Answered: Suppose L1, L2 are two regular…

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

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Suppose L1, L2 are two regular languages over the alphabet Σ = {a, b}. Consider the following two new languages:

shuffle1(L₁, L₂) = {a₁b₁ . . . a_nb_n : a₁ . . . a_n ∈ L₁, b₁ . . . b_n ∈ L₂, a_i, b_i ∈ Σ, a_i = b_i, i ∈ [1 . . . n]}

shuffle2(L₁, L₂) = {a₁b₁ . . . a_nb_n : a₁ . . . a_n ∈ L₁, b₁ . . . b_n ∈ L₂, a_i, b_i ∈ Σ, a_n−(i−1) = b_i, i ∈ [1 . . . n]}

Notice how shuffle1, shuffle2 are similar, but not quite the same. One of them is necessarily regular and the other is not necessarily regular.
Which is which? Prove your answer. If part of your proof relies on a construction, you do not need to prove its correctness.

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