Lec-02 Languages for finite automata FA

CSC312
Automata Theory
Lecture # 2
Languages

2
Alphabet: An alphabet is a finite set of symbols, usually
letters, digits, and punctuations.
Valid/In-valid alphabets: An alphabet may contain
letters consisting of group of symbols for example Σ=
{a, ba, bab, d}.
Remarks: While defining an alphabet of letters
consisting of more than one symbols, no letter should
be started with the letter of the same alphabet i.e.
one letter should not be the prefix of another.
However, a letter may be ended in a letter of same
alphabet.
Valid alphabet :
Invalid alphabet :
 
, ,
a ba c
 
Alphabet and Strings
 
, ,
a ab c
 

3
String or word: A finite sequence of
letters/alphabets
Examples: “cat”, “dog”, “house”, “read” …
Defined over an alphabet:
Language: A language is a set of
strings constructed from some
alphabet e.g. Urdu, English, Java, the
set of all binary strings
 
z
c
b
a ,
,
,
, 


Alphabets and Strings

Sentences are made up of certain
combinations of words.
Not all combinations of words lead to a valid English sentence.
So we see that some basic units are
combined to make bigger units.

Languages
How can you tell whether a given sentence
belongs to a particular languages
Black is cat the
The tea is hot
I like chocolates two much
Rules give a clue to forming as well as
validating sentences.

Formal vs. Informal Rules
Informal language -> abstract languages
Incoherent strings are also
understandable
Slang, idiom, dialect etc.
Raise ambiguity
Interpretation varies with region
I am through (BrE/AmE)
Same words have multiple meanings.
Like, light, base, etc.

Summary of Languages
Three aspects/specifications
Lexical
Defines valid words/units of a language
Syntactic
Defines rules for combining the units to
form valid sentences (computer programs
in context of machines)
Semantic
Concerned with the interpretation or
meaning of a sentence (what output to
produce in context of machines)
Affected by ambiguity the most.

Formal languages
Rules defined explicitly and clearly
No ambiguities
Universally uniform understanding
Lets the machine
Interpret an input uniformly every time. i.e.
always produces same output for a
particular input
Avoid crashes because of ambiguity.
Explicitly and categorically reject invalid
input

Formal Languages
Need uniformly understandable notation
Representations
Alphabet
Represents a finite set of fundamental units
of lanauges, e.g. for English ={a,b,….z.A,…
Z,}
∑ = {0,1}
∑ = {0,1,2,3,4,5,6,7,8,9}

Formal Languages
List of words
Set of all valid words of a given language,
e.g., a language English_Words that
contains all valid words of English would
have a  = {all entries of the dictionary
+ punctuation marks and blank space}
Denoted by 
Is  Finite or Infinite set.
Strings: A string a finite sequence of symbols
chosen from alphabet. For example
0111100 , 123045, abbbcdeg etc.

String Variable: A letter used for
denoting a string. The author uses w, x, y
and z as string variable. For example
w = 0111100 , x = 123045, z = abbbcdeg
Length of String: The number of positions
for symbols in the string. For simplicity
we can say that it is the number of
symbols in the string. For example
|w| = 7 , |x| = ? , |z| = ?

12
Alphabets and Strings
We will use small letters for alphabets:
Strings
abba
w
bbbaaa
v
ab
u



 
b
a,


b
aaabbbaaba
baba
abba
ab
a

13
String Operations
m
n
b
b
b
v
a
a
a
w


2
1
2
1


bbbaaa
abba
m
n b
b
b
a
a
a
wv 
 2
1
2
1

Concatenation
abbabbbaaa
Let we have following strings
Reverse
1
2a
a
a
w n
R

 abba
aaabbb

14
String Length
Length:
Examples:
n
a
a
a
w 
2
1

n
w 
1
2
4



a
aa
abba

15
Length of Concatenation
Example:
v
u
uv 

8
5
3
8
5
,
3
,











v
u
uv
aababaab
uv
v
abaab
v
u
aab
u

16
Empty String
A string with no letters:
Observations:
Note-1: A language that does not
contain any word at all is denoted by
 or { }. This language doesn’t contain
any word not even the NULL string.
i.e. { } ≠ {}
or or
  
0
w w w
abba abba abba

 
 

 
 

17
Empty String
Note-2: Suppose a language L doesn’t
contain NULL then
L = L + 
but L ≠ L + {}.
Important : NULL is identity element
with respect to concatenation.

