Arden's Theorem:
One of the applications of Arden's Lemma is, in the conversion of FA to RE.
Statement:
if P and Q are two regular expressions over Σ and P does not contain ε then
R= Q+ RP has a unique solution and it is given by
R=QP*
Prove:
R = Q + RP
= Q + (Q+ RP)P // repacing R with Q + RP
= Q + QP + RP2
= Q + QP + (Q+ RP)P2
= Q + QP + QP2 + RP3
.
.
.
.
.
= Q + QP + QP2 + QP3 + QPn +QPn Pn+1
=Q(ε + P+P2 + P3 + Pn +Pn Pn+1 )
= QPn
or
= QP*
1. Union : If L1 and If L2 are two regular languages, their union L1 ∪ L2 will also be regular.
For example, L1 = {an | n ≥ 0} and L2 = {bn | n ≥ 0}
L3 = L1 ∪ L2 = {an ∪ bn | n ≥ 0} is also regular.
2. Intersection : If L1 and If L2 are two regular languages, their intersection L1 ∩ L2 will also be regular.
For example,
L1= {am bn | n ≥ 0 and m ≥ 0} and L2= {am bn ∪ bn am | n ≥ 0 and m ≥ 0}
L3 = L1 ∩ L2 = {am bn | n ≥ 0 and m ≥ 0} is also regular.
3. Concatenation : If L1 and If L2 are two regular languages, their concatenation L1.L2 will also be regular.
For example,
L1 = {an | n ≥ 0} and L2 = {bn | n ≥ 0}
L3 = L1.L2 = {am . bn | m ≥ 0 and n ≥ 0} is also regular.
4. Kleene Closure : If L1 is a regular language, its Kleene closure L1* will also be regular.
For example,
L1 = (a ∪ b)
L1* = (a ∪ b)*
5. Complement : If L(G) is regular language, its complement L’(G) will also be regular. Complement of a language can be found by subtracting strings which are in L(G) from all possible strings.
For example,
L(G) = {an | n > 3}
L’(G) = {an | n <= 3}
Note :
Two regular expressions are equivalent if languages generated by them are same. For example, (a+b*)* and (a+b)* generate same language. Every string which is generated by (a+b*)* is also generated by (a+b)* and vice versa.
Power of a String:
Consider ∑ = {0, 1}. The following are the power of an alphabet over input alphabet ∑.
(L)∑ = {null}: zero length string
(L)∑ = {0, 1}: set of 1 length string
(L)∑ = {01, 10, 11}: set of 2 length string
Theory of Computation is a branch of computer science that deals with designing abstract self-propelled computing devices that follow a predetermined sequence of operations automatically.
Theory of Computation – Basic Terms:
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Symbols: |
Example: 0, 1 are symbols of binary digits
A, B, C, …., Z is symbols of letter alphabet.
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Alphabet: |
Example − ∑ = {a, b, c, d} is an alphabet set where ‘a’, ‘b’, ‘c’, and‘d’ are symbols.
∑ = {0, 1} is an alphabet of binary digits set where ‘0’, ‘1’ are symbols.
Rule:
If we have used any alphabet then it should not be prefix in the next combination or alphabet.
Σ = {a, b} ✓ Valid
Σ = {a, ba, c} ✓ Valid
Σ = {a, ab, c} ☓ Invalid
Σ = {0, 10} ✓ Valid
Σ = {1, 10, 11} ☓ Invalid
Power of an alphabet:
Consider ∑ = {a, b}. The following are the power of an alphabet over input alphabet ∑.
∑ = {null}: zero length string
∑ = {a, b}: set of 1 length string
∑ = {aa, ab}: set of 2 length string
Algebraic Structure
A non empty set S is called an algebraic structure with respect to binary operation if (a*b) belongs to S for all (a*b) belongs to S, i.e (*) is closure operation on ‘S’.
Ex : S = {1,-1} is algebraic structure under *
As 1*1 = 1, 1*-1 = -1, -1*-1 = 1 all results belongs to S.
But above is not algebraic structure under + as 1+ (-1) = 0 not belongs to S.
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String |
String is a finite sequence of symbols taken from ∑.
Ex : ‘ cabcad ’ is a valid string on the alphabet set ∑ = {a, b, c, d}
Set of a String :
∑* contains an empty string є. The set of non- empty string is denoted by ∑+.
