[PPT] - The State Automata Formalism Untimed models of discrete event PowerPoint Presentation

SLIDE 1

The State Automata Formalism

Untimed models of discrete event systems
Languages
Regular Expressions
Automata

– (Deterministic) Finite State Automata – Nondeterministic Finite State Automata – State Aggregation – Discrete Event Systems as State Automata

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 1/31

SLIDE 2

Untimed models

Level of specification: I/O System (state based, deterministic)
Time Base =

✁

(time

✂

progression index)

Dynamic but

– only sequence (order) of states traversed matters – not when in state or how long in state

Discrete Event: event set E

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 2/31

SLIDE 3

Languages – Regular Expressions – Automata

language L, defined over alphabet E (events)

✄

set of strings formed from E

Example: all possible input behaviours:

L

✂ ☎

ε

✆

ARR

✆

DEP

✆

ARR ARR DEP

✆✞✝ ✝ ✝ ✟

Regular expression: shorthand notation for a regular language

ARR DEP

✆

ARR

✠

DEP

✠ ✆ ✡

DEP

☛

ARR

☞ ✠

Concatenation, Alternatives (

☛

), Kleene closure (

✠

).

Finite State Automaton (model): generate/accept a language

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 3/31

SLIDE 4

Finite State Automaton

✡

E

✆

X

✆

f

✆

x0

✆

F

☞

E is a finite alphabet
X is a finite state set
f is a state transition function,

f : X

✌

E

✍

X

x0 is an initial state, x0

✎

X

F is the set of final states

Dynamics (x

✏

is next state):

x

✏ ✂

f

✡

x

✆

e

☞

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 4/31

SLIDE 5

FSA recognizes Language

extended transition function:

f : X

✌

E

✠ ✍

X f

✡

x

✆

ue

☞ ✂

f

✡

f

✡

x

✆

u

☞ ✆

e

☞

A string u over the alphabet E is recognized by a FSA

✡

E

✆

X

✆

f

✆

x0

✆

F

☞

if f

✡

x0

✆

u

☞ ✂

x where x

✎

F.

The language L

✡

A

☞

recognized by a FSA A

✂ ✡

E

✆

X

✆

f

✆

x0

✆

F

☞

is the set of strings

☎

u : f

✡

x0

✆

u

☞ ✎

F

✟

.

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 5/31

SLIDE 6

FSA graphical notation: State Transition Diagram

Init End_1 End_0 1 1 1

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 6/31

SLIDE 7

Simulation steps

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 7/31

SLIDE 8

Init End_1 End_0 1 1 1 Current State Init End_1 End_0 1 1 1 Current State Init End_1 End_0 1 1 1 Current State Init End_1 End_0 1 1 1 Current State

Rule 1 Rule 2 Rule 2 Rule 2 Final Action

"Accept Input"

input 0 input 1 input 0 end of input

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 8/31

SLIDE 9

FSA Operational Semantics

/ <ANY> <ANY> <ANY> Current State 2 4 3 1 / <COPIED> <COPIED> <COPIED> Current State 2 4 3 1 <ANY> 1 / <ANY> <ANY> <ANY> <ANY> Current State 2 4 3 5 1

/ <COPIED> <COPIED> <COPIED> <COPIED> Current State 2 4 3 5 1

::= ::= ::=

Rule 1 (priority 3) Rule 2 (priority 1) Rule 3 (priority 2) Locate Initial Current State State Transition Local State Transition condition: matched(4).input == input[0] action: remove(input[0]) condition: matched(4).input == input[0] action: remove(input[0])

<COPIED> Current State 3 1 2

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 9/31

SLIDE 10

Nondeterministic Finite State Automaton

NFA

✂ ✡

E

✆

X

✆

f

✆

x0

✆

F

☞

f : X

✌

E

✍

2X

Monte Carlo simulation (if probabilities added)
Transform to equivalent FSA (aka DFA)

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 10/31

SLIDE 11

Nondeterministic Finite State Automaton

Idle Coffee Tea drinkC drinkT thirsty thirsty Button Button TDrunk Cdrunk

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 11/31

SLIDE 12

Constructed Deterministic Finite State Automaton

Idle Coffee$Tea drinkC$drinkT thirsty Button Cdrunk TDrunk

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 12/31

SLIDE 13

Transformation Rules

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 13/31

SLIDE 14

Rule LHS

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 14/31

SLIDE 15

Rule RHS

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 15/31

SLIDE 16

Managing Complexity: State Aggregation

✡

E

✆

X

✆

f

✆

x0

✆

F

☞

R

✑

X R consists of equivalent states with respect to F

if for any x

✆

y

✎

R

✆

x

✒ ✂

y and any string u, f

✡

x

✆

u

☞ ✎

F

✓

f

✡

y

✆

u

☞ ✎

F x and y are equivalent for as far as “accepting/rejecting” input strings is

concerned.

