Type your answers for problems 5-9. Hand-drawn diagrams for problems 1-3 are acceptable. 1. For each of the languages L1 and L2 given below, show a state diagram of a deterministic finite state automaton (DFA) that accepts the language. The alphabet of the two languages is {a, b}. L1={x | x  (a+b)* contains 3n+2 a’s for n ≥ 0} L2={x | x  (a+b)* contains 2n+1 b’s for n ≥ 0} 2. Show a state diagram of a DFA that accepts the intersection of the two languages given in problem 1. 3. Show a state diagram of a push-down automaton (PDA) that accepts the language given below. The alphabet of the language is {a, b, c}. {a2i bk c3i | i, k > 0} 4. Circle one of the letters R, C or N for each language given below. Circle R if the language is regular (and of course also context-free), circle C if the language is context free but not regular, and circle N if the language is not context free. Note that xR denotes the string x written backward. 1. { ai bj ck | i > j > k ≥ 0} Circle one: R C N 2. { ai bj ck | i, j, k ≥ 0} Circle one: R C N 3. { ai bk | i, k  0, and i < k} Circle one: R C N 4. { ai bk | i, k  0} Circle one: R C N 5. { x an xR | n  0 and x  (a+b)*} Circle one: R C N 6. { x xR x x | x  (a+b)*} Circle one: R C N 7. { xR x | x  (a+b)*} Circle one: R C N 8. { a2i+1 b2k+1 | i, k  0} Circle one: R C N 9. ( | i , k  0} Circle one: R C N | i , j ≥ 0} Circle one: R C N ){ a2i b3k 10. { a5i bj 11. { a5i b2i | i  0 } Circle one: R C N 12. { ai bj ck | i > j and j, k ≥ 0} Circle one: R C N 5. Use the pumping lemma for regular languages to prove that the language L given below is not regular. The alphabet of L is ={a, b} L = { ai bk | 2i > k } You may write your proof in your preferred format, but your proof must precisely specify the information required for each of the five blanks in the following template. Note that the third blank requires an integer as an exponent. Use p as the pumping length. Define all variables that you use in your proof. Proof. The sentence s= _contradicts the pumping lemma because for all x, y, z * such that xyz=s, if |xy| p and |y| >0, then y must beand xy in L because _ . QED. z = is not You need to use the following concepts for problems 6 and 9. Two strings x and y are distinguishable with respect to a language L if there is a string d such that one of xd and yd is a sentence in L but the other is not. The string d is known as a distinguisher for x and y. A set of strings are pair-wise distinguishable if there is a distinguisher for each pair of strings in the set. 6. Show a set D of 5 strings that are pairwise distinguishable with respect to the language given below. For each pair of strings in the set D, give a distinguisher. The alphabet of L is  = {a}. L = {a5n | n ≥ 0 } 7. (Bonus credit) Without minimizing a DFA that accepts the language L given in problem 6 and without using the pumping lemma, prove that any DFA that accepts the language L must have 5 or m ...

1. Type your answers for problems 5-9. Hand-drawn diagrams for
problems 1-3 are acceptable.
1. For each of the languages L1 and L2 given below, show a
state diagram of a deterministic finite state automaton (DFA)
that accepts the language. The alphabet of the two languages is
{a, b}.
2. Show a state diagram of a DFA that accepts the intersection
of the two languages given in problem 1.
3. Show a state diagram of a push-down automaton (PDA) that
accepts the language given below. The alphabet of the language
is {a, b, c}.
{a2i bk c3i | i, k > 0}
4. Circle one of the letters R, C or N for each language given
below. Circle R if the language is regular (and of course also
context-free), circle C if the language is context free but not
regular, and circle N if the language is not context free. Note
that xR denotes the string x written backward.
1. { ai bj ck | i > j > k ≥ 0} Circle one: R C N
2. { ai bj ck | i, j, k ≥ 0} Circle one: R C N
C N
C N
C N
Circle one: R C N
Circle one: R C N

2. C N
9. (
|
i
, k
0} Circle one: R
C
N
|
i
, j ≥ 0} Circle one: R
C
N
){ a2i b3k
10. { a5i bj
C N
12. { ai bj ck | i > j and j, k ≥ 0} Circle one: R C N
5. Use the pumping lemma for regular languages to prove that
the language L given below is not regular. The alphabet of L is
L = { ai bk | 2i > k }
You may write your proof in your preferred format, but your
proof must precisely specify the information required for each
of the five blanks in the following template. Note that the third

3. blank requires an integer as an exponent. Use p as the pumping
length. Define all variables that you use in your proof.
Proof. The sentence s= _contradicts the pumping lemma because
because _ . QED.
z =
is not
You need to use the following concepts for problems 6 and 9.
Two strings x and y are distinguishable with respect to a
language L if there is a string d such that one of xd and yd is a
sentence in L but the other is not. The string d is known as a
distinguisher for x and y. A set of strings are pair-wise
distinguishable if there is a distinguisher for each pair of strings
in the set.
6. Show a set D of 5 strings that are pairwise distinguishable
with respect to the language given below. For each pair of
= {a}.
L = {a5n | n ≥ 0 }
7. (Bonus credit) Without minimizing a DFA that accepts the
language L given in problem 6 and without using the pumping
lemma, prove that any DFA that accepts the language L must
have 5 or more states.
8. Show an infinite set of strings that are pairwise
distinguishable with respect to the language A given below.
Also give a distinguisher for each pair of strings in the infinite

