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ARDEN'S THEOREM FOR REGULAR EXPRESSIONS

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    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 + ...
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SIMPLIFICATION OF CONTEXT-FREE GRAMMAR

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  Simplification of Context Free Grammar Context Free Grammar  has recursive structure. The languages that are accepted with Context Free Grammar are called  Context Free Languages . Context Free Grammar has one condition for production rules, i.e.,  on the left-hand side of each rule, there must be only single variable, and on the right-hand side, there may be combination of variables and terminals including  ?. In Context Free Grammar, sometimes all the productions rules and symbols are not needed for the derivation to solve. Some productions rules are never used during derivation of any string. Elimination of these types of productions or symbols is called  Simplification of Context Free Grammar. We will use the various simple methods to simplify the given context free grammar without changing the resulting language. The following types of productions rules are never used during derivation of any string from context free grammar: Useless productions Null...
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CONTEXT FREE GRAMMAR

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About C.F.G. :-   -> In   formal language   theory, a   context-free grammar   ( CFG ) is a   formal grammar   whose   production rules   are of the form {\displaystyle A\ \to \ \alpha } with  {\displaystyle A}  a  single   nonterminal  symbol, and  {\displaystyle \alpha }  a string of  terminals  and/or nonterminals ( {\displaystyle \alpha }  can be empty). A formal grammar is "context free" if its production rules can be applied regardless of the context of a nonterminal. No matter which symbols surround it, the single nonterminal on the left hand side can always be replaced by the right hand side. This is what distinguishes it from a  context-sensitive grammar . A formal grammar is essentially a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the first rule in the picture, {\displaystyle...
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