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Languages
Fall 2006
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Language: a set of strings
String: a sequence of symbols
from some alphabet
Example:
Strings: cat, dog, house
Language: {cat, dog, house}
Alphabet:
Fall 2006
  a, b, c,, z
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Languages are used to describe
computation problems:
PRIMES  {2,3,5,7,11,13,17,}
EVEN  {0,2,4,6,}
Alphabet:
Fall 2006
  {0,1,2,,9}
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Alphabets and Strings
An alphabet is a set of symbols
Example Alphabet:
  a, b
A string is a sequence of
symbols from the alphabet
Example Strings
a
ab
abba
aaabbbaabab
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u  ab
v  bbbaaa
w  abba
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Decimal numbers alphabet
102345
567463386
Binary numbers alphabet
100010001
Fall 2006
  {0,1,2,,9}
  {0,1}
101101111
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Unary numbers alphabet
  {1}
Unary number:
1
11
111
1111
11111
Decimal number:
1
2
3
4
5
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String Operations
w  a1a2 an
abba
bbbaaa
v  b1b2 bm
Concatenation
wv  a1a2 anb1b2 bm
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abbabbbaaa
7
w  a1a2  an
ababaaabbb
Reverse
w  an  a2 a1
R
Fall 2006
bbbaaababa
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String Length
w  a1a2  an
Length:
w n
Examples:
abba  4
aa  2
a 1
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Length of Concatenation
uv  u  v
Example:
u  aab, u  3
v  abaab, v  5
uv  aababaab  8
uv  u  v  3  5  8
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Empty String
A string with no letters is denoted:
 or 
Observations:   0
w  w   w
abba  abba  abba  abba
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Substring
Substring of string:
a subsequence of consecutive characters
String
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Substring
abbab
ab
abbab
abbab
abbab
abba
b
bbab
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Prefix and Suffix
abbab
Prefixes

a
ab
abb
abba
abbab
Fall 2006
Suffixes
abbab
bbab
bab
ab
b
w  uv
prefix
suffix

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Another Operation
w  ww

w




n
n
Example:
Definition:
abba  abbaabba
2
w 
0
abba   
0
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The * Operation
 * : the set of all possible strings from
alphabet 
  a, b
*   , a, b, aa, ab, ba, bb, aaa, aab,
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The + Operation
 : the set of all possible strings from

alphabet  except 
  a, b
*   , a, b, aa, ab, ba, bb, aaa, aab,

   * 

  a, b, aa, ab, ba, bb, aaa, aab,
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Languages
A language over alphabet
is any subset of  *
Examples:

  a, b
*   , a, b, aa, ab, ba, bb, aaa,
Language:

a, aa, aab
Language:
{ , abba, baba, aa, ab, aaaaaa}
Language:
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More Language Examples
Alphabet
  {a , b }
An infinite language
L  {a b : n  0}
n n

ab
aabb
aaaaabbbbb
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L
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abb  L
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Prime numbers
Alphabet
  {0,1,2,,9}
Language:
PRIMES  {x : x   and x is prime}
*
PRIMES  {2,3,5,7,11,13,17,}
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Even and odd numbers
Alphabet
  {0,1,2,,9}
EVEN  {x : x   and x is even}
*
EVEN  {0,2,4,6,}
ODD  {x : x   and x is odd}
ODD  {1,3,5,7,}
*
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Unary Addition
Alphabet:
  {1,, }
Language:
ADDITION  {x  y  z : x  1 , y  1 , z  1 ,
n
m
k
nm k}
11  111  11111  ADDITION
111  111  111  ADDITION
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Squares
Alphabet:
  {1, # }
Language:
SQUARES  {x # y : x  1 , y  1 , m  n }
n
m
2
11#1111  SQUARES
111#1111  SQUARES
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Note that:
Sets
  { }  {}
Set size
{}    0
Set size
{}  1
String length
 0
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Operations on Languages
The usual set operations
a, ab, aaaa  bb, ab  {a, ab, bb, aaaa}
a, ab, aaaa  bb, ab  {ab}
a, ab, aaaa  bb, ab  a, aaaa
Complement:
L   * L
a, ba   , b, aa, ab, bb, aaa,
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Reverse
Definition:
Examples:
L  {w : w  L}
R
R
ab, aab, baba  ba, baa, abab
R
L  {a b : n  0}
n n
L  {b a : n  0}
R
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n n
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Concatenation
Definition:
Example:
L1L2  xy : x  L1, y  L2 
a, ab, bab, aa
 ab, aaa, abb, abaa, bab, baaa
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Another Operation
Definition:
L 
LL

L
n
n
a, b  a, ba, ba, b 
aaa, aab, aba, abb, baa, bab, bba, bbb
3
0
Special case: L  
a , bba , aaa 0  
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L  {a b : n  0}
n n
L  {a b a b : n, m  0}
2
n n m m
2
aabbaaabbb L
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Star-Closure (Kleene *)
All strings that can be constructed from
Definition:
L
L*  L  L  L 
0
1
2
Example:
 ,

a, bb,



a, bb*  

aa
,
abb
,
bba
,
bbbb
,


aaa, aabb, abba, abbbb,
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Positive Closure
Definition:
L  L  L 

1
2
Same with L* but without the

a, bb,


 
a, bb  aa, abb, bba, bbbb,

aaa, aabb, abba, abbbb,


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