Homeworks for the course CS 5315/CS 6315, Spring 2021

1. (Due January 26) Prove that the function computing the sum 1 + 2 + ... + n is primitive recursive. This proof should follow the same pattern that we used in class to prove that addition and multiplication are primitive recursive:

- You start with a 3-dot expression
- First you write a for-loop corresponding to this function
- Then you describe this for-loop in mathematical terms
- Then, to prepare for a match with the general expression for
primitive recursion, you rename the function to f and the
parameters to n
_{1}, ..., m - Then you write down the general expression of primitive recursion for the corresponding k
- Then you match: find g and h for which the specific case of primitive recursion will be exactly the functions corresponding to initialization and to what is happening inside the loop
- Then, you get a final expression for the function 1 + 2 + ... + n
that proves that this function is primitive recursive, i.e., that
it can be formed from 0, π
^{k}_{i}, and σ by composition and primitive recursion.

2. (Due January 28) Prove that if P(n), Q(n), f(n), g(n), and h(n) are all primitive recursive, then the function described as

3. (Due February 2) Prove, from scratch -- i.e., using only the definition of the primitive recursive function (and not using any results that we had in class without proving them) -- that the function a / b + a % b is primitive recursive.

4. (Due February 2) Write a Java program corresponding to the
following primitive recursive function f =
σ(PR(π^{2}_{2},
σ(add(π^{4}_{1},
π^{4}_{2}, π^{4}_{3})))).
For this function f, what is the value of f(0, 2, 2)?

5. (Due February 2 for extra credit, due February 4 for regular credit) Let us define a function to be A-primitive
recursive if it can be obtained from 0, σ,
π^{k}_{i}, and Ackermann's function A(n) by
using composition and primitive recursion. Prove that there exists
a computable function which is not A-primitive recursive.
*Hint*: we need a minor modification of the first proof that there exists a computable function which is
not primitive recursive.

6. (Due February 4) Show that the following function f(a, b) is μ-recursive: f(a, b) = a + b when each of the two inputs a and b is either equal to 0 or equal to 1, and f(a, b) is undefined for other pairs (a, b).

7. (Due February 4) Prove that the following function is mu-recursive:

int j = 1; while(!(a * j <= m)) {j++;}

8. (Due February 4, for extra credit) Describe integer division a / b in terms of μ-recursion, similarly to how in the lecture, we describe subtraction in terms of μ-recursion.

9. (Due February 9) Design a Turing machine that computes a function f(n) which is equal to n − 2 when n ≥ 2 and to 0 when n = 0 or 1; it is OK to assume that the input n is given in unary code.

10. (Due February 11) In class, we designed a Turing machine for
computing π^{2}_{1}. Use this design as a sample
to design a Turing machine for computing
π^{3}_{1}. Trace, step-by-step, on an example,
how your Turing machine works. For example, you can take as input
the triple (1, 3, 2).

*Reminder:* The Turing machine for computing
π^{2}_{1} is based on the following idea:

- we go step-by-step until we find the first blank space,
- when we see blank, this means that we are outside the 1st number, and we are going inside the second number, so we continue going right,
- when we see blank again, this means that we reached the end of the second number, so we go back and erase 1s one by one until we reach the blank space separating the 2nd number from the 1st one,
- after that, we stop erasing and simply go back.

- start, -- → R, in1st
- in1st, 1 → R
- in1st, -- → R, in2nd, R
- in2nd, 1 → R
- in2nd, -- → L, erasing
- erasing, 1 → --, L
- erasing, -- → L, back
- back, 1 → L
- back, -- → halt.

11. (Due February 11 for extra credit, due February 16 for the
regular credit) In class, we designed a Turing machine for
computing π^{2}_{2}. Use this design as a sample
to design a Turing machine for computing
π^{3}_{2}. Trace, step-by-step, on an example,
how your Turing machine works. For example, you can take as input
the triple (1,3,2). Also, check also that your code works when one
of the numbers is 0, especially when the third number is 0.

