Transcript Languages and Finite Automata

Languages
Costas Busch - LSU
1
Language: a set of strings
String: a sequence of symbols
from some alphabet
Example:
Strings: cat, dog, house
Language: {cat, dog, house}
Alphabet:
  a, b, c,, z
Costas Busch - LSU
2
Languages are used to describe
computation problems:
PRIMES  {2,3,5,7,11,13,17,}
EVEN  {0,2,4,6,}
Alphabet:
  {0,1,2,,9}
Costas Busch - LSU
3
Computation is translated to set membership
Example computation problem:
Is number
x
prime?
Equivalent set membership problem:
x  PRIMES  {2,3,5,7,11,13,17,}?
Costas Busch - LSU
4
Alphabets and Strings
An alphabet is a set of symbols
Example Alphabet:
  a, b
A string is a sequence of
symbols from the alphabet
a
ab
Example Strings
abba
aaabbbaabab
Costas Busch - LSU
String variables
u  ab
v  bbbaaa
w  abba
5
Decimal numbers alphabet
102345
567463386
Binary numbers alphabet
100010001
  {0,1,2,,9}
  {0,1}
101101111
Costas Busch - LSU
6
Unary numbers alphabet
  {1}
Unary number: 1
11
111
1111
11111
Decimal number: 1
2
3
4
5
Costas Busch - LSU
7
String Operations
w  a1a2  an
abba
v  b1b2 bm
bbbaaa
Concatenation
wv  a1a2 anb1b2 bm
Costas Busch - LSU
abbabbbaaa
8
w  a1a2 an
ababaaabbb
Reverse
w  an a2a1
R
bbbaaababa
Costas Busch - LSU
9
String Length
w  a1a2 an
Length:
w n
Examples:
abba  4
aa  2
a 1
Costas Busch - LSU
10
Length of Concatenation
uv  u  v
Example:
u  aab, u  3
v  abaab, v  5
uv  aababaab  8
uv  u  v  3  5  8
Costas Busch - LSU
11
Empty String
A string with no letters is denoted:
 or 
Acts as a neutral element
Observations:   0
w  w   w
abba  abba  abba  abba
Costas Busch - LSU
12
Substring
Substring of string:
a subsequence of consecutive characters
String
Substring
abbab
abbab
ab
abba
abbab
b
abbab
bbab
Costas Busch - LSU
13
Prefix and Suffix
string abbab
Prefixes
Suffixes

abbab
a
bbab
ab
bab
abb
ab
b
abba
abbab
w  uv
prefix
suffix

Costas Busch - LSU
14
Exponent Operation
w  ww

w




n
n
Example:
Definition:
abba  abbaabba
2
w 
0
abba  
0
Costas Busch - LSU
15
The * Operation
 * : the set of all possible strings from
alphabet 
  a, b
*   , a, b, aa, ab, ba, bb, aaa, aab,
Costas Busch - LSU
16
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,
Costas Busch - LSU
17
Languages
A language over alphabet
is any subset of  *
Example:

  a, b
*   , a, b, aa, ab, ba, bb, aaa,
Language:

a, aa, aab
Language:
{ , abba, baba, aa, ab, aaaaaa}
Language:
Costas Busch - LSU
18
More Language Examples
Alphabet   {a , b }
An infinite language
L  {a b : n  0}
n n

ab
aabb
aaaaabbbbb
bbabb  L
L
Costas Busch - LSU
abb  L
19
Prime numbers
Alphabet
  {0,1,2,,9}
Language:
PRIMES  {x : x   and x is prime}
*
PRIMES  {2,3,5,7,11,13,17,}
Costas Busch - LSU
20
Even and odd numbers
Alphabet
  {0,1,2,,9}
Languages:
EVEN  {x : x   and x is even}
*
EVEN  {0,2,4,6,}
ODD  {x : x   and x is odd}
ODD  {1,3,5,7,}
*
Costas Busch - LSU
21
Addition
Alphabet:
(of unary numbers)
  {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
Costas Busch - LSU
22
Squares (of unary numbers)
Alphabet:
  {1, # }
Language:
SQUARES  {x # y : x  1 , y  1 , m  n }
n
m
2
11#1111  SQUARES
111#1111  SQUARES
Costas Busch - LSU
23
Two special languages
Language with
empty string
{ }
Empty language
{}
Size of a language (number of elements):
| {} | 0
|  | 1
| a , aa, ab| 3
| { , aa, bb , abba , baba } | 5
Costas Busch - LSU
24
Note that:
Sets
  { }  {}
Set size
{}    0
Set size
{}  1
String length
 0
Costas Busch - LSU
25
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,
Costas Busch - LSU
26
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
n n
Costas Busch - LSU
27
Concatenation
Definition:
Example:
L1L2  xy : x  L1, y  L2 
a, ab, bab, aa
 ab, aaa, abb, abaa, bab, baaa
Costas Busch - LSU
28
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
Special case:
0
L  
0
a , bba , aaa   
Costas Busch - LSU
29
Example
L  {a b : n  0}
n n
L  {a b a b : n, m  0}
2
n n m m
2
aabbaaabbb L
Costas Busch - LSU
30
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,
Costas Busch - LSU
31
Positive Closure
Definition:
L  L  L 

1
2
a, bb,


 
a, bb  aa, abb, bba, bbbb,

aaa, aabb, abba, abbbb,


Costas Busch - LSU
32