Formal Language and Automata
Theory
Prof. Dheeraj Shringi, Assistant Professor
Computer Science & Engineering
CHAPTER-1
Introduction of FLAT
What is FLAT ?
• It is the mathematical study of computing machine and its capabilities.
• It is a study of formal language and automata theory.
Alphabet
• The non empty finite set of symbols is called as an alphabet and it is denoted by
∑.
• Example
∑ = {a.b.c………z}
∑ = {0,1}
∑ = {1}
String
• Any finite sequence of symbol from alphabet ∑ , is called as string and is
represented by w.
• Example
∑ = {a,b}
W = a,ab,aa,bb
W= ab,aab,abc
∑ = {0,1}
W = 0,01,00,11,1
W = 102,2013
∑ = {1}
W = 1, 11, 111
W = 1,12, 121
Length of a String
• If w is any string over alphabet ∑ then the number of symbols involved in the
sequence of string is called as length of the string and denoted by |w|.
• Example
∑ = {a,b} W = a,ab,aa,bb ,|w|=1,2,2,2
∑ = {0,1} W = 0,01,00,11,1 |w| = 1,2,2,2
• Empty String
• A string of length zero or string without any symbols is known as empty
string and is denoted by €
w = € , |w| = 0
w. € = w = €.w
Substring
• Let u,w be the two strings over same alphabet ∑ then u is said to be substring of
w if u can be obtained from w.
• Any consecutive sequence of symbols over given string.
• If u is a substring of w then |u| <= |w|
• Every string is substring to itself.
• Empty string “€” is substring for every string.
• Example
W = TOC
Zero length Substring= € One length substring=T,O,C
Two length substring= TO,OC Three length substring=TOC
Types of Substring
• The substrings are classified into two types
1. Trivial Substring or improper Substring
2. Non-trivial or proper substring
• Trivial Substring or improper Substring
If w is any string over alphabet ∑ then the substring w itself and € is called as trivial
substring
• Non-trivial Substring or proper Substring
If w is any string over alphabet ∑ then any substring of w over w other than w itself and €
is called non trivial substring.
Facts about Substring
• If w is any string with distinct symbols and |w|= n
1. No of substrings = ∑n +1 = n(n+1)/2
2. No. of trivial string = 2
3. No. of non trivial substring = ∑n -1
4. No. of non empty subdtring= ∑n
5. No of substrings of distinct length = n
6. No. of strings of length n generated over alphabet ∑ = | ∑ |n
Prefix and Suffix
• Prefix
The sequence of starting or leading symbol is called as prefix.
• Suffix
The sequence of ending or trailing symbol is called as suffix.
• Example
• w=TOC , |w|=3
• Prefix : € , T , TO , TOC
• Suffix : TOC , OC , C , €
Facts about Prefix and Suffix
• If w is any string of length ‘n’ then
1. No. of prefix = No. of suffix = n+1
2. Trivial substrings are common for both prefix and suffix
3. Every prefix and suffix must be a substring but every substring need not be
prefix or suffix.
Power of an alphabet
• If ∑ is any alphabet then ∑k is the set of all the strings of length k.
• Example : ∑ = {0,1}
∑2 = {00,01,10,11}
∑3 = {000,001,010,011,100,101,110,111}
∑k = {all k-length stings}
• +ve closure(∑+)
∑+ = {w | |w|>=1}
• Kleen closure(∑*)
∑* = {w | |w|>=0}
• ∑* = ∑+ ꓴ €
Language
• The collection of strings from the alphabet ∑ is called language
• Example ∑ = {0,1}
• L ={00,01,10,11}
• L = { (01)n | n>=1 }
• L = { 0n 1m |m>=1,n>=1}
• If ∑ is any alphabet then ∑* is called as universal language
Formal Language
• The collection of strings where we can put some restriction in the formation of
string is called as formal language.
• Example ∑ = {0,1}
• L ={00,01,10,11}
• L = { (01)n | n>=1 }
• L = { 0n 1m |m>=1,n>=1}
Chomsky classification of Formal Language
• According to Chomsky the formal languages are classified as
1. Type 0 or Recursive Enumerable Languages
2. Type 1 or Context sensitive languages
3. Type 2 or Context free languages
4. Type 3 or Context Regular languages
Types of Language
• Empty Languages
The Language that does not contain any string even empty string is called as empty language
and is denoted by ф.
• Non Empty Languages
The Language that contain at least one string is called as non empty language .
• Finite Languages
The Language which contains finite number of strings and length of each string is finite is
called as finite language .
L = { 0n 1n |1<=n<=1}
• Infinite Languages
The Language which contains infinite number of strings and length of each string is finite is
called as infinite language .
L = { 0n 1n |n>=1}
Automata
• The mathematical system that can represent the formal language is called as
Automata i.e. The mathematical representation of formal language is called as
an automata.
• Types of Automata
1. Finite Automata(FA)
2. Push Down Automata(PDA)
3. Linear Bound Automata(LBA)
4. Turing Machine(LBA)
Expressive Power of an Automata
• The number of languages accepted by an automata is called as Expressive Power
of an Automata.
1. E(FA)=1
2. E(PDA)=2
3. E(LBA)=3
4. E(TM)=4
Grammar
• The collection of rules which are used to generate the string is called grammar.
• Grammar G is a collection of 4 tuples {V,T,P,S}
G= {V,T,P,S} ,Where
V= set of all Nonterminal symbol/variable
P= setoff all productions
T= set of all terminal symbols
S= starting symbol
Example
1. A XYZ (r1) V={A,X,Y,Z}
X a (r2) T={a,b,c}
Y b (r3) P={r1,r2,r3,r4}
Z c (r4) S=A
Chomsky classification of Grammar
• According to Chomsky the Grammar is classified as
1. Type 0 or Recursive Enumerable Grammar
2. Type 1 or Context sensitive Grammar
3. Type 2 or Context free Grammar
4. Type 3 or Context Regular Grammar
Other classification of Grammar
• Recursive grammar
The grammar g is said to be recursive if atleast one production contain
the same variable at both left hand side and right hand side of
production.
Example S asb|€
• Non Recursive grammar
The grammar g is said to be recursive if noproduction contain the same
variable at both left hand side and right hand side of production.
Example S ab|€
Derivation
• The process of deriving a string is called as derivation.
• The geometrical representation of derivation is called Derivation tree or parse
tree.
• Steps involved in derivation is called sentential form
• Example Derivation Parse Tree
1. A XYZ A XYZ A
X a A aYZ
Y b A abZ X Y Z
Z c A abc
a b c
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