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## Finite Automata

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### The automatic door at the grocery store has two pads: One in front of the door. ... Upper paddle. Lower gate. Lower paddle. Definition of a Finite Automaton ... – PowerPoint PPT presentation

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Title: Finite Automata

1
Finite Automata
• Lecture 4
• Section 1.1
• Wed, Sep 6, 2006

2
The Automatic Door
• The automatic door at the grocery store has two
• One in front of the door.
• One behind the door.
• The door is in one of two possible states
• Open
• Closed

3
The Automatic Door
• There are two independent input signals
• A person is or is not standing on the front pad.
• A person is or is not standing on the rear pad.
• There are four combinations of input signals.

4
The Automatic Door
• In terms of input signals and door states,
describe the behavior of the door.

5
The Automatic Door
• Express the behavior as a table.
• Express the behavior as a graph.

6
A Canal Lock
• Describe the operation of a canal lock designed
so that the gates can never be opened when the
water on the two sides is not at the same level.

7
A Canal Lock
• The working parts of the lock are
• Upper gate
• Lower gate

8
Definition of a Finite Automaton
• A finite automaton is a 5-tuple (Q, ?, ?, q0, F),
where
• Q is a finite set of states,
• ? is a finite alphabet,
• ? Q ? ? ? Q is the transition function,
• q0 is the start state, and
• F ? Q is the set of accept states.

9
Definition of a Finite Automaton
• If, at the end of reading the input string, the
automaton is in an accept state, then the input
is accepted.
• Otherwise, it is rejected.

10
Definition of a Finite Automaton
• Describe the automatic door formally.
• Describe the canal lock formally.
• An accept state is any state that doesnt cause a
disaster.

11
The Language of a Machine
• A given finite automaton accepts a specific set
of input strings.
• That is called the language of the automaton.
• A language is called regular if it is the
language of some finite automaton.

12
Examples
• Design a finite automaton that accepts all
strings that start with a and end with b.
• Design a finite automaton that accepts all
strings that contain an even number of as.

13
The Regular Operations
• We may define operations on languages
• Union
• A?? B x x ? A or x ? B.
• Concatenation
• A?? B xy x ? A and y ? B.
• Star
• A x1x2xk xi ? A and k ? 0.

14
Closure under Union
• Theorem If A and B are regular languages, then
so are
• A ? B
• A?? B
• A

15
Examples
• Let A x x contains an even number of as.
• Let B x x contains an even number of bs.
• Try to design finite automata for
• A ? B
• A?? B
• A