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
  pads
• One in front of the door.
• One behind the door.
• The door is in one of two possible states
• Open
• Closed

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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.

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The Automatic Door

• In terms of input signals and door states,
  describe the behavior of the door.

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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.

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A Canal Lock

• The working parts of the lock are
• Upper gate
• Upper paddle
• Lower gate
• Lower paddle

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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.

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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.

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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.

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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.

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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.

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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.

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Closure under Union

• Theorem If A and B are regular languages, then
  so are
• A ? B
• A?? B
• A

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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