2. Fix an alphabet Σ. For any string w with w|≥2, let skip(w) be the stringobtained by removing the first two symbols of w. Define two operators on languages:1(L) = {w € Σ*: skip (w) € L}, andf2(L) {skip(w) *: WELa) Consider L' = L(bba*) over the alphabet Σ= {a, b}. Write a regularexpression representing f₁(L'). Write another regular expression representingf₂(L').b) Claim: for every regular language L the language f1(L) is regular.Clearly state whether the claim is TRUE or FALSE, and then prove your answer.c) Claim: for every regular language L the language f2(L) is regular.Clearly state whether the claim is TRUE or FALSE, and then prove your answer.

Let's break down the problem to answer each part systematically. Problem Recap:We are given two…

Answered: 2. Fix an alphabet Σ. For any string w…

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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For any string w 6=/ E, the principal suffix of w is the string resulting by removing the first symbol from w. We will denote this string by ps(w). For any language L, define ps(L) := {ps(w) : w ∈ L ∧ w 6=/ E}. Show that if L is regular, then ps(L) is regular. (The underlying alphabet is arbitrary.) Show by giving rules to transform a regex for L into a regex for ps(L).
Answer itthe nfa. L1 = {u ∈ Σ∗| u ends with aa}.L2 = {u ∈ Σ∗| u ends and begins with different letters }.L3 = {u ∈ Σ∗| u contains abba}.L4 = {u ∈ Σ∗| u is of the form anbamfor n,m > 0}. Given the above languages:(a) Use the set operators ‘union’ and ‘complement’ to describe L5 = L1 ∩ L2.(b) Prove that L5 is regular.(c) Construct an NFA M that accepts L5, and prove its correctness.?
Let Σ={0,1}. Give a regular expression r over Σ such that L(r) = { w in Σ* | (w contains at least 2 symbols, that is, the length of w is at least 2) and (the first symbol of w is different from the last symbol of w) and (centre (w) is mimsy according to the following two definitions)}. For any string y containing at least 2 symbols, let centre (y) be the substring of y obtained by removing the first symbol and the last symbol of y. A string x over Σ is called mimsy if x is made up of either • at least two 0-symbols (and no 1-symbols), or an even number of 1-symbols (and no 0-symbols).
Consider the language L = {w010w | w ∈ Ld} Is this language recursively enumerable? Justify your answer
For each of the following regular expressions over Σ = {a, b}, state 5 strings that are in the language of the regular expression, and 5 strings that are not. 1. (a∗b)∗|a∗2. (Σ∗aaΣ∗)|(Σ∗bbΣ∗)|(ab)∗|(ba)∗|a|b
Suppose LS {a, b}* is defined as follows: A e L; for every x and y in L, the strings axb, bxa, and xy are in L. Show that L= AE B, the language of all strings x in {a, b}* satisfying na(x) = n¿(x).
Let L be the language L={aPb9cds | pq and q#r} over the alphabet Σ={a,b,c,d). Prove that L is not regular using the pumping lemma. Note: Follow the mathematical formalism given in the textbook. Explain your steps.
Using Kleene's theorem, build a FA that defines the language L over the alphabet Σ = {0,1}, as given by the regular expression L = J + K, where J is the language of strings that do not contain the substring 10, and K is the language of strings that contain an even number of 1's.
Consider the following regular expression over alphabet Σ = {a,b,c}: Σ∗a+Σ∗bc Provide two strings that are in the language L and two strings that are not in the language L. w∈L: w∈L: w /∈L: w /∈L:
Let L be the language L={aPb9cds | pq and q#r} over the alphabet Σ={a,b,c,d). Prove that L is not regular using the pumping lemma. Note: Follow the mathematical formalism given in the textbook. Explain your steps.

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