Copyright © 2011-2014 - Curt Hill Finite State Machines The Simplest and Least Capable Automaton.

Copyright © 2011-2014 - Curt Hill

Finite State Machines

The Simplest and Least Capable

Automaton

Automata• Lets imagine a simple machine as

a model of computation• It must have:• Input

– Assume a string of characters from some alphabet

• A control unit with an internal state• Output

– Output is determined by input and state


Transition Function• Each Finite State Automaton must

have a transition function• This transition function requires

two parameters:– The current state– The current input value

• The result is the new state• Any pair of parameters that has no

defined result generates an error state


Models• This simple automaton may be

modeled in several ways• A state chart

– On one side is the state– On the other is the inputs

• A graph– Each node represents a state– Each arc an input that causes the

state change


Examples


s0s1

s2

a b

b

a

a

b

s0

s1

s2

ba

s0

s0

s1

s1 s2

s2

Text Example• Rosen gives the example of a

vending machine• This is more complicated that we

can show in PowerPoint– 7 States– 5 possible inputs


FSM or FSA?• A Finite State Machine may also be

known as a Finite State Automaton• They come in two flavors

– Producing Output– Not producing Output

• Strangely we will deal with the Output producing first


More Formally• An output producing finite state

machine is 6 tupleM=(S,I,O,f,g,s0)

• S is a set of states• I is the set of inputs• O is the set of outputs• f is the transition function• g is the output function

• s0 is the initial stateCopyright © 2011-2014 - Curt Hill

The functions• The functions in the above are f and

g• The transition function, f, takes two

inputs– Current state– Input symbol

• The function then produces the new state

• The output function, g, takes the same two inputs, but produces the output symbol instead of the next state


Function Representation• We have already seen the table

representation of the transition function

• The output function has the same form, although the two are often combined into a single table

• In the graph form the arcs are labeled first by the input and then the output

• See next slideCopyright © 2011-2014 - Curt Hill

Output Example


s0s1

s2

a,0 b,2

b,1

a,1

a,2

b,0

s0

s1

s2

ba

s0

s0

s1

s1 s2

s2

s0

s1

s2

ba

0

0

1

1 2

2

Transition, f Output, g

Initial is s0

Example Continued


s0s1

s2

a,0 b,2

b,1

a,1

a,2

b,0

s0

s1

s2

ba

s0

s0

s1

s1 s2

s2

s0

s1

s2

ba

0

0

1

1 2

2


Initial is s0

What outputs with these inputs: ababbab aaabbbba aabaaaabab

Other Way• Given a FSM and an input we

should be able to show the output• Can we go the other way?

– Given a language can we compute the FSM?

• How do we specify that language?


Problem• We want an FSM that takes a bit

string and outputs a modified bit string

• If we number the bits from 1 – Odd bits are output unchanged– Even bits are reversed

• Lets pause a moment and consider what the state table would look like


Output Example


s0s1

1,1

1,0

s0

s1

10

s1

s0

s1

s0


Initial is s0

s0

s1

10

0

0

1

1

0,0

0,1

Finally• Several different machines• Mealy machines (G.H. Mealy) are

the output machines considered here

• Moore machines (E.F. Moore) gives an output based only on state– This is not Gordon Moore of Moore’s

Law


Exercises• 13.2

– 3, 5, 13, 19


Copyright © 2011-2014 - Curt Hill Finite State Machines The Simplest and Least Capable Automaton.

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