17.02.05 Lecture5 1

Timed Automata II

CS 5270 Lecture 5

17.02.05 Lecture5 2

Goals

• Timed Transition Systems:– Review– Definition– Timed behaviors.– Parallel composition– Reachability

• UPPAAL

17.02.05 Lecture5 3

Timed Transition Systems

• Timed Transition Systems =

Transition Systems + Clock Variables.• Clock variables.

– Used to record the passage of (real) time.– Act like Timers.– Can be read.– Transitions constrained (guarded) by current values

of clock variables.– Can be reset to 0 during a transition.

Continue to evolve (in real time) until reset.

17.02.05 Lecture5 4

Using Clock Variables

Hot; y On-ac; x OK

Off-acx 5Off-ac

Spec. :

Turn off ac if the temperature is OK or any time after 5 units of time has elapsed since turning it on.

Turn on ac within 2 time units after receiving Hot signal.

y ≤ 2

Using Clock Variables

Hot; y On-ac; x OK

Off-acx ¸ 5 Off-ac

y ≤ 2

Three components:

Action on-ac

Reset x In general, more than one clock can be reset.

Guard y ≤ 2 In general, more than one constraint can be used.

17.02.05 Lecture5 6

State Invariants

• A clock constraint is associated with each state: state invariant– The system can stay in the state only as long

as the state’s invariant is not violated.

• For time points which violate the invariant one expects an output transition to be enabled.– Otherwise a time deadlock.

The progress of time is blocked (in the model!).

17.02.05 Lecture5 7

Timed Transition Systems: Definition

• Clock Constraints:• X, a finite set of clocks (clock variables).

– x X

• (X), the set of clock constraints over X.– (X) = x ≤ c | x ≥ c | x < c | x >c | 1 2 | – c Q

Q , the set of non-negative rational numbers.

• 2X --- The set of subsets of X X = {x, y} 2X = { , {x}, {y}, {x, y} }

17.02.05 Lecture5 8

Timed Transition Systems

• TTS = (S, sin, Act, X, I, )

– S is a finite set of states.

– sin, initial state

– Act, a finite set of actions– X, a finite set of clock variables.– I : S (X), assigns a clock invariant to

each state.– S Act 2X (X) S , the

transition relation.

9

Example

s3 x > 2 y 3

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, add, {x}, True, s1) is a transition in TTS.

10

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, add, {x}, True, s1) is a transition in TTS.X = {x0, x1,..xn}True = x0 ≥ 0

X = {x, y} True = x 0

11

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s1, data, {y}, True, s2) is a transition in TTS.

(s2, ack.add, ;, x > 2, s3) is a transition in TTS.

(s3, ack.data, ;, y ≤ 5, s0) is a transition in TTS

17.02.05 Lecture5 12

Behaviors

s0 s1 s2

s3

s4

a b

c

d

s0 a s1 b s2 c s3

s0 a s1 b s2 d s4

Are both paths (runs).

17.02.05 Lecture5 13

Behaviors

s0 s1 s2

s3

s4

a ; x b

c

d

s0 a s1 b s2 c s3 is NOT a run!

s0 a s1 b s2 d s4 is a run.

Computing behaviors is difficult.

x = 1

x < 1

x ≥ 2

17.02.05 Lecture5 14

Behaviors

• TTS = (S, sin, Act, X, I, )

• We associate a “normal” transition system with TTS while taking time into account:– TSTTS = (S, sin, Act R, )

– R, non-negative reals S Act R S

• TSTTS is, almost always, an infinite transition system!

17.02.05 Lecture5 15

Behaviors

• TTS = (S, sin, Act, X, I, )

• TSTTS = (S, sin, Act R, )

• S = S V

• V --- Valuations– A valuation says what the current values of

each clock variable is. v : X R

17.02.05 Lecture5 16

Behaviors

s0 s1 s2

s3

s4

a ; x b

c

d

(s1, 0) (s2, 1.8) (s4, ) are t-states.

(s3, 5) is a t-state but not reachable.

x = 1

x < 1

x ≥ 2

17.02.05 Lecture5 17

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s1, (2, 5) ) is a state; (s1, V) V(x) = 2 V(y) = 5

(s2, V’) is a state. V’(x) = 15 V’(y) = 0 (s2, (15, 0) )

17.02.05 Lecture5 18

Behaviors

• TTS = (S, sin, Act, X, I, !)

