title=Problem Solving: Finite state machines
A finite state machine consists of states, inputs and outputs. The number of states is fixed; when an input is executed, the state is changed and an output is possibly produced. Finite state machines are widely used when designing computer programs, but also have their uses in engineering, biology, linguistics and other sciences thanks to their ability to recognise sequences.
A finite state machine expressed visually is a State transition diagram. It is used to show all the states, inputs and outputs. Each state is represented with a circle, and each transition with an arrow. Transitions are labelled with an input that causes a transition and possibly an output that results from it. A double circle signifies the accept state. Not all FSMs will have an accepting state and it is possible they could run forever.
A finite state machine can be with or without outputs:
Finite state automaton[edit]
Finite State Automaton  an FSM with no outputs
Looking at the above diagram we can see that it starts in state S1, an input of 1 will keep it in state one, 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 Answer:
For the FSM above which of these inputs are valid:
Answer:
Draw a finite state automata that will accept the word Banana whilst using only 3 states Draw a finite state automata that will accept the words:

Mealy Machines[edit]
Mealy Machine  an FSM with outputs
Some FSMs output values dependent on the state and the input values:
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 For the Mealy machine above, what do the following inputs output:
What does this machine do, what do the outputs tell you? Answer:
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:

Extension: nondeterminism 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: 
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:
Input  Current State  Next State  Output 

0  S_{1}  S_{2}  null 
1  S_{1}  S_{1}  null 
0  S_{2}  S_{1}  null 
1  S_{2}  S_{3}  null 
Exercise: State transition tables Answer:
Answer:
Draw the FSM for the following state transition table:
Draw the FSM for the following state transition table:
