Data Structure Algorithms Articles - Page 3 of 25

A context-free grammar is a quadruple G = (N, T, P, S), Where, N is a finite set of nonterminal symbols, T is a finite set of terminal symbols, N ∩ T = ∅, P is a finite set of productions of the form A → α, Where A ∈ N, α ∈ (N ∪ T)*, S is the start symbol, S ∈ N.Construct a Context free grammar for the language, L = {anbm| m ≠n}Case 1n > m − We generate a string with an equal number of a’s and b’s and add extra a’s on the left −S ... Read More

A Turing machine (TM) can be formally described as seven tuples −(Q, X, ∑, δ, q0, B, F)Where, Q is a finite set of states.X is the tape alphabet.∑ is the input alphabet.δ is a transition function:δ𝛿:QxX->QxXx{left shift, right shift}.q0 is the initial state.B is the blank symbol.F is the final state.A Turing machine T recognises a string x (over ∑) if and only when T starts in the initial position and x is written on the tape, T halts in a final state.T is said to recognize a language A, if x is recognised by T and if and ... Read More

A Deterministic Finite automata (DFA) is a five tuplesM=(Q, ∑, δ, q0, F)Where, Q − Finite set called states.∑ − Finite set called alphabets.δ − Q × ∑ → Q is the transition function.q0 ∈ Q is the start or initial state.F − Final or accept state.Let’s see the worst case number of states in DFA for the language A∩B and A*Let A and B be the two states, |A| = number of states = nA|B| = number of states = nBDFA = |A∩B|   =nA.nB|A ∪ B| =nA.nB|A*|=3/4 2nA|AB| = nA (2nB-2nB-1)NFAThe non-deterministic finite automata (NFA) also have five states ... Read More

Chomsky Hierarchy represents the class of languages that are accepted by the different machines.Chomsky hierarchyHierarchy of grammars according to Chomsky is explained below as per the grammar types −Type 0. Unrestricted grammars   Turing Machine (TM)Type 1. Context-sensitive grammars   Linear Bounded Automaton (LBA)Type 2. Context-free grammars   Pushdown Automaton (PDA)Type 3. Regular grammars   Finite Automaton (FA)Type-1 Context Sensitive Grammar (CSG)Type 1 grammar is also known as context sensitive grammarThe context sensitive grammar is used to represent context sensitive languageThe CSG follows some rules, which are as follows −The context sensitive grammar may have more than one symbol on the left hand side ... Read More

Pushdown Automata (PDA) are the finite automata (FAs), but with the ability to push and pop symbols to/from a stack.PDA accepts strings if there is a legal path from start state to acceptance state for input. Otherwise, the string is rejected.A PDA can be represented by a 7-tuple(Q, ∑, ℾ, q0, ha, ∆, δ)WhereThe PDA is to finite subsets of Q ☓ (ℾ ∪ {∆})*.Parentheses are balanced ifWhile reading string, number of opening parentheses >= number of closing parentheses.When string is read, number of opening parentheses = number of closing parentheses.Examples(())() − Balanced((()() − Not balanced)()(() − Not balancedThe context ... Read More

ProblemGenerate the push down automata (PDA) that recognizes the language E={aibj| i is not equal to j and I is not equal to 2j}.SolutionConsider the two languages as given below −L1={aibj|i,j>=0 and i>2j}L2={aibj|i,j>=0 and iaA   A->aaAb|aA|epsilonIn L2, the number of a's are less than double the number of b'sSo the CFG for L2 becomes as follows −   S2->Bb|aBb   B->Bb|aBb|aaBb|epsilon   S->S1|S2L1: {aibj:i>2j}L2:{aibj: i

All grammars are not always optimized, which means the grammar may consist of some extra symbols (non-terminals) which increase the length of grammar.So, we have to reduce the grammar by removing the useless symbols.PropertiesThe properties to reduce grammar are explained below −Each non-terminal and terminal of G appears in the derivation of some word in L.There should not be any production as X->Y where X and Y are non-terminals.If epsilon is not in language L then there need not be in the production X-> ε.The diagram given below depicts the use of reduce grammar−The productions of type S-> ε are ... Read More

The instantaneous description (ID) of a push down automata (PDA) is represented by a triple (q, w, s)Where, q is the state.w is unconsumed input.s is the stack contents.ID is an informal notation of how a PDA compares an input string and makes a decision that string is accepted or rejected.Turnstile NotationIt is used for connecting pairs of ID's that represent one or more moves of a PDA.The process of transition is denoted by the turnstile symbol "⊢"⊢ it represents one move.⊢ sign describes a sequence of moves.Example(P, b, T) ⊢ (q, w, a)While taking a transition from P to ... Read More

Context Free Grammars (CFG) are definitely recognized by Non-deterministic push down automata (NPDA), but Programming languages are translated to binary (Machine Code) via Deterministic PDA.This is because it has the following below mentioned impacts −If Programming languages were supposed to be translated via NPDA then for one given program instance we will have multiple versions of binary(Machine code) generated for the same program, which ideally shouldn't be the scenario.For a given program only 1 version of binary code should be generated and that should remain consistent across all OS Platforms.Outputs will vary significantly: If we have multiple object files, then ... Read More

ProblemDefine the language, E={aibj|i not equal to j and i not equal to 2j} and design an unambiguous context free grammar (CFG) in Chomsky normal form (CNF) that generates E.SolutionThe unambiguous CFG for the given language is as follows −S->AC|CBA->aA|aB->Bb|bC->aCb|aaCb|epsilonNow, convert this CFG into CNF. You can follow the below mentioned steps for the successful conversion.Step 1First add a new start symbol S0   S0->S   S->AC|CB   A->aA|a   B->Bb|b   C->aCb|aaCb|epsilonStep 2Next eliminate the epsilon symbol in the production other than the start symbol.   C->epsilon is a null productionAfter eliminating null production, the new productions are as follows −   S0->S   S->AC|CB   A->aA|a   B->Bb|b  ... Read More