• TSTTS = (S, sin, Act R, ) R, non-negative reals S Act R S

• S = S V

• sin = (sin, VZERO)

– VZERO (x) = 0 for every x in X.

17.02.05 Lecture5 19

Behaviors

• There will be two types of transitions.• Time pass move:

– (s, v) (s, v’) t units of time pass starting from V. V’ (x) = V(x) + t for every x.

• Instantaneous transition.– (s, v) (s’, v’)– Some transition in the timed automaton is taken at v;

the guard is satisfied; v’ is the same as v except for resets.

t

a

20

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s1, (0, 5)) (s2, (1, 6) )1

Time- passing move?

21

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s1, (0, 5)) (s1, (1, 7) )1

Time- passing move?

22

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s1, (0, 5)) (s1, (0, 5) )0

Time- passing move?

23

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s1, (0, 5)) (s1, (2, 7.7) )2

Time- passing move?

24

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, (3, 3) ) add

(s1, (0, 3)) a transition in TS?

25

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, (3, 3) ) add

(s3, (0, 3)) a transition in TS?

26

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, (3, 3) ) add

(s1, (0, 4)) a transition in TS?

27

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, (0, 0) ) add

(s1, (0, 0)) a transition in TS?

28

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, (0, 0) ) add

(s1, (0, 0))

data

(s2, (0, 0) )

Is this possible?

29

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s2, (3, 2)) (s3, (3, 2))ack.add

(s3, (5, 5)) (s0, (5, 5))ack.data

30

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

TSTTS will have (uncountably) infinite number of states and transitions.

(s0, (0, 0))

(s0, (.8, .8))

.8 …………..

(s0, (1, 1))

1

17.02.05 Lecture5 31

Timed Behaviors

• TTS = (S, S0, Act, X, I, !)

• TSTTS = (S, sin, Act R, )

• The timed behavior of TTS is defined as the behavior of TSTTS

– Runs– computations

32

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, (0, 0)) 1.6 (s0, (1.6, 1.6)) add (s1, (0, 1.6)) 2 (s1, (2, 3.6) data (s2, (2, 0)) ..…

17.02.05 Lecture5 33

Timed Behaviors

• Two consecutive time-passing moves can be amalgamated into one time passing move.

• (s1, (2,5)) 2 (s1, (4, 7)) 1.3 (s1, (5.3, 8.3)) can also be recorded as :– (s1, (2,5)) 3.3 (s1, (5.3, 8.3))

• (s, V) 1 (s, V+1) 2 (s, (V+1)+2) can also be recorded as:– (s, V) 1+2 (s, V+(1+2))

17.02.05 Lecture5 34

Timed Computations

• TTS = (S, S0, Act, X, I, !)

• TSTTS = (S, S0, Act [ R, ))

• (s0, V0) 0 (s0, V0’) a0 (s1, V1) 1 (s1, V1’) a1 (s2, V2) …… induces the timed computation– (a0, 0) (a1, 0+ 1) …… (an, 0+1+..n)

• Transition systems --- Computations• Timed Transition Systems ---

Timed Computations.

35

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, (0, 0)) 1.6 (s0, (1.6, 1.6)) add (s1, (0, 1.6)) 2 (s1, (2, 3.6) data (s2, (2, 0))

(add, ? ) (data, ?) ….

36

Example

s3 x > 2

s0 s1

s2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(s0, (0, 0)) 1.6 (s0, (1.6, 1.6)) add (s1, (0, 1.6)) 2 (s1, (2, 3.6) data (s2, (2, 0))

(add, 1.6 ) (data, 3.6) ….

17.02.05 Lecture5 37

Example

add ; x

data ; y

ack.add x > 2

ack.datay ≤ 5

(add, 1) (d, 10) (ack.a , 3) A timed computation?

17.02.05 Lecture5 38

Example

add ; x

data ; y

ack.add x > 2

ack.datay ≤ 5

(add, 1) (d, 10) (ack.a , 3) A tmed computation?

No! Time must increase monotonically.

17.02.05 Lecture5 39

Example

add ; x

data ; y

ack.add x > 2

ack.datay ≤ 5

(a, 1) (ack.a , 3) (d, 10) A timed computation?

40

Example

add ; x

data ; y

ack.add x > 2

ack.datay ≤ 5

(a, 1) (ack.a , 3) (d, 10) A timed computation?

No! The order of execution must respect the specification!

