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Communicating Timed Automata

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Channel machines (Communicating finite state machines) Computing in the common (real) time ... Finite automata connected by unbounded (FIFO) unidirectional channels ... – PowerPoint PPT presentation

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Title: 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?
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