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 16/31

SLIDE 17

State Aggregation Algorithm

1. Mark

✡

x

✆

y

☞

for all x

✎

F

✆

y

✒ ✎

F

2. For every pair

✡

x

✆

y

☞

not marked in previous step: (a) If

✡

f

✡

x

✆

e

☞ ✆

f

✡

y

✆

e

☞ ☞

is marked for some e

✎

E, then:

i. Mark

✡

x

✆

y

☞

ii. Mark all unmarked pairs

✡

w

✆

z

☞

in the list of

✡

x

✆

y

☞

. Repeat this step for each

✡

w

✆

z

☞

until no more markings possible. (b) If no

✡

f

✡

x

✆

e

☞ ✆

f

✡

y

✆

e

☞ ☞

is marked, the for every e

✎

E:

i. If f

✡

x

✆

e

☞ ✒ ✂

f

✡

y

✆

e

☞

then add

✡

x

✆

y

☞

to the list of f

✡

x

✆

e

☞ ✒ ✂

f

✡

y

✆

e

☞

Pair which remain unmarked are in equivalence set

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 17/31

SLIDE 18

digit sequence (123) detector FSA

x_1 x_12 x_123 x_3 x_2 1 1 1 3 2 3 3 2 2 2 1 1 3 3 2

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 18/31

SLIDE 19

State Reduced FSA

x_1 x_12 x_123 x_0 1 1 1 3 2 3 2 2 1 3 3 2

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 19/31

SLIDE 20

State Automata to model Discrete Event Systems

X is state space

✔ ✔

Q

All inputs are strings from an alphabet E (the events)

✔ ✔

X

State transition function x

✏ ✂

f

✡

x

✆

e

☞ ✔ ✔

δ

Allow X and E to be countable rather than finite
Introduce feasible events

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 20/31

SLIDE 21

State Automaton

✡

E

✆

X

✆

Γ

✆

f

✆

x0

☞

E is a countable event set
X is a countable state space
Γ

✡

x

☞

is the set of feasible or enabled events

x

✎

X

✆

Γ

✡

x

☞ ✎

E

f is a state transition function,

f : X

✌

E

✍

X, only defined for e

✎

Γ

✡

x

☞

x0 is an initial state, x0

✎

X

✡

E

✆

X

✆

Γ

✆

f

☞

mits x0 and describes a class of State Automata.

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 21/31

SLIDE 22

Feasible/Enabled Events

On transition diagram: not feasibe

✕

not marked

Meaning: ignore non-feasible events
Why not f

✡

x

✆

e

☞ ✂

x for non-feasible events ?

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 22/31

SLIDE 23

State Automata for Queueing Systems

Physical View Queue Cashier Departure Arrival Departure Queue Abstract View Cashier [ST distribution] [IAT distribution] Arrival

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 23/31

SLIDE 24

State Automata for Queueing Systems: customer centered

...

1 2 3 4 5

a d a d a d a d a d a d

E

✂ ☎

a

✆

d

✟

X

✂ ☎ ✆

1

✆

2

✆✞✝ ✝ ✝ ✟

Γ

✡

x

☞ ✂ ☎

a

✆

d

✟ ✆✗✖

x

✘ ✆

Γ

✡ ☞ ✂ ☎

a

✟

f

✡

x

✆

a

☞ ✂

x

✙

1

✆✗✖

x

✚

f

✡

x

✆

d

☞ ✂

x

✔

1

✆✗✖

x

✘

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 24/31

SLIDE 25

State Automata for Queueing Systems: server centered (with breakdown) I B

s c b r

D

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 25/31

SLIDE 26

State Automata for Queueing Systems: server centered (with breakdown)

E

✂ ☎

s

✆

c

✆

b

✆

r

✟

Events: s denotes service starts, c denotes service completes, b denotes breakdown, r denotes repair.

X

✂ ☎

I

✆

B

✆

D

✟

State: I denotes idle, B denotes busy, D denotes broken down.

Γ

✡

I

☞ ✂ ☎

s

✟ ✆

Γ

✡

B

☞ ✂ ☎

c

✆

b

✟ ✆

Γ

✡

D

☞ ✂ ☎

r

✟

f

✡

I

✆

s

☞ ✂

B

✆

f

✡

B

✆

c

☞ ✂

I

✆

f

✡

B

✆

b

☞ ✂

D

✆

f

✡

D

✆

r

☞ ✂

I

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 26/31

SLIDE 27

Interpretations/Uses

Generate all possible behaviours.
Accept all allowed input sequences

✕

code generation.

Verification of properties.

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 27/31

SLIDE 28

State Automata with Output

✡

E

✆

X

✆

Γ

✆

f

✆

x0

✆

Y

✆

g

☞

Y is a countable output set,
g is an output function

g : X

✌

E

✍

Y

✆

e

✎

Γ

✡

x

☞

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 28/31

SLIDE 29

State Automata for Adventure Games

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 29/31

SLIDE 30

State Automata (later: Statecharts) for Graphical User Interface Specification

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 30/31

SLIDE 31

Limitiations/extensions of State Automata

Adding time ?
Hierarchical modelling ?
Concurrency by means of

✌

States are represented explicitly
Specifying control logic, synchronisation ?

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Finite State Automata 31/31