CSE322 Definition and description of finite Automata Lecture #1

Definition An automaton is defined as a system where energy, materials and information are transformed, transmitted and used for performing some functions without direct participation of man.

Formal Definition Finite Automaton (FA) Finite Automaton (FA) : set of states : input alphabet : transition function : initial state : set of accepting states

Input alphabet Input alphabet

Set of States Set of States

Initial State Initial State

Set of Accepting States

Non-Deterministic Finite Automata

Nondeterministic Finite Automaton Alphabet =

Alphabet = Two choices

Alphabet = Two choices No transition No transition

First Choice

First Choice

First Choice

First Choice All input is consumed “accept”

Second Choice

Second Choice

Second Choice No transition: the automaton hangs

Second Choice Input cannot be consumed “reject”

An NFA accepts a string: when there is a computation of the NFA that accepts the string There is a computation: all the input is consumed and the automaton is in an accepting state

Example is accepted by the NFA: “accept” “reject” because this computation accepts

Rejection example

First Choice

First Choice “reject”

Second Choice

Second Choice

Second Choice “reject”

OR An NFA rejects a string: when there is no computation of the NFA that accepts the string. For each computation: All the input is consumed and the automaton is in a non final state OR The input cannot be consumed

Example is rejected by the NFA: “reject” “reject” All possible computations lead to rejection

Rejection example

First Choice

First Choice No transition: the automaton hangs

First Choice Input cannot be consumed “reject”

Second Choice

Second Choice

Second Choice No transition: the automaton hangs

Second Choice Input cannot be consumed “reject”

is rejected by the NFA: “reject” “reject” All possible computations lead to rejection

Language accepted:

Lambda Transitions

(read head does not move)

all input is consumed “accept” String is accepted

Rejection Example

(read head doesn’t move)

No transition: the automaton hangs

Input cannot be consumed “reject” String is rejected

Language accepted:

Another NFA Example

“accept”

Another String

“accept”

Language accepted

Another NFA Example

Language accepted (redundant state)

Remarks: The symbol never appears on the input tape Simple automata:

NFAs are interesting because we can express languages easier than FAs NFA FA

Formal Definition of NFAs Set of states, i.e. Input aplhabet, i.e. Transition function Initial state Accepting states

Transition Function

Extended Transition Function

Formally : there is a walk from to with label

The Language of an NFA