Deterministic Finite Automata And Regular Languages

Deterministic Finite Automaton (DFA) Input Tape String Output “Accept” or “Reject” Finite Automaton

Transition Graph initial state accepting state transition state

Alphabet For every state, there is a transition for every symbol in the alphabet

Initial Configuration head Input Tape Input String Initial state

Scanning the Input

Input finished accept

A Rejection Case Input String

Input finished reject

Another Rejection Case Tape is empty Input Finished reject

Language Accepted:

To accept a string: all the input string is scanned and the last state is accepting To reject a string: all the input string is scanned and the last state is non-accepting

Another Example Accept state Accept state Accept state

Empty Tape Input Finished accept

Another Example trap state Accept state

Input String

Input finished accept

A rejection case Input String

Input finished reject

Language Accepted:

Another Example Alphabet: Language Accepted:

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

Set of States Example

Input Alphabet :the input alphabet never contains Example

Initial State Example

Set of Accepting States Example

Transition Function Describes the result of a transition from state with symbol

Example:

Transition Table for symbols states

Extended Transition Function Describes the resulting state after scanning string from state

Example:

Special case: for any state

implies that there is a walk of transitions In general: implies that there is a walk of transitions states may be repeated

Language Accepted by DFA Language of DFA : it is denoted as and contains all the strings accepted by We say that a language is accepted (or recognized) by DFA if

For a DFA Language accepted by :

Language rejected by :

More DFA Examples Empty language All strings

Language of the empty string

= { all strings with prefix } accept

= { all binary strings containing substring }

= { all binary strings without substring }

Regular Languages Definition: A language is regular if there is a DFA that accepts it ( ) The languages accepted by all DFAs form the family of regular languages

Example regular languages: { all strings in {a,b}* with prefix } { all binary strings without substring } There exist automata that accept these languages (see previous slides).

There exist languages which are not Regular: There is no DFA that accepts these languages (we will prove this in a later class)

This Summary is an Online Content from this Book: Michael Sipser, Introduction to the Theory of Computation, 2ndEdition It is edited for Computation Theory Course 6803415-3 by: T.Mariah Sami Khayat Teacher Assistant @ Adam University College For Contacting: mskhayat@uqu.edu.sa Kingdom of Saudi Arabia Ministry of Education Umm AlQura University Adam University College Computer Science Department المملكة العربية السعودية وزارة التعليم جامعة أم القرى الكلية الجامعية أضم قسم الحاسب الآلي