4. set.
A={ an bn | n ≥ 0 }
9. (Bonus credit) Assume that an infinite set required by
problem
8 exists. Without using the pumping lemma, prove that the
language A given above is not regular.
Devin
The Reconstruction was successful due to the fact it allowed
African-Americans to become allies with radical republicans,
which then led to them challenging the white supremacy while
they participated more in the political system (Hine, 2014). The
Reconstruction also help create African-American schools for
the children. Also, due to their involvement it led to the federal
government enforcing laws that made all people equal and gave
them the same equal protection (Hine, 2014). The 13th
amendment nullified the slave trade and was the start of
abolishing slavery. The 14th amendment was established to
create equality between all races and ensure that all races are
equally protected by the laws. The 14th amendment is what
really help pave the way for African-Americans to be
recognized as citizens in the United States. The 15th amendment
gave all citizens (remembering that blacks are now considered
citizens) the right to vote. It clearly states that the right to vote
will not be denied based off of race and/or color (Hine, 2014).
The 13th, 14th and 15th amendment helped begin the movement
of ending slavery and for blacks to gain equality. In the
beginning the Supreme Court didn't really enforce these laws
too well. The Voting Rights Act of 1965 is what helped the
Supreme Court start to enforce these laws more strictly. The
white supremacy made many attempts to hinder blacks become
citizens and respecting their rights. They burnt down they
churches, they raided their schools and streets to stop them from
making it to school. Reconstruction helped blacks gain some

5. improvement in being looked at as citizens and their votes being
counted just as much as everyone else. However, the white
supremacy made many attempts to slow down the movement.
Some of those attempts worked, but some were not stopping the
motivation the blacks had during that time.
Preston
In my opinion from what I read is that the Reconstruction was a
total disaster and failure for BOTH sides! The African
American population suffered greatly during this period as did
many whites and in the end left the south in shambles.
The 13th amendment was supposed to abolish slavery, at least
on paper it did but in reality racism and slavery prevailed by
way of the KKK, extreme violence, corrupt white politicians
and the Black Codes, which were designed to keep blacks in
servitude to be used as a labor force.
The 14th amendment declared that all persons born or
naturalized in the United States to be both national and state
citizens and forbid the states from depriving any person of life,
liberty, or property without due process of law and granted them
equal protection under those same laws. Again, this is what it
said on paper,but it was major opposition to this amendment,
neither the republicans or the democrats wanted this to pass and
Then President Andrew Johnson went so far as to veto the bill
because it granted citizenship to free black slaves. The
amendment was eventually passed into law however. This was
really a sad time for African Americans.
The 15th amendment was enacted so that no person could be
denied the right to vote because of race or previous conditions
of servitude. African American men in the North were the
amendments immediate beneficiaries because before its
adoption, black men could only vote in 8 northern states.
None of these amendments were really enforced when they were
passed.and there were millions of loopholes that the whites used
to keep blacks in check. There were poll taxes enacted, literacy
tests, and property qualifications that had to be met in order to

6. vote. Blacks had to work for whites in their fields in order to
get a pinch of land for themselves.
The African-American Odyssey, volume 2, 6th edition (c) 2017
Anatoly
The intent of reconstruction was largely an effort to rebuild all
the infrastructure and political system the United States had
before the civil war, this period of reconstruction was also a
time a great pain feel by the citizen due to all the damage and
destruction brought by the civil war. Reconstruction was also a
time of confusing, because so many new factors had to be added
to our political system that had not been dealt with before such
as the term in which the confederate was going to be allowed
back into the Union, the place of now Freedman in society and
the rights they would have been some of many other factors.
Reconstruction failed vastly because of the lack of enforcement,
especially coming from the president himself. President Andrew
Johnson becomes the president after the assassination of
Lincoln and earn a lot of freedmen followers by claiming to be
the friend of the all the freedmen, claiming to be heavy handed
on those who are found to be traitors, but this was far from the
trust. Not only was he not the friend of African Americans, but
he was a strong believer of the inferiority of black people and
quickly began to placate white southerners, he then granted
pardons and amnesty to those mostly wealthy southerners, this
action were encouraging to former slave owners and whites who
opposed the union making it very difficult for reconstruction to
engaged properly.
The Thirteenth Amendment was created to simply free the
slaves, this amendment to the U.S. Constitution outlawed
slavery and involuntary servitude which allowed the slaves to
leave their masters in the first march to freedom. The

7. Fourteenth Amendment secured the legal rights of freedmen by
making a person born in the United States a citizen of the
United States and of the state in which he or she lived, this
amendment later became the pinnacle of immigration mandating
and laws. The Fifteenth Amendment granted the right to vote
for black people and all Americans. These Amendments were
not welcomed by many southern whites, as they could not
accept that blacks now had the same rights as they did, black
people who had been defined as an inferior race in every f