Languages and Finite Automata

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Finite Automata
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Finite Automaton

Input
String
Output
Finite Automaton
String
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Finite Accepter

Input
String
Output
Accept or Reject
Finite Automaton
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Transition Graph

Abba -Finite Accepter
initial state
final state accept
transition
state
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Initial Configuration

Input String
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Reading the Input

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Input finished
Output accept
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Rejection

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Input finished
Output reject
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Another Rejection

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Output reject
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Another Example
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Input finished
Output accept
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Rejection
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Input finished
Output reject
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Formalities

• Deterministic Finite Accepter (DFA)

set of states
input alphabet
transition function
initial state
set of final states
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Input Alphabet

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Set of States

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Initial State

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Set of Final States

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Transition Function

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Transition Function

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Extended Transition Function

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Observation There is a walk from to
with label
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Example There is a walk from to
with label
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Inductive Definition
Basis Induction
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Languages Accepted by DFAs

• Take DFA
• Definition
• The language contains
• all input strings accepted by
• strings that drive to a
  final state

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Example

accept
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Another Example

accept
accept
accept
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Formally

• For a DFA
• Language accepted by

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Observation

• Language rejected by

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More Examples

trap state
accept
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all strings with prefix
accept
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all strings without substring
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Regular Languages

• A language is regular if there is
• a DFA such that
• All regular languages form a language family

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Examples of regular languages
all strings with prefix
all strings with prefix
all strings without substring
There exist automata that accept these Languages
(see previous slides).
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Another Example

• The language
• is regular

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There exist languages which are not Regular
Example
There is no DFA that accepts such a language
(we will prove this later in the class)