A grammar is said to be regular, if the production is in the form - A → αB, A -> a, A → ε, for A, B ∈ N, a ∈ Σ, and ε the empty string A regular grammar is a 4 tuple - G = (V, Σ, P, S) V - It is non-empty, finite set of non-terminal symbols, Σ - finite set of terminal symbols, (Σ ∈ V), P - a finite set of productions or rules, S - start symbol, S ∈ (V - Σ)

1. Regular Grammar ( Theory of Computation)

2. A grammar is a 4-tuple G = (V,T,P,S)
 V: set of variables or nonterminals
 T: set of terminal symbols (terminals)
 P: set of productions
 S: a start symbol from V

3.  V = { S }, T = { 0, 1 }
 Productions:
S  
S  0S1
This grammar
represents strings
such as:
0011
000111
01


4.  Leftmost derivation: the leftmost variable is
always the one replaced when applying a
production
 Rightmost derivation: rightmost variable is
replaced

5.  A tree in graph theory is a set of nodes such
that
› There is a special node called the root
› Nodes can have zero or more child nodes
› Nodes without children are called leaves
› Interior nodes: nodes that are not leaves
 A parse tree for a grammar G is a tree such that
the interior nodes are non-terminals in G and
children of a non-terminal correspond to the
body of a production in G

6.  Yield: concatenation of leaves from left to
right
 If the root of the tree is the start symbol, and
all leaves are terminal symbols, then the
yield is a string in L(G)
 Note: a derivation always corresponds to
some parse tree

7.  Regular Language – Regular Language are
those languages which are accepted by finite
automaton.
 Regular Expression – Regular Expression
are used to denote Regular Languages.
 Regular Set – Any set that represent the
value of Regular Expression.

8.  In order to find out a regular expression of a
finite automaton we use Arden's theorem.
 Statement 1 - Let P & Q be two regular
expression.
 Statement 2 – If P doesn’t contain null string,
then R = Q + R P ( has a unique solution)
 i.e. , R= Q P*

9.  R = Q + R P
R = Q + ( Q + R P ) P
R = Q + Q P + R P 2
R = Q + Q P + ( Q + R P ) P 2
R = Q + Q P + Q P 2 + R P 3
R = Q ( Ф + P + P 2 + _ _ _ ) + R P n+ 1
R = Q ( P * ) + R P n + 1
R = Q P * + R P n + 1

10.  Union of two regular language are always be regular.
 Intersection of two regular language are always be
regular.
 Complement of two regular language are always be
regular.
 Difference of two regular language are always be
regular.
 Kleen closure operation over a regular language
always be a regular.

11.  Concatenation of two regular language are
always be regular.
 Reverse of two regular language are always
be regular.

12.  Pumping Lemma is used as a proof for
irregularity of a language. Thus, if a
language is regular, it always satisfies
pumping lemma. If there exists at least one
string made from pumping which is not in L,
then L is surely not regular.

13.  The opposite of this may not always be true.
That is, if Pumping Lemma holds, it does not
mean that the language is regular.

14.  It always consist of three variables i.e., x, y,
I, z.
 Where, X can be null or can hold value
 y can not be null and has a value in it.
 I is the power upon y which increment by
step .
 Z can be null or can hold value .

15.  x y z
Ф aa Ф
wxyiz i>0
Ф(aa)iФ = (aa)I
aaФL i=1
aaaФL i=2