Real-Time Synchronised Petri Nets

Giovanna Di Marzo SerugendoDino Mandrioli, Didier Buchs, Nicolas Guelfi

University of Geneva, Switzerland

PN’02 / 24th June 2002 Giovanna Di Marzo Serugendo

2

Motivations

Synchronised Petri Nets• CO-OPN

Real-Time• Time interval attached to transitions

Inhibitor Arcs• Maximum number of tokens in place

Real-Time Synchronised Petri Nets

3

Synchronised Petri nets

Concurrent Object-Oriented Petri Nets• Object-Orientation• External and Internal Transitions• Synchronisation Requests

• simple

• simultaneity, sequence, alternative operators

• transactional semantics (all or nothing)

• Abstract Data Types

This paper: • Object-based• Time stamped mono colored token

4

Real-Time

Time Interval• Attached to each transition • Merlin-Farber model• Instantaneous firing• Time of Firing:

• Bound by time interval in R+

• Relative to time when transition becomes enabled

tt, ,0 ', tt

5

move [5..15] put [2..10]

O1 O2

p1 p2

1

00

0

10

1

processing [1..9]

Real-Time Synchronised Petri Nets

Methods

Transition

Time IntervalSynchronisation

Objects

6

move [5..15] put [2..10]

O1 O2

p1 p2

1

00

0

10

1

processing [1..9]

Real-Time Synchronised Petri Nets

5X

]10..5[t

7

Real-Time Synchronised Petri Nets

move [5..15] put [2..10]

O1 O2

p1 p2

1

00

0

10

1

processing [1..9]

5

]14..6[t

X

8

move [5..15] put [2..10]

O1 O2

p1 p2

1

00

0

10

1

processing [1..9]

]10..6[t

Real-Time Synchronised Petri Nets

7

9

Semantics

How to deal with:• Merlin-Farber and Synchronisations

• Firing respecting Time interval

• Tokens production time

Synchronisation• Intersection of time intervals for simultaneity

• Correct time of production of tokens for sequence

• Non-determinism in the case of alternative

Inhibitor Arcs• Ensure inhibitor arc condition even with chains of synchronisations

10

Semantics

Structured Operational Semantics Rules (SOS)

4 Steps• Weak Transition System• Strong Transition System• Expanded Transition System (Synchronisation)• Observable Strong Time Semantics

• paths starting from initial marking

11

Transition System

Events• Methods and Transitions• Synchronisation /Simultaneous / Sequence / Alternative

Markings• Time Stamped Tokens

Time of Firing

}2{{},{}{}, put

2

12

Weak Transition System

Single Firings• synchronisation is not taken into account• time of firing occurs in time interval• inhibitor arc condition is verified• post-condition produces tokens stamped at time of firing

Elapsed Time Interval • does not impose transition firing at maximal bound of

interval

13

SOS Rules

move [5..15] put [2..10]

O1 O2

p1 p2

1

000

101

processing [1..9]

X

{}},0,0{{}},0,0,0{ put with move5

14

Strong Transition System

Elapsed Time Interval• Transition has to fire• Remove transitions that prevent other transitions to fire when

time elapses

Two transitions reaching maximal bound of firing at the same time:• both are allowed • one may still disable the other

15

Expanded Transition System

Synchronisation• move and put must fire at the same time• result takes into account both firings• Observable Transition: only move

Simultaneity• Both transitions occur at the same time• Observable Transition: e1 // e2

16

Expanded Transition System

Sequence• Transition e1 fires before e2 (t1 < t2)• Observable transition: e1 .. e2• Tokens stamped at t1 and t2• Tokens stamped at t2: not available before t2

Alternative• Transition e1 or transition e2 fires at t• Observable transition: e1 + e2• Tokens stamped at t• Result: due to e1 or e2

17

Synchronisation

{}},0,0{{}},0,0,0{ put with move

5

}5{{},{}{}, put

5

}5{},0,0{{}},0,0,0{ move

5

18

Strong Time Semantics

Retain observable transitions reachable from initial marking

Time increases

Semantics = paths of transitions starting from initial marking

19

Strong Time Semantics

{0,0,0},{}

move,5

{0,0},{5}

put,2

{0,0,0},{2}

{0,0},{}

processing,6 move,6

put,6

{0},{6}

{0,0},{6} ...

...

{0,0,0},{}

processing,3

...

{0,0},{5.5}move + put,5.5

...

......

time

20

Train

Railroad Crossing System Specification

• Trains• Level Crossing

Safety and Utility Properties• determination of time needed to operate the bar

21

Conclusion

Syntax and Semantics of Real-time Synchronised Petri Nets

Towards: • Object-Oriented Real-Time Petri Nets

True Operational Semantics already realised• first step towards reachability analysis

Future Work• Axiomatisation: formal verification of properties