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Finite Automata

- Lecture 4
- Section 1.1
- Wed, Sep 6, 2006

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

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.

The Automatic Door

- In terms of input signals and door states,

describe the behavior of the door.

The Automatic Door

- Express the behavior as a table.
- Express the behavior as a graph.

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.

A Canal Lock

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

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.

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.

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.

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.

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.

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.

Closure under Union

- Theorem If A and B are regular languages, then

so are - A ? B
- A?? B
- A

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