b?,c1
27
Reordering Technique
Untimed Case Reordering Technique
a!,c1
d?,c1
a?,c1
b!,c1
d

c1
?
?
a?,c1
?
d!,c1
b?,c1
28
Reordering Technique
Untimed Case Reordering Technique
a!,c1
d?,c1
a?,c1
b!,c1
d

c1
?
?
a?,c1
?
d!,c1
b?,c1
29
Reordering Technique
Untimed Case Reordering Technique
a!,c1
d?,c1
a?,c1
b!,c1

c1
?
?
a?,c1
?
d!,c1
b?,c1
30
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
  CONCUR98, BJLY
• A desired semantics
• language equivalent to the original one
• a state with finite control and a counter

• Reordering of the computation
• 1st phase there is at most one letter in the
  channel
• 2nd phase letters are not read

31
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 FSTTCS05, PK
• tA x ? tB y
• tA x ? 1 (tB y)
• x tA ? tB y
• x a clock of A, y a clock of B, ? ? lt,gt,

Semantics (?A, ?B) satisfies tA x ? tB y
? fr(?A(tA))-fr(?A(x)) ? fr(?B(tB))-fr(?B(y))
32
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

state (sA, sB, DA, DB, tA?tB , CDR, w, N)
unbounded place/counter
finite
33
One Counter Machines

• Counter number of as in the channel
• Control unit locations of A, B
• q1 C goto q2
• A s1
  s2 B s1
  s2
• q1 if C0 then goto q2 else C-- goto q3

?
a!
b?
s2
?
?
s2
b!
a?
s3
A s1
B s1
?
b?
s3
error
34
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
  anbanbanbanb
• Each pair anban 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 as n 2c3d
• C doubling/halving of the number of as (anba2n
  is context free), D multiplication/division by
  three
• Test for zero modulo two/three

35
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?