Prove that the intersection of two context-free languages is not necessarily context-free. This is unlike the case of regular languages which are closed under intersections. *Hint*: Do something similar to what we did on Activity #21 to show something about two of the languages below and then use what we proved in class about the remaining language. Write several sentences to explain your reasoning.\[ L_1 = \{ w | w \text{ has the form } a^n b^n c^m \text{ for some whole numbers } n \text{ and } m \} \]\[ L_2 = \{ w | w \text{ has the form } a^m b^n c^n \text{ for some whole numbers } n \text{ and } m \} \]\[ L_3 = \{ w | w \text{ has the form } a^n b^n c^n \text{ for some whole number } n \} \]

Answered: Prove that the intersection of two…

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

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Prove that the intersection of two context-free languages is not necessarily context-free. This is unlike the case of regular languages which are closed under intersections. *Hint*: Do something similar to what we did on Activity #21 to show something about two of the languages below and then use what we proved in class about the remaining language. Write several sentences to explain your reasoning.

\[ L_1 = \{ w | w \text{ has the form } a^n b^n c^m \text{ for some whole numbers } n \text{ and } m \} \]

\[ L_2 = \{ w | w \text{ has the form } a^m b^n c^n \text{ for some whole numbers } n \text{ and } m \} \]

\[ L_3 = \{ w | w \text{ has the form } a^n b^n c^n \text{ for some whole number } n \} \]

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We delve into the complex behavior that distinguishes context-free languages from regular languages as we investigate the intersection of context-free languages. Unlike regular languages, which display closure under intersection, context-free languages cannot be uniformly affirmed to do so. To demonstrate this departure, we look at two context-free languages, L1 and L2, which are defined by different string patterns. The purpose is to show that the intersection of L1 and L2 may not comply to the context-free language class, highlighting the complexities and limitations of these language structures.

Let L1 = {ww has the form aⁿbⁿc^m for some whole numbers n and m} and L2 = {w/w has the form a^mbⁿcⁿ for some whole numbers n and m}.

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