Languages

1. Courtesy: Costas Busch - LSU 1
Languages
Slides have been modified by Shaukat Wasi

2. Courtesy: Costas Busch - LSU 2
Language: a set of strings
String: a sequence of symbols
from some alphabet
Example:
Strings: cat, dog, house
Language: {cat, dog, house}
 zcba ,,,, Alphabet:

3. Courtesy: Costas Busch - LSU 3
Languages are used to describe
computation problems:
},17,13,11,7,5,3,2{ PRIMES
},6,4,2,0{ EVEN
}9,,2,1,0{ Alphabet:

4. Courtesy: Costas Busch - LSU 4
Computation is translated to set membership
Is number prime?x
}?,17,13,11,7,5,3,2{ PRIMESx
Example computation problem:
Equivalent set membership problem:

5. Courtesy: Costas Busch - LSU 5
Alphabets and Strings
abbaw
bbbaaav
abu



 ba,
baaabbbaaba
abba
ab
a
Example Strings
Example Alphabet:
An alphabet is a set of symbols
A string is a sequence of
symbols from the alphabet
String variables

6. Courtesy: Costas Busch - LSU 6
}9,,2,1,0{ Decimal numbers alphabet
102345 567463386
}1,0{Binary numbers alphabet
100010001 101101111

7. Courtesy: Costas Busch - LSU 7
}1{Unary numbers alphabet
1Unary number:
Decimal number: 1
11
2
111
3
1111
4
11111
5

8. Courtesy: Costas Busch - LSU 8
String Operations
m
n
bbbv
aaaw


21
21


bbbaaa
abba
mn bbbaaawv  2121
Concatenation
abbabbbaaa

9. Courtesy: Costas Busch - LSU 9
12aaaw n
R

naaaw 21 ababaaabbb
Reverse
bbbaaababa

10. Courtesy: Costas Busch - LSU 10
String Length
Length:
Examples:
naaaw 21
nw 
1
2
4



a
aa
abba

11. Courtesy: Costas Busch - LSU 11
Length of Concatenation
Example:
vuuv 
853
8
5,
3,




vuuv
aababaabuv
vabaabv
uaabu

12. Courtesy: Costas Busch - LSU 12
A string with no letters is denoted:
Acts as a neutral element
Observations:
Empty String
 or
abbabaababbaabba
www





 0

13. Courtesy: Costas Busch - LSU 13
Substring
Substring of string:
a subsequence of consecutive characters
String Substring
bbab
b
abba
ab
abbab
abbab
abbab
abbab

14. Courtesy: Costas Busch - LSU 14
Prefix and Suffix
Prefixes Suffixes
abbab
abbab
abba
abb
ab
a


b
ab
bab
bbab
abbab uvw 
prefix
suffix
string

15. Courtesy: Costas Busch - LSU 15
Exponent Operation
Example:
Definition:
  
n
n
wwww 
  abbaabbaabba 2
0
w
  0
abba

16. Courtesy: Costas Busch - LSU 16
The * Operation
: the set of all possible strings from
alphabet
*


 
 ,,,,,,,,,
,
*
aabaaabbbaabaaba
ba



17. Courtesy: Costas Busch - LSU 17
The + Operation
: the set of all possible strings from
alphabet except



 
 ,,,,,,,,,
,
*
aabaaabbbaabaaba
ba


 *
 ,,,,,,,, aabaaabbbaabaaba


18. Courtesy: Costas Busch - LSU 18
Languages
A language over alphabet
is any subset of *

 
 ,,,,,,,,
,
*
aaabbbaabaaba
ba


 
 
},,,,,{
,,
{}
aaaaaaabaababaabba
aabaaa


Languages:

Example:

19. Courtesy: Costas Busch - LSU 19
More Language Examples
An infinite language }0:{  nbaL nn
aaaaabbbbb
aabb
ab

L
Labb
},{ baAlphabet
Lbbabb

20. Courtesy: Costas Busch - LSU 20
prime}isand:{ *
xxxPRIMES 
}9,,2,1,0{ 
},17,13,11,7,5,3,2{ PRIMES
Prime numbers
Alphabet
Language:
Rule: Numbers divisible by 1 and itself

21. Courtesy: Costas Busch - LSU 21
even}isand:{ *
xxxEVEN 
}9,,2,1,0{ 
Even and odd numbers
Alphabet
odd}isand:{ *
xxxODD 
},6,4,2,0{ EVEN
},7,5,3,1{ ODD
Languages:

22. Courtesy: Costas Busch - LSU 22
Addition (of unary numbers)
},,1{ 
}1,1,n
,1z,1y,1x:{ mn


mnkm
zyxADDITION k
ADDITION 1111111111
ADDITION 111111111
Alphabet:
Language:
}111111111111111111111111{ ...,,,,ADDITION 

23. Courtesy: Costas Busch - LSU 23
Squares (of unary numbers)
}#,1{
}11{ 2
nm,,y: xx#ySQUARES mn

SQUARES# 111111
SQUARES# 1111111
Alphabet:
Language:
}11111111111111111111{ , ...#,#,#SQUARES

24. Courtesy: Costas Busch - LSU 24
 
 
5|},,,,{|
3|,,|
1||
0|{}|




babaabbabbaa
abaaa


Two special languages
Size of a language (number of elements):
or}{ }{
Empty language
Language with
empty string

25. Courtesy: Costas Busch - LSU 25
Note that:
}{}{ 
0}{ 
1}{ 
0
Sets
Set size
Set size
String length

26. Courtesy: Costas Busch - LSU 26
Operations on Languages
The usual set operations:
   
   
      difference,,,,
onintersecti}{,,,
union},,,{,,,
aaaaaabbbaaaaaba
ababbbaaaaaba
aaaabbabaabbbaaaaaba



LL  *
   ,,,,,,, aaabbabaabbaa 
Complement:

27. Courtesy: Costas Busch - LSU 27
Reverse
Definition:
Examples:
}:{ LwwL RR

   ababbaabababaaabab R
,,,, 
}0:{
}0:{


nabL
nbaL
nnR
nn

28. Courtesy: Costas Busch - LSU 28
Concatenation
Definition:
Example:
 2121 ,: LyLxxyLL 
  
 baaabababaaabbaaaab
aabbaaba
,,,,,
,,,


29. Courtesy: Costas Busch - LSU 29
Another Operation
Definition:
Special case:
 
n
n
LLLL 
     
 bbbbbababbaaabbabaaabaaa
babababa
,,,,,,,
,,,, 3

 
   



0
0
,, aaabbaa
L

30. Courtesy: Costas Busch - LSU 30
}0:{  nbaL nn
}0,:{2
 mnbabaL mmnn
2
Laabbaaabbb
Example
2
?????? L

31. Courtesy: Costas Busch - LSU 31
Star-Closure (Kleene *)
All strings that can be constructed from
Definition:
Example:
 210*
LLLL
 















,,,,
,,,,
,,
,
*,
abbbbabbaaabbaaa
bbbbbbaabbaa
bba
bba

L
0
L
1
L
2
L
3
L

32. Courtesy: Costas Busch - LSU 32
Positive Closure
Definition:  321
LLLL
 











,,,,
,,,,
,,
,
abbbbabbaaabbaaa
bbbbbbaabbaa
bba
bba
1
L
2
L
3
L
Note that: 
 LLL 0*

33. CLASS ACTIVITY#01
Courtesy: Costas Busch - LSU 33