Communicating Timed Automata: Channel Systems with Dense Time for Real-Time Tasks

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

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

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 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 b 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 b b 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 b b b 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 b b 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 d … c1 b?,c2 b b 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 d … c1 b?,c2 b b d?,c1 … 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 B C • Negative results carry over • Positive results – do not carry over (previous proofs do not work in the timed setting)

Communicating Timed Automata – Semantics • State: (sA, sB, A, B, w) • sA, sB – 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. A B

Communicating Timed Automata – Results • 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 B A B C • Non-context free context sensitive languages, e.g., (anbanb)* • Petri nets with two unbounded places (eager reading: Turing machines)

Untimed Case – Reordering Technique • Equivalent to finite automata A B … • 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 b … 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

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 • 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 [FSTTCS’05, PK] • tA – x ◊ tB – y • tA – x ◊ 1 – (tB – y) • x – tA ◊ tB – y • … • x – a clock of A, y – a clock of B, ◊  {<,>,=} 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 a’s in the channel • Control unit – locations of A, B • q1: C++; goto q2 A: s1 s2 B:s1 s2 • q1: if C=0 then goto q2 else C--; goto q3  a! b? s2   s2 b! a? s3 A: s1 B: s1  s3 b? 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 a’s: n = 2c∙3d • C: doubling/halving of the number of a’s (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?

Communicating Timed Automata: Channel Systems with Dense Time for Real-Time Tasks