Posts

Showing posts with the label kleene theorem

ARDEN'S THEOREM FOR REGULAR EXPRESSIONS

-
Image
    Identities for Regular Expression:- Given R, P, L, Q as regular expressions, the following identities hold − ∅* = ε ε* = ε RR* = R*R R*R* = R* (R*)* = R* RR* = R*R (PQ)*P =P(QP)* (a+b)* = (a*b*)* = (a*+b*)* = (a+b*)* = a*(ba*)* R + ∅ = ∅ + R = R   (The identity for union) R ε = ε R = R   (The identity for concatenation) ∅ L = L ∅ = ∅   (The annihilator for concatenation) R + R = R   (Idempotent law) L (M + N) = LM + LN   (Left distributive law) (M + N) L = ML + NL   (Right distributive law) ε + RR* = ε + R*R = R* Arden's Theorem:- In order to find out a regular expression of a Finite Automaton, we use Arden’s Theorem along with the properties of regular expressions. Statement  − Let  P  and  Q  be two regular expressions. If  P  does not contain null string, then  R = Q + RP  has a unique solution that is  R = QP* Proof  − R = Q + (Q + RP)P  [After putting the value R = Q + RP] = Q + ...
Post a Comment
Read more

PIGEONHOLE PRINCIPLE AND APPLICATIONS OF PUMPING LEMMA

-
Image
  Pigeonhole principle for regular language:- a hole or small recess for pigeons to nest a small open compartment (as in a desk or cabinet) for keeping letters or documents a neat category which usu. fails to reflect actual complexities. pigeonhole principle:   if  n  objects are put into  m  containers, where  n  >  m , then at least one container must hold more than one object. The pigeonhole can be used to prove that certain infinite languages are not regular. (Remember, any finite language  is  regular.) As we have informally observed, dfas "can't count." This can be shown formally by using the pigeonhole principle. As an example , we show that L = {a b :  n  > 0} is not regular. The proof is by contradiction. Suppose L is regular. There are an infinite number of values of  n  but M(L) has only a finite number of states. By the pigeonhole principle, there must be distinct values of  i  and...
1 comment
Read more