Theory of Computation: Finite state machines

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PAPER 1 - ⇑ Theory of computation ⇑

← Decision tables Finite state machines Algorithm design →


Finite state machines[edit]

A finite state machine is a form of abstraction (WHY/HOW?). It models the behaviour of a system by showing each state it can be in and the transitions between each state.

Consider an elevator :

[picture of elevator to be inserted here]


Possible states of the system:

'static on flour 1', 'moving up', 'static on flour 2', 'moving down'


[picture of states inserted here]


The transitions and their inputs:

  • from 'static on flour 1' to 'moving up' triggered by 'up button'
  • from 'moving up' to 'static on flour 2' triggered by 'arrival 2'
  • from static on flour 2 to 'moving down' triggered by 'down button'
  • from 'moving down' to 'static on flour 1' triggered by 'arrival 1'

Form the above we can draw the finite state machine for the elevator.

[picture of completed finite state machine to be inserted here]

We can also produce a state transition table (DEFINE THIS LANGUAGE BOX?):

[picture of state transition diagram to be inserted here]


Representing a system as a finite state machine is very powerful because the model allows us to demonstrate the behaviour very clearly. We can prove that the system is robust and will not behave in any unexpected manner. Consider the example of the elevator: by modelling the system as a finite state machine, it is possible to show that the elevator is not able to move up and down without stopping. This is because the design clearly shows that it is impossible to transition from the state 'moving up' to the state 'moving down'.

Applications of finite state machines are found in many sciences. Mainly engineering, biology and most commonly in linguistics, where they are used to describe languages.


A finite state automata accepting binary input

Looking at the above diagram we can see that it starts in state S1, an input of 1 will keep it in state S1, and an input of 0 will move it to state S2. Once in S2 an input of 1 will keep it there, and an input of 0 will switch it back to S1. This means that the following inputs are valid:

110011001
001110011

It might appear to accept any binary value, but this isn't true. The only state it can accept in is state S1. This places the following rule on all accepted inputs: "A combination of binary digits involving an even number of zeros". This is useful for parity checks. If I try the following:

110011011

I am stuck in state S2 and the FSM has not accepted. Can you create a FSM to only accept Binary numbers with odd numbers of 1s?

Exercise: Finite State Automaton

CPT-FSM-abcd.svg

For the FSM above which of these inputs are valid:

  1. aaacdb
  2. ababacdaaac
  3. abcdb
  4. acda
  5. acdbdb

Answer :

  1. aaacdb (CORRECT)
  2. ababacdaaac(CORRECT)
  3. abcdb (ERROR no input that accepts b then c)
  4. acda (ERROR S1 is not a accepting state)
  5. acdbdb (CORRECT)

CPT-FSM-0123456789+-.svg

For the FSM above which of these inputs are valid:

  1. 987654321+994-0
  2. 5-5+2*4
  3. 9+8+7+6+5+4+3+2+1
  4. 0+1+2+1+
  5. 99+88-77

Answer :

  1. 987654321+994-0 (CORRECT)
  2. 5-5+2*4 (ERROR no input *)
  3. 9+8+7+6+5+4+3+2+1 (CORRECT)
  4. 0+1+2+1+ (ERROR S3 is not a accepting state)
  5. 99+88-77 (CORRECT)

Draw a finite state automata that will accept the word Banana whilst using only 3 states

Answer :

CPT-FSM-FSA-Banana.svg

Draw a finite state automata that will accept the words:

  • Tim
  • Grit
  • Grrrrrim

Answer :

CPT-FSM-FSA-Grim.svg

Mealy Machines[edit]

Mealy Machine - an FSM with outputs


Some FSMs output values dependant on the state and the input values:

CPT-FSM-Mealy-01.svg

The above Mealy Machine outputs a value for each input. You can tell which is which by: input / ouput. So for the following input:

000101010

It outputs

000010101

Shifting all the bits right, and dividing the binary number input by two.

Exercise: Mealy Machines
CPT-FSM-Mealy-02.svg

For the Mealy machine above, what do the following inputs output:

  1. 50
  2. 20,10,20
  3. 10,10,10,20

What does this machine do, what do the outputs tell you?

Answer :

  1. 0
  2. 30,20,0
  3. 40,30,20,0

This machine could be used to track the money going into a vending machine, letting you know how much you have left to pay on a 50p chocolate bar

What is the difference between a mealy machine and a finite state automaton?

Answer :

  • mealy machines output values
  • finite state automata do not
Extension: non-determinism

In this section we are learning about deterministic finite automaton. This means that for a state and a valid input there is only one possible transition to take. There are such things a nondeterministic finite automaton where, for a given input there are multiple paths (or none) that could be taken:

In state p, for input 1 there are two possible transitions

State transition tables[edit]

A state transition table follows every state and input. Inputs are usually placed on the left, and separated from the outputs, which are on the right. Here's a simple example of a state machine with two states, and a binary input:

CPT-FSM-01.svg

Input Current State Next State Output
0 S1 S2 null
1 S1 S1 null
0 S2 S1 null
1 S2 S3 null
Exercise: State transition tables

Create a state transition table for the following FSM:

DFAexample.svg

Answer :

Input Current State Next State Output
0 S1 S2 null
1 S1 S1 null
0 S2 S1 null
1 S2 S2 null

Create a state transition table for the following FSM:

CPT-FSM-Mealy-01.svg

Answer :

Input Current State Next State Output
0 S0 S2 0
1 S0 S1 0
0 S1 S2 1
1 S1 S1 1
0 S2 S2 0
1 S2 S1 0

Draw the FSM for the following state transition table:

Input Current State Next State Output
Timer Green Green null
Button Green Yellow null
Timer Yellow Red null
Timer Red Green null

Answer :

CPT-FSM-FSA-Traffic.svg

You might have already cottoned on. This represents a traffic light system

Draw the FSM for the following state transition table:

Input Current State Next State Output
a q0 q0 null
b q0 q1 null
a q1 q2 null
b q1 q1 null
a q2 q1 null
b q2 q1 null

Answer :

Deterministicfiniteautomaton.svg