Loading...

PPT – Communicating Timed Automata PowerPoint presentation | free to view - id: 22dbfd-NzIwN

The Adobe Flash plugin is needed to view this content

Communicating Timed Automata

- Pavel Krcál
- Wang Yi
- Uppsala University
- CAV06

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

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

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

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!

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

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

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

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

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

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

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

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

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

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

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)

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.

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)

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

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

Reordering Technique

Untimed Case Reordering Technique

a!,c1

d?,c1

a?,c1

b!,c1

c1

?

?

a?,c1

?

d!,c1

b?,c1

Reordering Technique

Untimed Case Reordering Technique

a!,c1

d?,c1

a?,c1

b!,c1

a

c1

?

?

a?,c1

?

d!,c1

b?,c1

Reordering Technique

Untimed Case Reordering Technique

a!,c1

d?,c1

a?,c1

b!,c1

c1

?

?

a?,c1

?

d!,c1

b?,c1

Reordering Technique

Untimed Case Reordering Technique

a!,c1

d?,c1

a?,c1

b!,c1

b

c1

?

?

a?,c1

?

d!,c1

b?,c1

Reordering Technique

Untimed Case Reordering Technique

a!,c1

d?,c1

a?,c1

b!,c1

b

c1

?

?

a?,c1

?

d!,c1

b?,c1

Reordering Technique

Untimed Case Reordering Technique

a!,c1

d?,c1

a?,c1

b!,c1

c1

?

?

a?,c1

?

d!,c1

b?,c1

Reordering Technique

Untimed Case Reordering Technique

a!,c1

d?,c1

a?,c1

b!,c1

d

c1

?

?

a?,c1

?

d!,c1

b?,c1

Reordering Technique

Untimed Case Reordering Technique

a!,c1

d?,c1

a?,c1

b!,c1

d

c1

?

?

a?,c1

?

d!,c1

b?,c1

Reordering Technique

Untimed Case Reordering Technique

a!,c1

d?,c1

a?,c1

b!,c1

c1

?

?

a?,c1

?

d!,c1

b?,c1

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

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))

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

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

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

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?