1. Pumping Lemma:
a. Is the following language a regular language? (Use pumping
lemma in proof.) L = {0n 1n 2n | n ≥ 0}
b. What is (i.e., define) the CFL pumping lemma?
Use the pumping lemma to show that the following languages are
not regular
(c) (5 pts) Let Σ = {0, 1, −, =} and
SUB = {x = y − z | x, y, z are binary integers, and x is the
result of the subtraction of z from y}. For example: 1 = 1 − 0, 10
= 11 − 01 are strings in SUB but not 1 = 1 − 1 or 11 = 11 − 10.
Let a < c < b, and let f be defined on [a,b]. Show that f
∈ R[a,b] if and only if f ∈ R[a, c] and f ∈ R[c, b]. Moreover,
Integral a,b f = integral a,c f + integral c,b f .
Use the pumping lemma to show that the following languages are
not regular.
A a. A1={0^n 1^n 2^n | n≥0}
b. A2 = {www | w ∈ {a,b}∗}
A c. A3 ={a^2^n | n≥0} (Here, a^2^n means a string of 2^n
a’s.)
A ={a3n |n > 0 }
Do not use Pumping Lemma for Regular Expression to prove the
following. You may think of Closure Properties of Regular
Languages
1. Fix an alphabet. For any string w with |w| ≥ 2, let middle(w)
be the string obtained by removing the first and last symbols of w.
That is, Given L, a regular language on Σ, prove that f1(L) is
regular, where
f1(L) = {w : middle(w) ∈ L}
a.) Prove the following: Lemma. Let a and b be integers. If both
a and b have the form 4k+1 (where k is an integer), then ab also
has the form 4k+1.
b.)The lemma from part a generalizes two products of integers of
the form 4k+1. State and prove the generalized lemma.
c.) Prove that any natural number of the form 4k+3 has a prime
factor of the form 4k+3.
Let A and B two events defined on a sample space . Show that A
and B
are independent if and only if their characteristic functions 1A
and 1B are independent random variables.