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Languages and Finite Automata

orhow to talk to machines...

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A language is a set of strings

String:A sequence of letters (a word)

Examples: “cat”, “dog”, “house”, …

Defined over an alphabet:set of symbols (letters)

Languages

zcba ,,,,

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Alphabets and StringsWe will use small alphabets

Strings

abbaw

bbbaaav

abu

ba,

baaabbbaaba

baba

abba

ab

a

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String Operations

Concatenation

Reverse

abbaw

aaaw n

21

bbbaaav

bbbv m

21

abbabbbaaawv

bbbaaawv mn

2121

aaabbbv

bbbvR

mR

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String Length

Length:

Examples:

naaaw 21

nw

1

2

4

a

aa

abba

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Recursive Definition of LengthFor any letter:

For any string :

Example:

1a

1wwawa

4

1111

111

11

1

a

ab

abbabba

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Length of Concatenation

Example:

vuuv

853

8

5,

3,

vuuv

aababaabuv

vabaabv

uaabu

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Proof of Concatenation LengthClaim:

Proof:By induction on the length

Induction basis:

is only one symbol

From definition of length:

vuuv

v

1v

vuuuv 1

v

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Inductive hypothesis:

for all with

Inductive step: we will prove

for

vuuv

v nv ,,2,1

1nv

vuuv

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Inductive StepWrite , where

From definition of length:

From inductive hypothesis:

Thus:

wav 1, anw

1

1

wwa

uwuwauv

wuuw

vuwauwuuv 1

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Empty StringA string with no letters:

Observations:

abbaabbaabba

www

0

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SubstringDefinition:

A substring of a string is any sequence of consecutive characters

Example: Substrings

w

abbab

bbab

b

abba

ab

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Prefix and Suffix

Prefixes Suffixes

abbab

abbab

abba

abb

ab

a

b

ab

bab

bbab

abbabuvw

prefix

suffix

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Another Operation

Example:

Definition for any :

n

n wwww

abbaabbaabba 2

w 0w

0abba

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The * Operation : the set of all possible strings from alphabet

Example:

*

,,,,,,,,,*

,

aabaaabbbaabaaba

ba

*

,,,,,,,, aabaaabbbaabaaba

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LanguageA language is any subset of

Examples:

A string is called “sentence”

*

,,,,,,,,,*

,

aabaaabbbaabaaba

ba

},,,,,{

,,,

aaaaaaabaababaabba

aabaaa

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Another ExampleAn infinite language

0: nbaL nn

aaaaabbbbb

aabb

ab

L Labb

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Operations on LanguagesThe usual set operations

Complement:

aaaaaabbbaaaaaba

ababbbaaaaaba

aaaabbabaabbbaaaaaba

,,,,

}{,,,

},,,{,,,

LL *

,,,,,,, aaabbabaabbaa

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Reverse

Definition:

Examples:

LwwL RR :

ababbaabababaaabab R ,,,,

0:

0:

nabL

nbaL

nnR

nn

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Concatenation

Definition:

Example:

2121 ,: LyLxxyLL

baaabababaaabbaaaab

aabbaaba

,,,,,

,,,

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Another OperationDefinition:

Example:

Special case:

n

n LLLL

bbbbbababbaaabbabaaabaaa

babababa

,,,,,,,

,,,, 3

0

0

,, aaabbaa

L

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Example

0: nbaL nn

0,:2 mnbabaL mmnn

2Laabbaaabbb

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Star-Closure (Kleene *)

Definition:

Example:

210* LLLL

,,,,

,,,,

,,

,

*,

abbbbabbaaabbaaa

bbbbbbaabbaa

bbabba

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Positive Closure

Definition:

*

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L

LLL

,,,,

,,,,

,,

,

abbbbabbaaabbaaa

bbbbbbaabbaa

bba

bba

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Finite Automata

Input

String

Output

String

FiniteAutomaton

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Finite Accepter

Input

“Accept” or“Reject”

String

FiniteAutomaton

Output

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Transition Graph

0q 1q 2q 3q 4qa b b a

initialstate

final state“accept”state

transition

5q

a a bb

ba,Abba -Finite Accepter

ba,

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Initial Configuration

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

Input Stringa b b a

ba,

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Reading the Input

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b b a

ba,

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0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b b a

ba,

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0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b b a

ba,

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0q 1q 2q 3q 4qa b b a

Output: “accept”

5q

a a bb

ba,

a b b a

ba,

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Rejection

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b a

ba,

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0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b a

ba,

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0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b a

ba,

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0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

Output:“reject”

a b a

ba,

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FormalitiesDeterministic Finite Accepter (DFA)

FqQM ,,,, 0

Q

0q

F

: set of states

: input alphabet

: transition function

: initial state

: set of final states

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Input Aplhabet

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

ba,

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Set of States

Q

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

543210 ,,,,, qqqqqqQ

ba,

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Initial State

0q

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

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Set of Final States

F

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

4qF

ba,

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Transition Function

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

QQ :

50

10

,

,

qbq

qaq

ba,

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Transition Function

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b

0q

1q

2q

3q

4q

5q

1q 5q

5q 2q

2q 3q

4q 5q

ba,5q5q5q5q

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Extended Transition Function

*

QQ *:*

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40

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,*

,*

,*

qabbaaq

qabbaq

qabq

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

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Recursive Definition

)),,(*(,*

,*

awqwaq

qq

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

2

1

0

0

0

0

,

,,

,,,*

),,(*

,*

q

bq

baq

baq

baq

abq

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Languages Accepted by DFAsTake DFA

Definition:The language accepted by contains all input strings accepted by

In other words: = { strings that drive to a final

state}

M

ML MM

M ML

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Example

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

abbaML M

accept

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Another Example

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

abbaabML ,, M

acceptacceptaccept

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Formally

For a DFA

Language accepted by :

FqQM ,,,, 0

M

FwqwML ,*:* 0

alphabet transitionfunction

initialstate

finalstates

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Observation Language accepted by :

Language rejected by :

FwqwML ,*:* 0

M

FwqwML ,*:* 0

M

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More Examples

a

b ba,

ba,

0q 1q 2q

0: nbaML n

accept trap state

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ML = { all substrings with prefix }ab

a b

ba,

0q 1q 2q

accept

ba,3q

ab

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ML = { all strings without substring }001

0 00 001

1

0

1

10

0 1,0

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Regular LanguagesDefinition:

A language is regularif there is a DFA such that

All the regular languages form a family

LM

MLL

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ExampleThe languageis regular:

*,: bawawaL

a

b

ba,

a

b

ba

0q 2q 3q

4q