Formal Languages
Give a regular expression for each of the following
languages:
L2a = {w ? {0,1}* | w corresponds to the binary encoding of
non-negative integers that are evenly divisible by 4
L2b = {w ? {a,b}* | w contains at least one 'a' and exactly two
b's}
L2c = {w ? {0, 1, 2}* | w starts with a 2, ends with a 1 and
contains an even number of 0's}.
Are the following languages over {a, b} regular? If they are
then prove it. If they are not prove it with the Pumping Lemma
a) {ap | p is a prime number}
b) {xax | x Î{a,b}*} (start by listing some strings
in, not in, the language
The first five questions give languages over {0, 1}. In each
case decide whether the language is regular or not, and prove your
answer is correct.
5. The set of strings in which the number of 0's is a perfect
square.
Consider the set of vectors S = {(1, 0, 1),(1, 1, 0),(0, 1,
1)}.
(a) Does the set S span R3?
(b) If possible, write the vector (3, 1, 2) as a linear
combination of the vectors in S. If not possible, explain why.
Consider the following statements,
[ 0 , 1 ] × [ 0 , 1 ] with the dictionary order is
complete.
[ 0 , 1 ] × [ 0 , 1 ) with the dictionary order is
complete.
[ 0 , 1 ) × [ 0 , 1 ] with the dictionary order is
complete.
Where the dictionary order on R × R is given by ( a , b
) < ( x , y ) if either a...
Consider the following equation: W =P(1−u)
Suppose that the markup of goods prices over marginal cost is
5%, the real wage is 0.952 and the natural rate of unemployment is
4.8%. What happens to the natural rate of unemployment when the
markup of prices over costs increases to 10%? Graph the result and
explain the logic of your answer.
3. Are the following languages A and B over the alphabet Σ = {a,
b, c, d} regular or nonregular? • For a language that is regular,
give a regular expression that defines it. • For a nonregular
language, using the pumping lemma prove that it is not regular. (a)
A = {d 2j+1c k+1 | j ≥ k ≥ 0} · {c r+2b 2s+3 | r ≥ 0 and
s ≥ 0} (b) B = {a 2j+2b k+3c j+1...