1. Let Σ be an alphabet. Prove or disprove that for every language L over alphabet Σ, if L2 = L, then L+ = L.

Answered: 1. Let Σ be an alphabet. Prove or…

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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Similar questions

Consider alphabet Σ = {0,1}. Prove or disprove that for all languages L₁,L₂, L over Σ, if L₁ CL C L2 and L1,L2 are decidable, then L is decidable.
Consider alphabet Σ = {0,1}. Prove or disprove that for all languages L1, L2, L over Σ, if L₁ CLCL2 and L₁, L₂ are decidable, then L is decidable.
Suppose L is a language over {a,b}, and there is a fixed integer k such that for every x ∈ Σ*, xz ∈ L for some string z with |z| ≤ k. Does it follow that there is an FA accepting L? Why or why not?
Given a language L over the alphabet Σ, define f(L) = {vo :v € L, o Σ, and v=uo for some u € [*} Prove that f(L) is regular if L is regular.
Suppose L is a non empty regular language, over Σ, such that every w E L satisfies |w] =k, for some ke N. Show that there cannot exist a DFA M = (E, Q, 8, q0, F), where |Q| = k.
4. Consider alphabet Σ = {0,1} and language Lo₁ = {we*: w=0"1" for some nonnegative n€Z}. Prove or disprove that for each language Lover Σ, if Ln Lo₁ = 0 and L is regular, then L is finite.
If A is any language, let ?− 1 2 be the set of all first halves of strings in A so that ?− 1 2 = (x | for some y, |x| = |y| and xy ϵ A}. Show that if A is regular, the so is ?− 1 2 .
A language L over an alphabet Σ is co-finite, if Σ∗ \L is empty or finite. LetCOFNFA = {〈N 〉|N is a NFA accepting a co-finite language}.Show that COFNFA is decidable.
Consider alphabet Σ = {0,1} and language Lo₁ = {weΣ* : w=0"1" for some nonnegative n€ Z}. Prove or disprove that for each language L over Σ, if L n Lo₁ = 0 and L is regular, then L is finite.
If A is any language, let A 1/3 – 1/3 be the set of all strings in A with their middle thirds removed so that A 1/3 – 1/3 = {x z| for some y, |x| = |y = |z| and xyz ∈ A}. Show that, if A is regular, then A 1/3 – 1/3 is not necessarily regular.
Let >={a,b,c}. Show that (b) The set of all languages over Σ is uncountable
Let Σ = {a,b,c}. Prove that the following languages over alphabet Σ are not regular.

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