18
Substring
Substring of string:
a subsequence of consecutive characters
String Substring
bbab
b
abba
ab
abbab
abbab
abbab
abbab

19
Prefix and Suffix
Let the string is
Prefixes Suffixes
abbab
abbab
abba
abb
ab
a


b
ab
bab
bbab
abbab uv
w 
prefix
suffix

20
Repeat Operation
- w repeated n time; that is,
Example:
Definition:

 
 
n
n
w
ww
w 
  abbaabba
abba 
2


0
w
  

0
abba
n
w

21
The * Operation
: the set of all possible strings from
alphabet , called closure of alphabets also known as
Kleene star operator or Kleene star closure.
i.e. infinitely many words each of finite length.
*


 
 

,
,
,
,
,
,
,
,
,
*
,
aab
aaa
bb
ba
ab
aa
b
a
b
a






22
The + Operation
: the set of all possible strings from
alphabet except , also known as Kleene plus operator.
Note : are infinite



 
 

,
,
,
,
,
,
,
,
,
*
,
aab
aaa
bb
ba
ab
aa
b
a
b
a










*
 

,
,
,
,
,
,
,
, aab
aaa
bb
ba
ab
aa
b
a



* and 
 

23
Languages
A language is a set of strings OR
A language is any subset of , usually
denoted by L. It may be finite or infinite.
Example:
Languages:
If a string w is in L, we say that w is a
sentence of L.
*

 
 

,
,
,
,
,
,
,
,
*
,
aaa
bb
ba
ab
aa
b
a
b
a





 
 
}
,
,
,
,
,
{
,
,
aaaaaa
ab
aa
baba
abba
aab
aa
a



24
Note that:
}
{
}
{ 



0
}
{ 


1
}
{ 

0


Sets
Set size
Set size
String length

25
Another Example
An infinite language }
0
:
{ 
 n
b
a
L n
n
aaaaabbbbb
aabb
ab

L
 L
abb

26
Operations on Languages
The usual set operations
Complement:
   
   
     
aaaa
a
ab
bb
aaaa
ab
a
ab
ab
bb
aaaa
ab
a
aaaa
bb
ab
a
ab
bb
aaaa
ab
a
,
,
,
,
}
{
,
,
,
}
,
,
,
{
,
,
,






L
L 

 *
   

,
,
,
,
,
,
, aaa
bb
ab
aa
b
ba
a 


27
Reverse
Definition:
Examples:
Concatenation
Definition:
Examples:
}
:
{ L
w
w
L R
R


   
abab
baa
ba
baba
aab
ab R
,
,
,
, 
{ : 0}
{ : 0}
n n
R n n
L a b n
L b a n
 
 
 
2
1
2
1 ,
: L
y
L
x
xy
L
L 


  
 
, , ,
, , , , ,
a ab ba b aa
ab aaa abb abaa bab baaa


28
Repeat Operation
Definition:
L concatenated with itself n times.
Special case:


 
n
n
L
LL
L 
     
 
bbb
bba
bab
baa
abb
aba
aab
aaa
b
a
b
a
b
a
b
a
,
,
,
,
,
,
,
,
,
,
, 3


 
   
0
0
, ,
L
a bba aaa





29
More Examples
}
0
:
{ 
 n
b
a
L n
n
}
0
,
:
{
2

 m
n
b
a
b
a
L m
m
n
n
2
L
aabbaaabbb

30
Star-Closure (Kleene *)
Definition:
Example:


 2
1
0
* L
L
L
L 
 
















,
,
,
,
,
,
,
,
,
,
,
*
,
abbbb
abba
aabb
aaa
bbbb
bba
abb
aa
bb
a
bb
a


31
Positive Closure
Definition:
Note: L+
includes  if and only if L includes 
 





*
2
1
L
L
L
L 


 













,
,
,
,
,
,
,
,
,
,
,
abbbb
abba
aabb
aaa
bbbb
bba
abb
aa
bb
a
bb
a

32
Lexicographical Order
Assume that the symbols in are
themselves ordered.
Definition: A set of strings is in
lexicographical order if
-The strings are grouped first according to
their length.
-Then, within each group, the strings are
ordered “alphabetically” according to the
ordering of the symbols.


Ex: Let the alphabet be
The set of all strings in Lexicographical
order is
, a, b, aa, ab, ba, bb, aaa, …., bbb, aaaa, …,
bbbb, ….
33
Lexicographical Order
{ , }
a b
 

Lec-02 Languages for finite automata FA

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