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Show that the following languages are not context-free: A. The language of palindromes over {0, 1} containing equal numbers of 0's and 1's B. The language of strings over {1, 2, 3, 4} with equal numbers of 1's and 2's, and also equal numbers of 3's and 4's.

Show that the following languages are not context-free: A. The language of palindromes over {0, 1} containing equal numbers of 0's and 1's B. The language of strings over {1, 2, 3, 4} with equal numbers of 1's and 2's, and also equal numbers of 3's and 4's.

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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**Proving Non-Context-Free Languages** In this section, we will explore two languages and provide proof that they are not context-free. **A. The Language of Palindromes Over {0, 1} Containing Equal Numbers of 0's and 1's** This language consists of palindromes where the number of 0's is equal to the number of 1's. A palindrome is a string that reads the same forwards and backwards. Examples include strings like "010" and "0110". To prove that this language is not context-free, one commonly applies the Pumping Lemma for context-free languages. However, since this is an educational website, you might explore this proof in a more detailed lecture or section on context-free languages and the Pumping Lemma. **B. The Language of Strings Over {1, 2, 3, 4} With Equal Numbers of 1's and 2's, and Also Equal Numbers of 3's and 4's** This language consists of strings formed using the symbols 1, 2, 3, and 4, where the count of 1's and 2's are equal, and the count of 3's and 4's are also equal. Examples of such strings include "1234", "1122", and "3344". Similar to part A, to prove that this language is not context-free, you would typically use the Pumping Lemma for context-free languages, demonstrating that any assumed context-free grammar for the language would lead to contradictions upon pumping. ### Graphs and Diagrams No graphs or diagrams are included in this text. However, if you prefer visual demonstrations, diagrams depicting the structure of palindromes or the counting of symbols in strings could be beneficial. In case of elaborating on these proofs or concepts, visual aids like parse trees or pumping lemma illustrations could be included to enhance understanding.
Transcribed Image Text:**Proving Non-Context-Free Languages** In this section, we will explore two languages and provide proof that they are not context-free. **A. The Language of Palindromes Over {0, 1} Containing Equal Numbers of 0's and 1's** This language consists of palindromes where the number of 0's is equal to the number of 1's. A palindrome is a string that reads the same forwards and backwards. Examples include strings like "010" and "0110". To prove that this language is not context-free, one commonly applies the Pumping Lemma for context-free languages. However, since this is an educational website, you might explore this proof in a more detailed lecture or section on context-free languages and the Pumping Lemma. **B. The Language of Strings Over {1, 2, 3, 4} With Equal Numbers of 1's and 2's, and Also Equal Numbers of 3's and 4's** This language consists of strings formed using the symbols 1, 2, 3, and 4, where the count of 1's and 2's are equal, and the count of 3's and 4's are also equal. Examples of such strings include "1234", "1122", and "3344". Similar to part A, to prove that this language is not context-free, you would typically use the Pumping Lemma for context-free languages, demonstrating that any assumed context-free grammar for the language would lead to contradictions upon pumping. ### Graphs and Diagrams No graphs or diagrams are included in this text. However, if you prefer visual demonstrations, diagrams depicting the structure of palindromes or the counting of symbols in strings could be beneficial. In case of elaborating on these proofs or concepts, visual aids like parse trees or pumping lemma illustrations could be included to enhance understanding.
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