Finite State
Machines
A Graph Automaton
Copyright © 2011 Curt Hill
Introduction
• We generally think of graphs as
models of real world things
– Airline flights
– Pre-requisite courses
• We may also use them to model
computations
Copyright © 2011 Curt Hill
Names
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•
•
•
•
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Not all of these are equivalent
Finite State Machines
Finite State Automata
Deterministic Finite Automata
Non-deterministic Finite Automata
Push Down Automata
Turing Machines
Copyright © 2011 Curt Hill
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
Copyright © 2011 Curt Hill
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
Copyright © 2011 Curt Hill
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
Copyright © 2011 Curt Hill
Examples
a
b
s0
s0 s1
s1
s2
s0
s2
s1
s2
b
a
s1
b
a
s0
s2
b
a
Copyright © 2011 Curt Hill
We need some more
• Every Finite State Machine needs a
couple of other things
• The initial state – where does the
machine start
• One or more terminal states – where
may the machine end
• Did you notice that it accepts a
language?
Copyright © 2011 Curt Hill
Exercise
• Recall the FSM that we just saw
– I will show again on next slide
• Lets assume that the initial and
terminal state is s0
• Find three strings that this
automaton accepts
• What is shortest string it may accept
• What is the longest string it may
accept
Copyright © 2011 Curt Hill
Exercise
a
b
s0
s0 s1
s1
s2
s0
s2
s1
s2
b
a
s1
b
a
s0
s2
b
a
Copyright © 2011 Curt Hill
Which are not legal?
•
•
•
•
•
•
aaabb
bbaa
abaab
bababba
abababa
Bbbbabab
a
b
s0
s0 s1
s1
s2
s0
s2
s1
s2
b
a
s1
b
a
s0
s2
b
Copyright © 2011 Curt Hill
a
Time Out
• It is now time to talk about Regular
Expressions
• Any language specified by this type
of FSM may be represented by a
regular expression
Copyright © 2011 Curt Hill