Derivation is a sequence of production rules. It is used to get input strings. During parsing, we have to take two decisions, which are as followsWe have to decide the non-terminal which is to be replaced.We have to decide the production rule by which the non-terminal will be replaced.Two options to decide which non-terminal has to be replaced with the production rule are as follows −Left most derivationRight most derivation.Let us understand these two options in detail.Left Most DerivationIn the leftmost derivation, the input is scanned and then replaced with the production rule from left side to right. So, we ... Read More

A context free grammar (CFG) is a forma grammar which is used to generate all the possible patterns of strings in a given formal language.It is defined as four tuples −G=(V, T, P, S)G is a grammar, which consists of a set of production rules. It is used to generate the strings of a language.T is the final set of terminal symbols. It is denoted by lower case letters.V is the final set of non-terminal symbols. It is denoted by capital lettersP is a set of production rules, which is used for replacing non-terminal symbols (on the left side of ... Read More

Some basic difference between a Non-deterministic finite automata (NFA) and Deterministic Finite Automata (DFA) is that in DFA each state has an output for each of the inputs, whereas this is not necessary for NFA.Also, in NFA the input can be \epsilon but in DFA this is not the case.Moreover, in NFA one state can go to a different state on the same output but this is not the case with DFA.Usually, we convert an NFA into a DFA by making the state transition diagram for the given NFA and accordingly vary that state diagram for the DFA. From this ... Read More

There are two Pumping Lemmas (PL), which are defined for Regular Languages and Context - Free Languages.Pumping lemma for Regular languagesIt gives a method for pumping (generating) many substrings from a given string.In other words, we say it provides means to break a given long input string into several substrings.Lt gives necessary condition(s) to prove a set of strings is not regular.TheoremFor any regular language L, there exists an integer P, such that for all w in L|w|>=PWe can break w into three strings, w=xyz such that.(1)lxyl < P(2)lyl > 1(3)for all k>= 0: the string xykz is also in ... Read More

Closure property is a technique to understand the class of the resulting language when we are performing an operation on two languages of the same class.That means, suppose L1 and L2 belong to regular language and if regular language is closed under operation ∪, then L1∪L2 will be a Regular language. But if RL is not closed under ∩, that doesn't mean L1∩L2 won't be a RL.For a class to be closed under an operation, it has to hold true for all languages in that class. So, if a class is not closed under an operation, we cannot say anything ... Read More

ProblemGenerate RE for a given finite automata by using the state elimination method.SolutionThere are two methods for converting a Deterministic Finite Automata (DFA) to Regular expression (RE).Arden's Theorem Method.State Elimination Method.The given automata is as follows−Step 1Initial state is A that has an incoming edge.So, we have to create a new initial state qi as follows−Step 2Initial state is B that has an incoming edge.So, we have to create a new final state qf as given below−Step 3Now, try to eliminate all intermediate states one by one.First eliminate state AThere is a path from state qi to B via A.So, ... Read More

ProblemGiven a Finite Automata, we need to convert the Finite Automata into equivalent Regular Expression using the State Elimination Method.Step 1 − Initial state q1 has incoming edge. So, create a new initial state qi.Step 2 − Final state q2 has outgoing edge. So, create a new final state.Step 3 − Start eliminating intermediate states one after another.If there is path going to qi to q2 via q1So after eliminating q1 we can connect a direct path from qi to q2 having costε .c*.a=c*aThere is a loop on q2 using state qiSo after eliminating state q1 we can draw a ... Read More

There are two methods for converting a Deterministic Finite Automata (DFA) to Regular expression (RE). These methods are as follows −Arden's Theorem Method.State Elimination Method.Now, let us learn about the state elimination method used in TOC.State Elimination MethodStep 1Initial state of DFA does not have any incoming edge.If there exists any incoming edge to the initial state, then we need to create a new initial state which has no incoming edge to it.An example about the relation between incoming edge and initial state is given below−Step 2There must be only one final state in DFA.If there exist multiple final states ... Read More

ProblemConstruct a regular expression for the given finite automata.SolutionStep 1 − Let us write down the equation.q1=q1a+ εq1 is the start state and ε will be added as the input a which is coming to q1 from q1.Hence, we writeState=source state of input * input coming to itStep 2 − Similarly, q2=q1b+q2bq3=q2aStep 3 − Let us simplify q1 first.q1=q1a+ εwe can re-write it as follows−q1= ε +q1aWhich is similar to R=Q+RP which has a solution as follows−R=QP*Step 4 − Assuming R=q1, Q= ε, P=aWe get q1 = ε.a* since ε.R=Rq1=a*Step 5 − Substituting values of q1 in q2 we get, ... Read More

There are two methods for converting a Deterministic Finite Automata (DFA) to Regular expression (RE). These methods are as follows −Arden's Theorem Method.State Elimination Method.Let us understand the Arden's Theorem method in detail.Arden's TheoremLet P and Q be the two regular expressions.If P does not contain null string, then the following equation in R, viz R = Q + RP, Which has a unique solution by R = QP*Here, The finite automata (FA) does not have epsilon moves.It must have only initial state q1.Its states are q1, q2, q3, ....qn. The final state may be some qi where i QP* ... Read More