**Problem Statement:**Show that the language \( L = \{ ww^R w : w \in \{a, b\}^* \} \) is not a context-free language.**Explanation:**The language \( L \) consists of strings that are in the form of \( ww^R w \), where \( w \) is any string composed of the characters 'a' and 'b'. The notation \( w^R \) signifies the reverse of the string \( w \).To prove that this language is not context-free, you would typically use the Pumping Lemma for context-free languages. The essence of the argument involves demonstrating that for any sufficiently long string in the language, you cannot divide the string into parts that satisfy the conditions of the Pumping Lemma without breaking the properties needed to remain in the language.**Detailed Steps for a Proof Idea (Pumping Lemma):**1. **Assume \( L \) is context-free:** Suppose, for contradiction, that \( L \) is context-free. By the Pumping Lemma, there exists some pumping length \( p \) such that any string \( s \) in \( L \) of length at least \( p \) can be split into five parts, \( s = uvxyz \), satisfying certain conditions.2. **Choose a specific string:** Consider the string \( s = a^p b^p a^p b^p \) which is clearly in \( L \) since it can be represented in the form \( ww^R w \).3. **Apply the Pumping Lemma:** - According to the lemma, we can write \( s = uvxyz \) with: - \( |vxy| \leq p \) - \( |vy| \geq 1 \) - And for all \( i \geq 0 \), the string \( uv^ixy^iz \) should also be in \( L \). 4. **Reach a contradiction:** By considering different placements for \( vxy \) and \( u \) and \( z \), show that no matter how you pump (i.e., increase \( i \)), you cannot maintain the structure \( ww^R w \), leading to a contradiction.Thus, the language \( L \) cannot be context-free.

proof by pumping lemma: 1)assume given language is context-free and there exists a PDA for this.…

Engineering

Computer Science

Show that L = {ww*w:w E {a, b}*} is not a context-free language.

Show that L = {www:w E {a, b}} is not a context-free language.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

See similar textbooks

Related questions

Q: Write a program that asks the user to enter integers as the width and length of a rectangle that…

A: I give the code in C++ along with output and code screenshot

Q: Make a C++ program that incorporates recursion to solve the following problem. The program has to…

A: In this question we have to write a C++ code to print the pattern from the character of letters to…

Q: OCaml Code: Please fix all the errors needed for interpreter.ml and use the test cases to test the…

A: The sv2str function is defined after the str2com function in the provided code, which causes the…

Q: A) Consider the programming language Lisp. Write a Lisp function that calculates the following…

A: Writing function: The macro named defun is used for defining functions. The defun macro requires…

Q: with Ada.Numerics.Generic_Elementary_Functions; package body Number_Theory is --…

A: ANSWER:

Q: Please Help In JAVA Here is the skeleton of a code for insertion sorting in a functional…

A: The given below code for insertion sorting in a java language.

Q: Can you give me an example on how in OpenGL for C++, I can implement a rolling sphere on a plane.…

A: It seems like you're providing information on how to parametrize points on a sphere using sector…

Q: I need help writing a lisp function using memory allocation methods the first function should…

A: According to the information given:- We aehv to follow the instruction in order to writing a lisp…

Q: Describe the language L over the alphabet. i.e. {y,t,z,tt,tz...}: 1. ʎ U (t + z) U (t + z)2 2.…

A: Given the alphabet ∑={a, t, z}

Q: Write these codes in c language please. Thank you in advance. 1. Define a function…

A: According to the Bartleby guideline, we are supposed to answer only one question at a time. Kindly…

Q: write a Common Lisp function that creates a list containing exactly c instances of m.

A: In Common Lisp, functions are defined using the `defun` keyword, followed by the function name and…

Q: Java Program: Create a semantic analysis that is recursive with blocks, MathOps, and throws well…

A: The semantic analyzer will check for type errors (e.g., using a boolean where an int is expected)…

Q: Java Programming: Make sure to have parsing statements for parseFor(), parseWhile(), parseIf(),…

A: Coded using Java.

Q: Make a C++ program that incorporates recursion to solve the following problem. The program has to…

A: Algorithm to Generate Concentric Squares Pattern1. Start2. Define a class ConcentricSquares: -…

Q: Write a Common Lisp function that does the following: returns (circle i) if i is a number returns…

A: The task at hand is to design a Common Lisp function that can distinguish between different types of…

Q: Create an Account class in C++ that implements this UML class diagram. Account - id: int - balance:…

A: #include <iostream>using namespace std; class Account{ private: int id;…

Q: If L is a regular language, then any subset of L is a regular language. If L is a regular language,…

A: A regular language is a language that can be expressed with a regular expression and generated by…

Q: have to create a small library in the C programming language to do Minifloat The minifloat should…

A: SUMMARY: - hence we discussed all the points.

Q: Remember the "zip" function of Haskell? It can be defined in Haskell as: zip : [a] -> [b] -> [(a,b)]…

A: Code screenshot :-

Q: Write a program in C that initializes one each of the basic C data types and then print them out.…

A: Initialization of Variables:Declare and initialize variables of various C data types, including…

Concept explainers

Bare bones programming language

Bare bones programming language is a low-level programming language that gives a detailed stepwise description of a program. It has a tight program design and in-depth understanding of the toolchain. It is based on the target environment and is used for safe coding practices. C programming language is a type of bare bones programming language.

Question

**Problem Statement:**

Show that the language \( L = \{ ww^R w : w \in \{a, b\}^* \} \) is not a context-free language.

