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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
    pads
  • 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
  • Upper paddle
  • Lower gate
  • Lower paddle

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
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