Languages and Finite Automata

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Formal Languages Finite Automata Hinrich
Schütze IMS, Uni Stuttgart, WS 2007/08 Slides
based on RPI CSCI 2400 Thanks to Costas Busch
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Chomsky Hierarchy

• Wikipedia

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Finite Automaton

Input
String
Output
Accept or Reject
Finite Automaton
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Transition Graph

initial state
accepting state
transition
state
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Initial Configuration

Input String
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Reading the Input

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Input finished
accept
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Rejection

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Input finished
reject
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Acceptance or Rejection?

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Initial State

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Rejection

reject
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Language?
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Acceptance Example
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Input finished
accept
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Rejection Example
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Input finished
reject
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Languages Accepted by FAs

• FA
• Definition
• The language contains
• all input strings accepted by
• strings that bring
• to an accepting state

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Example L(M) ?

accept
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L(M) abba

accept
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L(M) ?

accept
accept
accept
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accept
accept
accept
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L(M) ?

trap state
accept
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trap state
accept
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Formal Definition

• Finite Automaton (FA)

set of states
input alphabet
transition function
initial state
set of accepting 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 Accepting 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 if there is a walk from to
with label then
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Example There is a walk from to
with label
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Recursive Definition
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Language Accepted by FAs

• For a FA
• Language accepted by

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Observation

• Language rejected by

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L(M) ?

accept
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all strings with prefix
accept
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L(M)?

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Example

all strings without substring
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L(M) ?

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Regular Languages

• Definition
• A language is regular if there is
• FA such that
• Observation
• All languages accepted by FAs
• form the family of regular languages

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Examples of regular languages
all strings with prefix
all strings without substring
There exist automata that accept these Languages
(see previous slides).
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There exist languages which are not Regular
Example
There is no FA that accepts such a language
(we will prove this later in the class)