**Explanation:**

The language \( L \) consists of strings that are in the form of \( ww^R w \), where \( w \) is any string composed of the characters 'a' and 'b'. The notation \( w^R \) signifies the reverse of the string \( w \).

To prove that this language is not context-free, you would typically use the Pumping Lemma for context-free languages. The essence of the argument involves demonstrating that for any sufficiently long string in the language, you cannot divide the string into parts that satisfy the conditions of the Pumping Lemma without breaking the properties needed to remain in the language.

**Detailed Steps for a Proof Idea (Pumping Lemma):**

1. **Assume \( L \) is context-free:**
Suppose, for contradiction, that \( L \) is context-free. By the Pumping Lemma, there exists some pumping length \( p \) such that any string \( s \) in \( L \) of length at least \( p \) can be split into five parts, \( s = uvxyz \), satisfying certain conditions.

2. **Choose a specific string:**
Consider the string \( s = a^p b^p a^p b^p \) which is clearly in \( L \) since it can be represented in the form \( ww^R w \).

3. **Apply the Pumping Lemma:**
- According to the lemma, we can write \( s = uvxyz \) with:
- \( |vxy| \leq p \)
- \( |vy| \geq 1 \)
- And for all \( i \geq 0 \), the string \( uv^ixy^iz \) should also be in \( L \).

4. **Reach a contradiction:**
By considering different placements for \( vxy \) and \( u \) and \( z \), show that no matter how you pump (i.e., increase \( i \)), you cannot maintain the structure \( ww^R w \), leading to a contradiction.

Thus, the language \( L \) cannot be context-free.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.

Similar questions

-precision number? 4. What is the gap between 201 and the next larger double-precision number?
I need help please with this question regarding Sorting Analysis in C++, The question is: Why is Radix sort unrelated to the F(n)=O(nlog⁡n) family of sorting algorithms?
code in ada with Ada.Numerics.Generic_Elementary_Functions; package body Number_Theory is -- Instantiate the library for floating point math using Floating_Type. package Floating_Functions is new Ada.Numerics.Generic_Elementary_Functions(Floating_Type); use Floating_Functions; function Factorial(N : in Factorial_Argument_Type) return Positive is begin -- TODO: Finish me! -- -- 0! is 1 -- N! is N * (N-1) * (N-2) * ... * 1 return 1; end Factorial; function Is_Prime(N : in Prime_Argument_Type) return Boolean is Upper_Bound : Prime_Argument_Type; Current_Divisor : Prime_Argument_Type; begin -- Handle 2 as a special case. if N = 2 then return True; end if; Upper_Bound := N - 1; Current_Divisor := 2; while Current_Divisor < Upper_Bound loop if N rem Current_Divisor = 0 then return False; end if; Upper_Bound := N / Current_Divisor; end loop;…
What are the dangers of C’s approach to encapsulation.
the code is in ada with Ada.Numerics.Generic_Elementary_Functions; package body Number_Theory is -- Instantiate the library for floating point math using Floating_Type. package Floating_Functions is new Ada.Numerics.Generic_Elementary_Functions(Floating_Type); use Floating_Functions; function Factorial(N : in Factorial_Argument_Type) return Positive is begin -- TODO: Finish me! -- -- 0! is 1 -- N! is N * (N-1) * (N-2) * ... * 1 return 1; end Factorial; function Is_Prime(N : in Prime_Argument_Type) return Boolean is Upper_Bound : Prime_Argument_Type; Current_Divisor : Prime_Argument_Type; begin -- Handle 2 as a special case. if N = 2 then return True; end if; Upper_Bound := N - 1; Current_Divisor := 2; while Current_Divisor < Upper_Bound loop if N rem Current_Divisor = 0 then return False; end if; Upper_Bound := N / Current_Divisor; end loop;…
Consider the following recursive definition to define a language L over the alphabet S={0,1}: Base Step: 1∈L Recursion Step: For each w in L, 0w0 is in L. Each string in L can be derived from the base step together with a finite number of applications of the recursion step. a) List two string in L. b) List two strings not in L. c) Write set notation to describe L.
binary search c++ recursive
C++ Attached is the question
for Haskell: explain what the following function someFunc does: someFunc m n = take m ((map (\f -> f n) . map (*)) [1..])
Write a program in c++ to find the magnitude of a vector whose end points in 3D are given by the user. The end points are to be stored in a class. The magnitude should be returned by a friend function of the class
JUnit: P2J1Test.javapublic static long fallingPower(int n, int k)Python has the integer exponentiation operator ** conveniently built in the language, whereas Javaunfortunately does not offer that operator that would be mostly useless anyway in a language withfixed size integers that silently hide the overflows easily produced by integer exponentiation. (Inboth languages, the caret character ^ denotes the bitwise exclusive or operation that has bupkis todo with integer exponentiation.)In the related operation of falling power that is useful in many combinatorial formulas and denotedsyntactically by underlining the exponent, each term that gets multiplied into the product is alwaysone less than the previous term. For example, the falling power 83 is computed as 8 * 7 * 6 = 336.Similarly, the falling power 105 equals 10 * 9 * 8 * 7 * 6 = 30240. Nothing essential changes even ifthe base n is negative; the falling power (-4)5 is computed as -4 * -5 * -6 * -7 * -8 = -6720.This method should…
Please help with C++Priority Queue question in C++ language. Sample output also in image. Thanks.