Equivalent states search

algorithm for models with

continuous time

Dalius Makackas and Regina Miseviciene

Kaunas University of Technology, Faculty of Informatics,

Kaunas, Lithuania

Introduction

• A real-time system’s accuracy depends not only on the logical result of

computations, but also on the time at which the results are produced

• The most commonly used are following formal notations: Time Petri Nets,

Discrete Event System Specification (DEVS) , Timed Automata, Piece-linear

Aggregate (PLA), Finite State Machine and others.

Problem

• Verifying correctness of real time systems, reachable states analysis method is

amply developed and used. However, this method, described in scientific

literature, cannot avoid the endless increase of reachable state space.

• This presentation presents an equivalent nodes search algorithm enabling to

reduce the number of nodes in the reachable states graph.

A goal of presentation

• A goal of this presentation is to present an algorithm, which transforms

the graph of the infinite reachable states into the graph with the finite node

numbers.

• The algorithm is designed for real-time systems specified by Piece-linear

aggregate method.

This presentation is organized following

• Formal definition of Piece-linear aggregate;

• Transformation algorithm of the reachable states graph for the continuous

time models

• Illustrative example

Piece-Linear Aggregate Method (1)

A system specified by the Piece-linear aggregate method is understood as a set of

interacting piece-linear aggregates. Each aggregate is defined by a set of states

...},{ 21 zzZ , a set of input signals ...},{ 21 xxX , a set of output signals

...},{ 21 yyY , a set of internal E and external E events, a set of transition

ZZE H : and output YZE G : operators.

Piece-Linear Aggregate Method (2)

The aggregate method generates time-point sequences ...},{ 10 ttT and state

)...}(),({ 10 tztz transitions in these time points. The state ))(),(()( tztvtz v consists of

two components: discrete )(tv and continuous )(tzv . Each element )(twi of a

continuous component )...)(),(()( 21 twtwtzv indicates a time when an event ie

occurs. The event changes j elements of discrete and continuous component of state

according to the law: ))(,()( tzthth vj

vj , ))(,()( tzthth w

jwj .

Piece-Linear Aggregate Method (3)

In the aggregate model it is also defined the concept of the operation. This function

takes the following values:

t. time at pasive is it

t; time at ended it

t; time at active is it

teOtOO ee

,1

,0

,1

,

Behavior definition

,,,,,,222111000

IesIesIes (8)

Other notation is used: 33221100

3210

IeIeIeIessss , (9)

Equal behaviors

Two behaviors ,,,,, 1

2

1

1

1

1

1

1

1

0

1

0

1

01sIesIes and ,,,,, 2

2

2

1

2

1

2

1

2

0

2

0

2

02sIesIes

are called equal when 21

jjss ( 1

je matches 2

je ), and their occurrence

intervals are equal 21

jjII for all j .

Definitions

Definition 3. The behavior is called periodical when its traces are periodical.

The system detailed functioning can be shown by the graph when all the behaviors in

the set are periodical or finite.

Definition 4. Two states i

s andj

s , are called equivalent in the behavior

,,,,,,,,,,,,,,,21112111111000

jjjjjjjiiiiiii

sIesIessIesIesIesIes ,

when 2121

:, TTITITji

.

The search algorithm of the equivalent states

The system behavior is chosen

For the state is from the behavior the

next equivalent state js , with the same

discrete component is determined.

Checking whether the same active

operations and corresponding continuous

components values coincide.

An Example

1

Mass service system

2

Queue

Two-channel mass service system

The system specification consists of the components:

a set of inputs X and a set of outputs Y ;

a set of events EEE , where E ; 321 ,, eeeE , 1e - a new message

arrived, 2e - a first channel service is completed, 3e - a second channel service is

finished;

controlling sequences 2101 ,, e , 2102 ,, e , 2103 ,, e ;

a discrete component tnt , where tn is a number of messages in a queue.

a continuous component tewtewtewtz ,,,,, 321 ;

a parameter s - is a maximum length of the queue.

Reachable state tree fragment

Equivalent states

2111

;,,6,4;0:3 Rtt , where 6401021

tttR

43121 ttt

6133

;,,6,4;0:9 Rtt , where

436412101061

ttttttR 64131ttt

43343 ttt .

Transformed graph

5

21l 22l 23l

51l

8

4

41l

7

2

3

31l

6

1

11l

61l

62l

10

Conclusions

• Verifying correctness of real time systems, a reachable states graph analysis method

is widely developed. Because of the infinite number of the nodes in the reachable

states graph is impossible to analyze the graph.

• Presented algorithm enables to minimize the nodes number of the reachable state

graph using the equivalent relationships. Using the relations, the reachable states

graph can be transformed into the graph with the finite nodes number.

• It is especially important for continuous time models when transforming the graph

into the smaller one and solving the same problems as in the original graph.