1. Finite State Machines k State Transition Diagrams
Peter A. Smith & Alexander M. Fedorec

2. Three Different Views of a System
Structural Model (e.g. entity, class, deployment diagrams...)
Showing the components, subsystems, modules, entities or objects and how are they related/connected
Functional Model (e.g. use case model, DFD, HIPO…)
Showing the capabilities and functions required of the system. Normally expressed in terms of inputs, outputs and processes
Behavioural Model (e.g. activity, state diagram…)
Shows system dynamics. Represents how the system changes state/mode and the events and conditions that cause the changes to happen.
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3. Systems Behaviour Systems can classified by the way they change:
Discrete
Finite number of discrete internal states,
Finite number of distinct inputs and outputs.
Operates in discrete time steps
Measures map to natural numbers (positive integers)
Continuous
Internal states, inputs and outputs continuously variable
Measures map to the real numbers
e.g. analogue watch vs. digital watch
coins in a purse vs. fluid in a bottle?
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4. Finite State Machines An FSM describes a system behaviourally as
Finite number of discrete states and
Finite number of permissible transitions between states
e.g. Traffic lights
3 lights each of which can be in one of two states (on or off)
8 possible states (all lights off, one on, any two on, all three on)
Only 4 legitimate states and 4 valid transitions: Red on  Red & Amber  Green  Amber on  Red
Can be modelled as “State Transition Graph” (STG))
States – labelled circles or rounded boxes. Each circle represents one possible state that the FSM can be in.
Transitions – Arcs connecting states labelled with events that trigger state transitions
Also known as Finite State Automata (FSAs)
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5. Example FSM: Traffic Lights
All Possible States
and Transitions
And so on….
Red
Red & Amber
All lights
off
Green & Red
All lights on
Amber
Green
Green & Amber
Q. Given 8 states how many possible transitions are there?
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6. Example FSM: Traffic Lights
Valid States and Legal Transitions
Red
Red
Red & Amber
Red & Amber
All lights
off
Green
All lights
off
Green & Red
All lights on
Amber
Amber
Green
Green & Amber
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7. FSMs with Output: Transducers
State transitions are triggered by inputs or caused by events (e.g. pushing a button, a clock tick, change temperature)
When machines change state they normally do something and the actions often generate output.
Automata with input and output are sometimes called transducers because of their connection to electronics
input/event
shutdown
POWER
POWER
idle
help F1
state
transition
new state
shutdown
light off
POWER
POWER
idle
light on
help
Display “stay calm” F1
state
output/action
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8. Example FSM: Traffic Lights Valid States, legal transitions and output
Red light S2
Red + Amber
State Output
S1 Stop Red light
S2 Ready Red + Amber lights
S3 Go Green light
S4 Caution Amber light
Note: system greatly simplified. Real system will have input signals:
Timing devices
Pedestrian buttons
Road induction loops
Traffic cameras S3
Amber S4
Green
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9. Other Example Finite State Machines
Car injection systems
Communications systems
Digital Computers
Computer language translation e.g. from high level to machine code
Electronic Games
Fetch-execute cycle
Operating systems
Vending machines
Washing machines
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10. Input or state changing event (c )
Mealy & Moore Machines
Transducer output can either be on the state nodes (“Moore machine” e.g. traffic lights example) or on the transition arcs (“Mealy machine”).
Mealy and Moore machines can be shown to be equivalent:
State (q4 )
Output or action (t) q4 t a c b
a/t
c/t
b/t
Input or state changing event (c )
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11. FSM with one input and one output.
FSM asserts its output (i.e. outputs a ‘1’) when it recognizes the input bit sequence: "1011".
Machine will keep checking for the proper bit sequence and does not reset to the initial state after it has recognized the string.
E.g. the input string = " " will cause the output to go high twice: = " "
FSM implemented as a Moore Machine (output on state):
S0 0
S1 0
S2 0
S3 0
S4 1
...1
...10
...101 1
FSM implemented as a Mealy Machine (output on transition): S0 S1 S2 S3
...1
...10
...101
0/0
1/1
1/0
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12. UML Notation For FSMs Active Idle Action on state
BACKGROUND INFORMATION FOR INTREST
UML Notation For FSMs
event
Action
disconnect
Idle
do: play dial tone
Active
off hook
on hook
Action on state
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Note UML is not part of the CSA syllabus

13. UML Notation for FSMs “Statechart Diagrams”
BACKGROUND INFORMATION FOR INTREST
UML Notation for FSMs “Statechart Diagrams”
State transition
State
Initial State
event
Final State
Action
disconnect
Idle
Active
off hook / play dial tone
on hook
Action on state transition
A.M.Fedorec 10/11/ :25
Note UML is not part of the CSA syllabus

14. Example Telephone Idle Talking PlayingDialTone Connecting Dialing
BACKGROUND INFORMATION FOR INTREST
Example Telephone
Idle
Event/action
on hook
off hook / play dial tone
Talking
on hook
PlayingDialTone
digit
connected
digit
on hook
Dialing
Connecting
complete
on hook
Note UML is not part of the CSA syllabus
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15. Nested States in UML Idle Active Talking PlayingDialTone Connecting
BACKGROUND INFORMATION FOR INTREST
Nested States in UML
Idle
off hook / play dial tone
Active
Initial state
PlayingDialTone
Dialing
Connecting
Talking
digit
complete
connected
on hook
Note UML is not part of the CSA syllabus
A.M.Fedorec 10/11/ :25

