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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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MINIMIZATION OF D.F.A.

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  -> In  automata theory   (a branch of  theoretical computer science ),  DFA minimization  is the task of transforming a given  deterministic finite automaton   (DFA) into an equivalent DFA that has a minimum number of states. Here, two DFAs are called equivalent if they recognize the same  regular language . Several different algorithms accomplishing this task are known and described in standard textbooks on automata theory. -> Minimal D.F.A. -  For each regular language, there also exists a   minimal automaton   that accepts it, that is, a DFA with a minimum number of states and this DFA is unique (except that states can be given different names).   The minimal DFA ensures minimal computational cost for tasks such as   pattern matching . There are two classes of states that can be removed or merged from the original DFA without affecting the language it accepts. ->Unreachable states   are the stat...
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N.D.F.A TO D.F.A. CONVERSION

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  Difference between D.F.A. and N.D.F. Each transition leads to exactly one state called as deterministic A transition leads to a subset of states i.e. some transitions can be non-deterministic. Accepts input if the last state is in Final Accepts input if one of the last states is in Final. Backtracking is allowed in DFA. Backtracking is not always possible. Requires more space. Requires less space. Empty string transitions are not seen in DFA. Permits empty string transition. For a given state, on a given input we reach a deterministic and unique state. For a given state, on a given input we reach more than one state. DFA is a subset of NFA. Need to convert NFA to DFA in the design of a compiler. δ : Q × Σ → Q For example − δ(q0,a)={q1} δ : Q × Σ → 2 Q For example − δ(q0,a)={q1,q2} DFA is more difficult to construct. NFA is easier to construct. DFA is understood as one machine. NFA is understood as multiple small machines computing at the same time. Steps for converting N.D.F.A. t...
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