FSM, Petri nets and State charts

1
FSM, Petri nets and State charts
2
Final State machine

• FSM - Final state machine (S, I, f)
• S The states of the system
• I The set of transitions from one state to
  another
• F SI -gt S, describes the legal transitions in
  the system
• Simple, but has limited computational power
• FSM has no memory
• Using states as memory is inefficient
• State space explosion for large problems

3
Reader-Writer
write
consume
P1
P2
C1
C2
produce
read
read
read
0
1
2
write
write
2 slot buffer
4
write
write
lt1,p1,c1gt
lt2,p1,c1gt
lt0,p1,c1gt
consume
consume
consume
produce
produce
produce
read
read
lt2,p1,c2gt
lt0,p1,c2gt
lt1,p1,c2gt
produce
produce
produce
read
read
consume
write
write
consume
lt0,p2,c2gt
lt1,p2,c2gt
lt2,p2,c2gt
5
FSM questions

• Using FSMs, describe a lighting system consisting
  of one lamp and two buttons. If the lamp is off,
  pushing either button causes the lamp to switch
  on and conversely
• Describe a system with two lamps and one button.
  When the light is off, pushing the button causes
  the first light to go on. Pushing the button
  again causes the second lamp to go on and the
  first to go off. Pushing the button again causes
  both lamps to go on, and pushing it once more
  causes both lamps to go off
• Describe a system that accepts an identifier that
  consists of letters and digits and begins with a
  letter

6
Control of Chemical Plant

• Describe a simple control of a chemical plant
  where temperature and pressure must be monitored
  for safety reasons. When either reaches a
  critical level the plant should turn off. The
  system restarts when the cause of the failure is
  fixed
• Enhance the previous scheme. When one of the
  conditions occur, a recovery action is
  automatically invoked . If, after a while, the
  recovery action succeeds the system resets to
  normal state. Otherwise the system must turned
  off. If the system tries to overcome one kind of
  anomaly and the second one occurs the system must
  be turned off

7
State Transition Diagram - Definitions

• A state is a condition in which the thing being
  modeled stays for some period of time, during
  which it behaves in the same way
• A transition is a change of state, usually caused
  by some particular event
• An event is something which happens outside the
  thing which is being considered, possibly
  requiring some action to be taken. Events may be
  caused by the arrival of some data, or some
  simple stimulus. Some events may convey data as
  input

8
State Transition Diagram - Definitions

• A guard on a transition is some condition which
  must be be true for the event to cause a
  transition
• Actions are what is done by the thing being
  modeled in response to an event
• An activity is something which is done by the
  thing being modeled, while it is in a particular
  state
• General form of a transition
• event(list of input data) guard / action

9
Transitions
10
Digital watch

• Two buttons ModeButton and IncButton. Pressing
  either of these generates an event that may cause
  a transition
• Three states Display, SetHours, SetMinutes
• Display is the start state. Event ModeButton
  causes a transition from Display to SetHours,
  from SetHours to SetMinutes and from SetMinutes
  to Display
• Event IncButton causes the action Increase
  Hours or Increase Minutes (depends on the
  state)
• How can we add an activity?

11
Window Shield wiper

• The operation of the windshield wipers of a car
  is determined by a stick with positions OFF
  (speed 0), INT, LOW  (speed 20) and HIGH (speed
  40). At the end of the stick a button determines
  the speed when INT 1 (speed 2), 2 (speed 4), 3
  (speed 6)
• Draw a state transition diagram and a state chart
  diagram describing the system
• Add activities and actions

12
Petri nets definition

• PN ( S, T, F, MI )

• PN describe the state of the system. Transitions
  describe actions or events and the presence of a
  token in a place denotes existence of some
  condition or state

13
(No Transcript)
14
How to avoid starvation
15
Deadlock situation
P1
P2
.
.
t2
t1
P3
P4
P5
..
t3
t4
P6
P7
t3
t4
P8
P9
2
2
t5
t6
16
Warehouse Automation Example

• Specifies requirements of a system
• P1 gt a truck arrives at the loading dock
• P2 gt paperwork is processed and inventory
  checked
• P3 gt conveyor belts move from the loading dock
  to the warehouse (transporting people, robots,
  forklifts, etc)

. .
P1
T1
P2
P3
P4
P5
T2
T3
T4
T5
17

• P4 gt conveyor belt move back from warehouse to
  loading dock with merchandise.
• P5 gt goods loaded into the truck
• Concurrent handling of multiple trucks can be
  modeled by multiple tokens
• But a token is present in P3, then another token
  cannot be present in P4 (the conveyor belts
  cannot move in two opposite direction at the same
  time)
• Therefore, the above model can be modified as
  follows to specify the above requirements of the
  system.

