Found 346 Articles for Data Structure Algorithms

The set of all strings (over terminal symbols) which can be derived from the start symbol is the language generated by the grammar G.Example 1Let grammar G be defined by the set of terminals T = {a, b}, the only non-terminal start symbol S and the set of production rules. Hence, the grammar G would be as follows −S → ∧, S → aSbOr in shorthand, it is as mentioned below −S → ∧ | aSbL(G) = {∧, ab, aabb, aaabbb, . . . }DefinitionIf G is called as a grammar with start symbol S and set of terminals T, ... Read More

Type 3 grammars are regular grammars that describe regular / formal languages.These grammars contain production rules consisting of the following −Only one non-terminal at the left hand side, The right hand side has a single terminal and may or may not be followed by non terminals.ExampleA → ε , A → a, A → b, A → aA etc.TypesThere are two types of regular grammars namely −Right linear / Right regular grammarLeft linear / Left regular grammarLet us learn about these two types of grammar in detail.Right linear grammarThis is a regular grammar with the production rules of the formA ... Read More

Let us understand the concept of grammar in the theory of computation (TOC).Introduction to GrammarA language over an alphabet ∑ is a set of strings from ∑.Grammar is a set of rules used to define a language. In short, it is the structure of the strings in the language.To describe a grammar for a language, two collections of alphabets (symbols) are necessary. These are explained below −Terminals are those symbols from which all strings in the language are made – symbols of a ‘given’ alphabet for a generated language. These are usually the lower case letters.Non-terminal are ‘temporary’ symbols (disjoint ... Read More

It is easy to see that for any language L the following simple properties hold −L · {∧} = {∧} · L = LL · ∅ = ∅ · L = ∅Now let’s see the commutativity and associativity of the operation of concatenation.Properties of products – commutativityThe operation of concatenation is not commutative. In other words, the order matters!Given two languages L and M, it’s usually true thatL · M ≠ M · LExampleIf L = {ab, ac} and M = {a, bc, abc}, then the productL · M is the languageL · M = {aba, abbc, ababc, aca, acbc, ... Read More

Basically a push down automata (PDA) is as follows −“Finite state machine+ a stack”PDA has three components, which are as follows −An Input tape.A control unit.A Stack with infinite size.A PDA can be formally described as seven tuples(Q, Σ, S, δ, q0, I, F)Where, Q is finite number of statesΣ is input alphabetS is stack symbolΔ is the transition function: QX(Σ∪{e})XSXQq0 is the initial state (q0 belongs to Q)I is the initial state top symbolF is a set of accepting states (F belongs to Q)ProblemConstruct a non-deterministic PDA (NPDA) for accepting the languageL = {a^m b^{2m} | m>=1}.SolutionThe strings which ... Read More

A push down automata (PDA) can be formally described as seven tuples(Q, Σ, S, δ, q0, I, F)Where, Q is finite number of statesΣ is input alphabetS is stack symbolΔ is the transition function: QX(ΣU{e})XSXQq0 is the initial state (q0 belongs to Q)I is the initial state top symbolF is a set of accepting states (F belongs to Q)ProblemConstruct PDA for anbmc(n+m) n, m≥1SolutionSo, the strings which are generated by the given language are as follows−L={abcc, aabccc, aaabbccccc, ….}That is to add the number of a's and b's, and that will equal the number of c's.So for every a's and ... Read More

There are three common ways of creating a new language from two languages −UnionIntersectionProductLanguages are sets of strings, so they can be combined by the usual set operations of union and intersection.IntersectionIf L1 and L2 are languages over ∑, then L1 ∩ L2 is the language of strings in both L1 and L2 .For example, If L = {aa, bb, ab} and M = {ab, aabb} then, The intersection is as follows −L ∩ M = {ab}AndUnionIf L1 and L2 are languages over the alphabet ∑, then the language L1 ∪ L2 is the language of all strings in at ... Read More

L = {ww’ | wcw’, w={0, 1}*} where w’ is the reverse of w.This is the language of all palindromes, both odd and even over the alphabet {0, 1}.For the construction of all length palindromes, let us use the Non-deterministic push down automata (NPDA).To construct the wcw’ we need to check if the string is of odd length and if reaches the middle element ‘c’ then process it and move to the next state without making any change in stack.ExampleGiven string is 1 1 0 0 1 1 1 1 0 0 1 1Result − ACCEPTEDGiven string is: 1 0 ... Read More

A Turing machine is a computational model, like Finite Automata (FA), Pushdown automata (PDA), which works on unrestricted grammar. The Turing machine is the most powerful computation model when compared with FA and PDA.Formally, a Turing machine M can be defined as follows −M = (Q, X, ∑, δ, q0, B, F)Q represents the finite, non-empty set of states.X represents the set of tape alphabets.∑ represents the non-empty set of input alphabets.δ is the transition function, whose mapping is given as, δ : Q x X → Q x X x {Left_shift, Right_shift}.q0 is the initial state of the machineB ... Read More

DPDA is the short form for the deterministic push down automata (DPDA).ProblemConstruct DPDA for anbncm where m, n>=1SolutionSo, the strings which are generated by the given language are −L={abc, aabbc, aaabbbcc, ….}That is we have to count equal number of a’s, b’s and different number of c’sLet’s count the number of a's which is equal to the number of b's.This can be achieved by pushing a's in STACK and then we will pop a's whenever "b" comes.Then for c nothing will happen.Finally, at the end of the strings if nothing is left in the STACK then we can say that ... Read More