16. Next state, output given input = 0
State Tables
11/10/2018
A FSM can be expressed in tabular form as the (state, input) pairs in terms of the next state, and action for each
e.g. A circuit starts takes as input a string of ‘0’s and ‘1’s. After it inputs its first ‘1’ it will output a ‘1’ on the second, fourth, sixth... ‘0’ it inputs until it inputs another ‘1’ when it will go back to the reset state.
reset q1 q2
1/0
0/-
0/0
0/1
State Table
Current state
Input = 0
Input = 1 R
R, -
q1, 0 q1
q2, 0
R, 0 q2
q1, 1
State Table
 A FSM can be expressed in tabular form as the (state, input) pairs in terms of the next state, and action for each. 
Next state, output given input = 0 1
State Diagram
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A.M.Fedorec OOP107APIClasses

17. FSM Example (to do on whiteboard)
A coffee vending machine dispenses coffee at 20p per cup.
It accepts 5p, 10p and 20p coins
No change is given
Inexhaustible supply of water cups, coffee powder etc.
Modifications:
A return coins button may be pressed at any time
Tea is dispensed at 25p per cup
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18. Coffee Machine Example
Accept 5p, 10p or 20p coins, Dispense coffee when input 20p Refund if return coin button pressed
20p / coffee
returnCoins / refund
returnCoins / refund q0
Actions
coffee machine dispenses coffee
refund machine returns coins
Events
5p customer enters 5p coin
10p customer enters 10p coin
20p customer enters 20p coin
returnCoins customer presses ‘return’ button
error
20p
10p q1 5p
5p / coffee q2
10p
10p / coffee
InsufficientMoney q3
10p 5p 5p
Input
State 5p
10p
20p
returnCoins q0 q1 q2
q0 , coffee q3
error
q0 , refund
Basic States
q0 no money in machine
q1 5p in machine
q2 10p in machine
q3 15p in machine
error more than 20p in machine
InsufficientMoney
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19. Minimal Tables
If given the same inputs two separate states produce the same outputs and take the system to the same next state then one is redundant
Current state
Input = 0
Input = 1 qa
qb, 1
qc, 0 qb
qa, 0
qb, 0 qc qa qc
1/0
0/1 qb
0/0
Two separate states
Redundant
Given same input
Same next state and output qa qb
1/0
0/0
0/1
Current state
Input = 0
Input = 1 qa
qb, 1
qb, 0 qb
qa, 0
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20. Subjects of State Diagrams in UML
BACKGROUND INFORMATION FOR INTREST ONLY
Subjects of State Diagrams in UML
All classifiers in UML have identity, state and behaviour and can therefore be represented by a statechart (however dumb entity objects may stay in only one state for their entire life, eg. a static string)
Examples:
Use Cases
Student registering interest for a job with the IT placement office involves recording their details if it is the first time they have applied for a job
Classes
Student object changes from level 5 to sandwich placement or level 6 etc.
Other Examples:
Windows
Edit-Paste is only valid if there is something in the clipboard
Devices, Controllers, Transactions, Application, Coordinators
A.M.Fedorec 10/11/ :25
Note UML is not part of the CSA syllabus

21. Concurrent States in UML
BACKGROUND INFORMATION FOR INTREST
Concurrent States in UML
Idle
maintain
Maintenance
Testing
Testing devices
Self Diagnosis
Commanding
Waiting
Command
keyPress
[not continue]
[continue]
Note UML is not part of the CSA syllabus
A.M.Fedorec 10/11/ :25

22. Concurrent States in UML Example 2
BACKGROUND INFORMATION FOR INTREST ONLY
Concurrent States in UML Example 2
© 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Note UML is not part of the CSA syllabus
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23. Non-Deterministic FSA
In automata theory, a finite state machine is called a deterministic finite automaton (DFA), if
each of its transitions is uniquely determined by its source state and input symbol, and
reading an input symbol is required for each state transition.
A nondeterministic finite automaton (NFA), or nondeterministic finite state machine, does not need to obey these restrictions. In particular, every DFA is also an NFA. P Q 1
0,1
Nondeterministic since in state p reading a 1 can lead to p or to q.
From: Nondeterministic finite automaton,
accessed Feb 2016
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24. Summary Systems have state
Events such as inputs cause the system to change state
State change may trigger an output on the state transition or in the new state.
A state transition graph describes the legal states and valid transitions a system can make
This can also be described with a state table
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25. Further Reading For Computer Scientists
Cohen, D., Introduction to Computer Theory 2e, Wiley, 1997
For Software Engineers
Harel, D., Politi, M. Modeling Reactive Systems with Statecharts, McGraw-Hill, 1998
For CSNE
Pedroni, V.A., Finite State Machines in Hardware: Theory and Design, MIT, 2013
For fun:
Stefan Hollos, S., Hollos, J.R., Finite Automata and Regular Expressions: Problems and Solutions, Abrazol Publishing, 2013
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