.
. .
P1
T1
P2
P3
P4
P5
T2
T3
T4
T5

• Now a token can enter P3 only if there is no
  token in P4

18

• Similarly to specify the requirements that only
  paperwork of one truck can be processed at a time
  and merchandise of only one truck can be loading
  into the truck at a time, the above model can be
  modified as follows

.
. .
P1
T1
P2
P3
P4
P5
T2
T3
T4
T5
.
.
19
PN - Questions

• Using PNs, describe a lighting system consisting
  of one lamp and two buttons. If the lamp is off,
  pushing either button causes the lamp to switch
  on and conversely
• Describe a system with two lamps and one button.
  When the light is off, pushing the button causes
  the first light to go on. Pushing the button
  again causes the second lamp to go on and the
  first to go off. Pushing the button again causes
  both lamps to go on, and pushing it once more
  causes both lamps to go off

20
Dining Philosophers

• We have 5 philosophers sitting around a round
  table. Each philosopher can eat or think. In
  order to eat a philosopher has a spaghetti plate
  and he needs to use two forks, which lie to his
  right and his left
• Two close philosophers cant eat at the same time
• Describe a PN that describes the above problem,
  which avoids deadlocks
• Describe a PN that describes the above problem,
  which allows deadlocks

21
PN Extensions 1

• A priority function pri from transitions to
  natural numbers, pri T ? N
• When several transitions are enabled, only the
  ones with maximum priority are allowed to fire
• Among those transitions that are allowed to fire
  at the same time, the one to fire is chosen
  non-deterministically

22
PN Extensions 2

• A pair of constants lttmin, tmaxgt is associated
  with each transition
• Once a transition is enabled, it must wait for at
  least tmin to elapse before it can fire
• If enabled, it must fire before tmax has elapsed,
  unless it is disabled by the firing of another
  transition before tmax

23

P

P

1


4

P
P


2

3
T1



T3
T2




tmin 0
tmin 1
tmin 2



tmax 5
tmax 4
tmax 3



1


priority

2
priority
priority
3

• At t0, i.e, when the above system begins to
  execute, all the transitions above are enabled.
  However, only T3 can fire at t0. At t1,
  transition T1 is also ready to fire. If T3 has
  not fired before t1, then T1 cannot fire until
  T3 has fired, since T3 has higher priority than
  T1. If T3 does not fire till t4, then T1 can
  never fire since T1 must fire before t4

24




P
P
P




4
1
P
3

2

T1

T2
T3





tmin 0
tmin 1
tmin 2



tmax 5
tmax 4
tmax 3



1


priority

2
priority
priority
3
If T3 fires before t1, T1 can fire at t1 or
anytime after t1 but before t2. This is because
at t2, T2 can fire and T2 has higher priority
than T1, therefore it will prevent T1 from
firing. If T1 fires before t2, then T2 gets
disabled and can never fire.
25




P
P
P




4
1
P
3

2

T1

T2
T3





tmin 0
tmin 1
tmin 2



tmax 5
tmax 4
tmax 3



1


priority

2
priority
priority
3
If T2 fires before T1 fires, it disables T1.
Therefore, T1 can never fire after T2 fires. If
T3 does not fire before t2, then it cannot fire
till t3 since T2 can fire at t2 and T2 has
higher priority than T3. If T2 fires before t3,
then T1 is disabled, but now T3 can fire anytime
before t5. After t3, if T3 has not fired yet,
it can fire anytime before t5 irrespective of
whether T2 has fired or not.
26
PN Extensions 3

• Tokens modified to carry a value
• Transitions are modified to have associated
  predicates and functions
• A transition with k input places and h output
  places is enabled if there exists a k-tuple of
  tokens, such that the predicate associated with
  the transition is satisfied by the values of the
  tokens
• The production of one token for each output place
  is determined by a function

27
PN extensions - Questions

• Describe a message dispatcher
• The dispatcher receives messages from two
  different input-channels. it checks the parity of
  each message. If the parity is wrong, it sends a
  "nack" through a reply-channel (there is one such
  reply-channel for each input channel)
• If the parity is right, it places the message in
  a buffer. The buffer may store ten messages. When
  the buffer is full, the dispatcher sends the
  whole contents of the buffer to a processing unit
  through another channel. No message can be placed
  in a full buffer

28
PN extensions - Questions

• Add suitable priorities to the previous Petri
  Net. If the dispatcher is in a condition where it
  can either receive a message from an
  input-channel or forward the buffer contents to
  the processor, then it must order its priorities
  as follows First get message then forward the
  buffer contents