17.02.05 Lecture5 41

Example

add ; x

data ; y

ack.add x > 2

ack.datay ≤ 5

(a, 1) (d, 1) (ack.add, 10) A time computation?

17.02.05 Lecture5 42

Example

x > 2

add ; x

data ; y

ack.add

ack.datay ≤ 5

(a, 1) (d, 1) (ack.add, 10) A timed computation?

Yes! Transitions are assumed to take zero time.

Must rule out an infinite number of occurrences in finite time.

17.02.05 Lecture5 43

Zeno Computations

x < 1b

a

… (b, ½) ….(b, 3/4) ….(b, 15/16) …. (b, 31/32)….

We must define computations of timed transition systems carefully!

17.02.05 Lecture5 44

Parallel Composition

• TTS = TTS_1 || TTS_2 || …… || TTS_n

• Same principle as before:– Do common actions together– Take union of clock variables.– Take conjunction of the guards!

17.02.05 Lecture5 45

Reachability of Control States

• For a finite TS it is trivial to decide whether s is reachable in TS.

• For finite TTS, whether s is reachable in TTS is not easy to decide because TSTTS is an infinite object!

• But this can be done and this verification process can be automated.

• More involved (liveness) properties can also be verified effectively but not always efficiently.

17.02.05 CS5272 - 18 Feb 2005Lecture5 46

An introduction toUPPAAL

17.02.05 Lecture5 47

What is UPPAAL?

• A toolbox for modeling, simulation and verification of real-time systems– Developed jointly by Uppsala University and Aalborg

University– Often used for real-time controllers, communication

protocols, etc.

• Consists of 3 main parts– a system description– a simulator– a verifier

17.02.05 Lecture5 48

System Description

17.02.05 Lecture5 49

System Description

• A system in UPPAAL: a set of concurrent processes

• A process: a timed-automaton with:– Clocks and data variables– data variables: int, bool, array of int/bool

• Scopes: global or local (to a process) – Both clocks and data variables! – Guards (on transitions) and invariants (on locations)

are conditions on data variables and clocks

17.02.05 Lecture5 50

System Description

• Synchronization actions are described using communication channels

• Two processes communicate with each other via global data variables or synchronization on communication channels

17.02.05 Lecture5 51

System Description

• A system description in UPPAAL is composed of– Global declarations– Process templates– Process assignments– A system definition

17.02.05 Lecture5 52

System Description

Overview of UPPAAL

17.02.05 Lecture5 53

Global declarations

All the clocks, variables, constants and channels declared here are visible to all the processes in the system

17.02.05 Lecture5 54

Global declarations

• clock x, y;– two clocks x and y

• const a 1;– a constant a with value 1

• int a; – an integer variable a in the default range from -32768 to 32767

• int[1,10] b := 2;– an integer variable b in the range 1,…,10 that is initialized to 2

• int a[2][3] := {{0,1,2},{3,4,5}};– a two dimensional integer array a default range and its

initialization• bool a[3], b[5];

– two bit arrays a and b, with 3 and 5 elements respectively.

17.02.05 Lecture5 55

Global Declarations

• chan d;– A communication channel d– Actions performed by two processes when they synchronize with

each other on channel d are denoted as d! (sending) and d? (receiving)

• urgent chan d;– When two processes are able to synchronize on channel d, they

have to do it immediately. – No guard is allowed on transitions labeled with synchronization

actions (eg. d!,d?) on urgent channels • broadcast chan e;

– Allows 1-to-many synchronization– A transition with label e! emits a broadcast on the channel e. – Any enabled transition with label e? will synchronise with the

emitting process. No guard is allowed on these transitions

17.02.05 Lecture5 56

Process Templates

• Define a common control structures for the processes that are alike– Each process is an instantiation of a template

• Each template can have – symbolic variables and constants as parameters

Parameters are syntactically similar to declarations, but with no initializations

Eg. clock x, y; int i1, i2; chan a, b

– local clocks and variablesDeclared in the same way as global clocks and variables

17.02.05 Lecture5 57

Process Templates

Draw the structure of the template here!

Declare symbolic constants and variables here! Name of the template

17.02.05 Lecture5 58

Process Templates

All clocks, variables, constants declared here are visible to process template P only

17.02.05 Lecture5 59

Process Templates

An example of process template

17.02.05 Lecture5 60

Process Templates

• Locations– 3 types: normal (with/without invariant),

committed(C), urgent (U)– When the system enters a committed location,

it must leave the location immediately:Just a way of breaking up the effect of a transition

into a sequence of steps.

