Question 4 Prove that the following language is not regular. ? =
{ 0 ?1 ? | ?, ? ≥ 0, ? ≠ 2? + 1 }
Question 5 Prove that the following language is not regular. ? =
{ ? ∈ { 0, 1, 2} ∗ | #0 (?) + #1 (?) = #2 (?) } where #? (?)
denotes the number of occurrences of symbol a in string w.
Show that a language that is described by a regular expression
can also be generated by a context-free grammar. As a hint,
consider each mechanism, such as using a basic form or forming the
regular expression re1 ∪ re2 from the
expressions re1 and re2, and show how this
can be transformed into grammar rules that can be added to the
grammars generating the collection of strings denoted by
re1 and re2.
Given that x =0 is a regular singular point of the given
differential equation, show that the indicial roots of the
singularity differ by an integer. Use the method of Frobenius to
obtain at least one series solution about x = 0.
xy"+(1-x)y'-y=0
(a) Show that x= 0 is a regular singular point.
(b) Find the indicial equation and the indicial roots of it.
(c) Use the Frobenius method to and two series solutions of each
equation
x^2y''+xy'+(x^2-(4/9))y=0
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 }
Determine whether or not the following languages are regular. If
the language is regular then give an NFA or regular expression for
the language. Otherwise, use the pumping lemma for regular
languages or closure properties to prove the language is not
regular.
1) L = { 0 n1 k : k ≤ n ≤ 2k}
2) L = { 0 n1 k : n > 0, k > 0 } È { 1 k0 n : k > 0, n...