Deterministic Finite Automata And Regular Languages Costas Busch

Deterministic Finite Automata And Regular Languages Costas Busch - LSU 1

Deterministic Finite Automaton (DFA) Input Tape String Output Finite Automaton Costas Busch - LSU “Accept” or “Reject” 2

Transition Graph initial state transition Costas Busch - LSU accepting state 3

Alphabet For every state, there is a transition for every symbol in the alphabet Costas Busch - LSU 4

Initial Configuration Input Tape Input String Initial state Costas Busch - LSU 5

Scanning the Input head Costas Busch - LSU 6

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Input finished accept Last state determines the outcome Costas Busch - LSU 9

A Rejection Case Input String Costas Busch - LSU 10

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Input finished reject Last state determines the outcome Costas Busch - LSU 13

Another Rejection Case Tape is empty Input Finished (no symbol read) reject Costas Busch - LSU 14

This automaton accepts only one string Language Accepted: Costas Busch - LSU 15

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 Costas Busch - LSU 16

Another Example Accept state Costas Busch - LSU Accept state 17

Empty Tape Input Finished accept Costas Busch - LSU 18

Another Example Accept state Costas Busch - LSU trap state 19

Input String Costas Busch - LSU 20

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Input finished accept Costas Busch - LSU 23

A rejection case Input String Costas Busch - LSU 24

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Input finished reject Costas Busch - LSU 27

Language Accepted: Costas Busch - LSU 28

Another Example Alphabet: Language Accepted: Costas Busch - LSU 29

Formal Definition Deterministic Finite Automaton (DFA) : set of states : input alphabet : transition function : initial state : set of accepting states Costas Busch - LSU 30

Set of States Example Costas Busch - LSU 31

Input Alphabet : the input alphabet never contains empty string Example Costas Busch - LSU 32

Initial State Example Costas Busch - LSU 33

Set of Accepting States Example Costas Busch - LSU 34

Transition Function Describes the result of a transition from state with symbol Costas Busch - LSU 35

Example: Costas Busch - LSU 36

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Transition Table for states symbols Costas Busch - LSU 39

Extended Transition Function Describes the resulting state after scanning string from state Costas Busch - LSU 40

Example: Costas Busch - LSU 41

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Special case: for any state Costas Busch - LSU 44

In general: implies that there is a walk of transitions states may be repeated Costas Busch - LSU 45

Language Accepted by DFA Language accepted by DFA : it is denoted as and contains all the strings accepted by We also say that recognizes Costas Busch - LSU 46

For a DFA Language accepted by : Costas Busch - LSU 47

Language rejected by : Costas Busch - LSU 48

More DFA Examples Empty language All strings Costas Busch - LSU 49

Language of the empty string Costas Busch - LSU 50

= { all strings with prefix } accept Costas Busch - LSU 51

= { all binary strings containing substring } Costas Busch - LSU 52

= { all binary strings without substring } Costas Busch - LSU 53

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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 Costas Busch - LSU 55

Example regular languages: { all strings in {a, b}* with prefix } { all binary strings without substring } There exist DFAs that accept these languages (see previous slides). Costas Busch - LSU 56

There exist languages which are not Regular: There are no DFAs that accept these languages (we will prove this later) Costas Busch - LSU 57