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1 Languages
1
Languages
2
A language is a set of strings
String: A sequence of letters
Examples: “cat”, “dog”, “house”, …
Defined over an alphabet:
Languages
zcba ,,,,
3
Alphabets and StringsWe will use small alphabets:
Strings
abbaw
bbbaaav
abu
ba,
baaabbbaaba
baba
abba
ab
a
4
String Operations
m
n
bbbv
aaaw
21
21
bbbaaa
abba
mn bbbaaawv 2121
Concatenation
abbabbbaaa
5
12aaaw nR
naaaw 21 ababaaabbb
Reverse
bbbaaababa
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String Length
Length:
Examples:
naaaw 21
nw
1
2
4
a
aa
abba
7
Recursive Definition of LengthFor any letter:
For any string :
Example:
1a
1wwawa
4
1111
111
11
1
a
ab
abbabba
8
Length of Concatenation
Example:
vuuv
853
8
5,
3,
vuuv
aababaabuv
vabaabv
uaabu
9
Proof of Concatenation LengthClaim:
Proof: By induction on the length
Induction basis:
From definition of length:
vuuv
v
1v
vuuuv 1
10
Inductive hypothesis:
for
Inductive step: we will prove
for
vuuv
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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SubstringSubstring of string:
a subsequence of consecutive characters
String Substring
bbab
b
abba
ab
abbab
abbab
abbab
abbab
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Prefix and Suffix
Prefixes Suffixesabbab
abbab
abba
abb
ab
a
b
ab
bab
bbab
abbab uvw
prefix
suffix
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Another Operation
Example:
Definition:
n
n wwww
abbaabbaabba 2
0w
0abba
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The * Operation : the set of all possible strings from alphabet
*
,,,,,,,,,*
,
aabaaabbbaabaaba
ba
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The + Operation : the set of all possible strings from alphabet except
,,,,,,,,,*
,
aabaaabbbaabaaba
ba
*
,,,,,,,, aabaaabbbaabaaba
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LanguageA language is any subset of
Example:
Languages:
*
,,,,,,,,*
,
aaabbbaabaaba
ba
},,,,,{
,,
aaaaaaabaababaabba
aabaaa
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Another Example
An 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:
Special case:
n
n LLLL
bbbbbababbaaabbabaaabaaa
babababa
,,,,,,,
,,,, 3
0
0
,, aaabbaa
L
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More Examples
}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
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Finite Automaton
Input
String
Output
String
FiniteAutomaton
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Finite Accepter
Input
“Accept” or“Reject”
String
FiniteAutomaton
Output
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Transition Graph
initialstate
final state“accept”state
transition
Abba -Finite Accepter
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
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Initial Configuration
1q 2q 3q 4qa b b a
5q
a a bb
ba,
Input Stringa b b a
ba,
0q
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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,
33
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b b a
ba,
34
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,
Input finished
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Rejection
1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b a
ba,
0q
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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,
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,
Input finished
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Another Example
a
b ba,
ba,
0q 1q 2q
a ba
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a
b ba,
ba,
0q 1q 2q
a ba
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a
b ba,
ba,
0q 1q 2q
a ba
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a
b ba,
ba,
0q 1q 2q
a ba
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a
b ba,
ba,
0q 1q 2q
a ba
Output: “accept”
Input finished
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Rejection
a
b ba,
ba,
0q 1q 2q
ab b
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a
b ba,
ba,
0q 1q 2q
ab b
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a
b ba,
ba,
0q 1q 2q
ab b
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a
b ba,
ba,
0q 1q 2q
ab b
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a
b ba,
ba,
0q 1q 2q
ab b
Output: “reject”
Input finished
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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 Alphabet
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
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
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Set of Final States
F
0q 1q 2q 3qa b b a
5q
a a bb
ba,
4qF
ba,
4q
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Transition Function
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
QQ :
ba,
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10 , qaq
2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q 1q
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50 , qbq
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
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0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
32 , qbq
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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 *:*
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
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20 ,* qabq
3q 4qa b b a
5q
a a bb
ba,
ba,
0q 1q 2q
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40 ,* qabbaq
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
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50 ,* qabbbaaq
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
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50 ,* qabbbaaq
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
Observation: There is a walk from to with label
0qabbbaa
1q
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Recursive Definition
)),,(*(,*
,*
awqwaq
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
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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 contains all input strings accepted by
= { strings that drive to a final state}
M
MLM
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 Languages
A language is regular if there is a DFA such that
All regular languages form a language family
LM MLL
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ExampleThe languageis regular:
*,: bawawaL
a
b
ba,
a
b
ba
0q 2q 3q
4q