– urgent location: Only 0 units of time can be spent in this location.

17.02.05 Lecture5 61

Process Templates

• A guard– A conjunction of timing and data expressions – Written as a list of expressions, separated by commas

x < y, x <= 5, (a[0] + 1) != (a[1]*5)

• Timing expression: x ~ e or x – y ~ e– x, y : clocks; e: an integer expression– ~ { <=, >=, ==, <, > }

• Data expression: – involves constants and variables– same as in C/C++/Java

17.02.05 Lecture5 62

Process Templates

• Invariants– Same as guards, with no lower bounds on clocks

• Updates– A list of assignment expressions– Each assignment is of the form: x := e or a := e’

In x: = e, almost always e is 0. x: a clocka: a variablee: an integer expressione’: a data expression

– Evaluated sequentiallya := 1, b := 2*a sets a to 1, b to 2.

17.02.05 Lecture5 63

Process Assignments

• Declare instances of process templates• Each assignment has the form P := T(A);

– P : a process name– T : a process template name– A : an argument list

• Examples:– P := Q();

Process P is an instance of template Q that has no parameter

– S:=R(x,1); Process S is an instance of template R that has two

parameters - a clock and a constant

17.02.05 Lecture5 64

Process Assignments

Instantiate your processes here!

17.02.05 Lecture5 65

System Definition

• Declare the set of processes in the system

• Eg. system P, Q;– A system consists of two processes P and Q

• Each process in the system must be either– a process that appears on the left-hand-side

of a process assignment, or – a template with no parameter

17.02.05 Lecture5 66

Simulator

17.02.05 Lecture5 67

Simulator

• A validation tool that– allows users to interact and observe the behavior of

the system during execution– visualise executions (i.e. symbolic traces) generated

by the verifier

• Interface: divided into 4 sub-windows– Simulation control– Variable window– Process window– Message Sequence Chart

17.02.05 Lecture5 68simulator control variable window

process window

Message sequence chart

17.02.05 Lecture5 69

Simulator Control

• Used to control the simulation and to select the (symbolic) state or transition to be visualized – Upper part: for step-by-step simulation

Displays all enabled transitionsEnables users to select an enabled transition to be

fired

– Lower part: displays the generated traceA trace is an alternating sequence of location

vectors and transitions.

17.02.05 Lecture5 70

Simulation Controluse the mouse to select the transition to be fired

press “Next” causes the system to fire the selected transition

press “Reset” resets the simulated system to its initial state

17.02.05 Lecture5 71

Simulation Controlthe simulation always progresses from this highlighted element

press “Replay” to replay the trace starting from the currently selected element.

press “Prev” to highlight the element immediately preceding the current selection in the trace.

press “Open” to open a file selector window for loading a trace from file.

control the speed of simulation

17.02.05 Lecture5 72

Variable window

• Displays the values of data and clocks in the current state or transition selected in the trace

17.02.05 Lecture5 73

Process window

• Displays the process instances of the system– The current location of each automaton is

marked with a red token– The transitions currently selected in the

simulation control panel are highlighted

17.02.05 Lecture5 74

the selected transition

17.02.05 Lecture5 75

Message Sequence Chart

• Displays an MSC view of the generated trace – a vertical lines for each

process – a horizontal line for each

synchronisation point.

synchronization point

17.02.05 Lecture5 76

Verifier

17.02.05 Lecture5 77

Verifier

• Used to check invariant and liveness properties of the system

• Requirement Specification Language

A[ ] p For all paths, p always holds

E<> p There exists a path where p will eventually hold

E[ ] p There exists a path where p always holds

A<> p For all paths, p will eventually hold

p --> q Whenever p holds, q will eventually hold

p : an expression that has no side-effect or state formula (conjunction of state locations)

17.02.05 Lecture5 78

Verifier

• Examples• A[] not deadlock

true if the system is not deadlocked.

– E<> p1.cs and p2.cs true if the system can reach a state where both process

p1 and p2 are in their locations cs.

• A[ ] p1.cs imply not p2.cstrue if whenever process p1 is in its location cs, the

process is not in its location cs.

17.02.05 Lecture5 79

The selected property to be verifiedPress to verify the selected property

Insert a new property

Remove the selected property

Show the comments in overview window

the system is deadlock free

verification results are shown here!