*Reminder:* The Turing machine for computing
π^{2}_{2} is based on the following idea:

- first, we place 1 in the very first cell, to make sure that we will know when to stop when we get back,
- then, one by one, we eliminate all the ones from the 1st number,
- then, we go to the second number and continue going until we reach the first blank space after its end,
- we need to move the second number closer to the starting cell,
- moving a unary number one step to the left means that we erase the last 1, and add a 1 before this number; this will keep the same number of ones, but we get one step closer to the starting cell of the Turing machine,
- so, once we reach the blank space after the second number, we go back one step, erase the 1 symbol, and start going left,
- we go left until we reach the end of the number, i.e., the first blank space, which we replace by 1,
- if directly to the left of the replaced black space is a symbol 1, this means that we are at the starting cell of the Turing machine, thus we have moved the number already; now all we need to do is replace this symbol 1 with blank space and stop,
- on the other hand, if directly to the left of the replaced blank space is an empty cell, this means that we need to again go right and repeat the same move-one-step-to-the-left procedure.

This Turing machine has the following main rules:

- start, # → 1, erase1st, R,
- erase1st, 1 → --, R,
- erase1st, -- → right, R,
- right, 1 → R,
- right, -- → erase, L,
- erase, 1 → --, left, L,
- left, 1 → L,
- left, # → 1, checking, L,
- checking, -- → right, R,
- checking, 1 → -- , halt,

- erase, -- → finish, L,
- finish, # → L,
- finish, 1 → -- , halt.

12. (Due February 11 for extra credit, due February 16 for the regular credit) In class, we described a Turing machine that computes g(n) = n + 1. In Homework 9, you designed a Turing machine that computes a function f(n) which is equal to n − 2 when n ≥ 2 and 0 when n = 0.

In class, we described the general algorithm for designing a Turing machine that computes the composition of two functions. The assignment is to use this general algorithm to design a Turing machine that computes the composition g(f(n)). Trace, step-by-step, on an example, how your Turing machine works. For example, you can take as input n = 2.

*Reminder:* The Turing machine for computing g(n) = n + 1 for
a unary input n is based on the following idea:

- we go step-by-step until we find the first blank space,
- then, we replace this blank space with 1 and go back.

- start, -- → working, R

(we start going to the right), - working, 1 → R

(we see 1, so we continue going), - working, -- → 1, back, L

(we see a blank space, so we replace it with 1 and start going back), - back, 1 → L

(while we see 1s, we continue going back), - back, -- → halt

(once we reach the very first cell, we stop).

- we go step-by-step until we find the first blank space,
- then, we go back, replace the last two 1s with blanks, and go back all the way.

13. (Due February 11 for extra credit, due February 16 for the
regular credit) Similarly to a Turing machine that we had in class,
that copies a number in unary code, design a Turing machine that
copies a binary number. Test it on the example of a binary number
1101 (stored as 1011). The result should be 1011_1011, with a blank
space in between. *Hint*: instead of marking 1s, mark both 0s
and 1s; instead of the state carry1in1st, we can now have two
different states: carry0in1st and carry1in1st.

14. (Due February 11 for extra credit, due February 16 for the regular credit) Sketch an example of a Turing machine for implementing primitive recursion (i.e., a for-loop), the way we did it in class, on the example of the following simple for-loop

sum = a; for(int i = 1; i <= b; i++) {sum = sum + a;}No details are required, but any details will give you extra credit.

15. (Due February 11 for extra credit, due February 16 for the regular credit) Sketch an example of a Turing machine for implementing mu-recursion, the way we did it in class, on the example of a function μm.(m = a). No details are required, but any details will give you extra credit.

16. (Due February 18) Write a method that emulates a general Turing machine. The input to this method should include:

- the
number N of states q
_{0}, ..., q_{N − 1}; we assume that q_{0}is the start state, that the last-but-one state q_{N − 2}is the accept state, and the last state q_{N − 1}is the reject state; - the number
M of symbols s
_{0}, ... s_{M − 1}; we assume that s_{0}is the blank state _; - an integer array
state[n][m] that describes to what state the Turing machine moves
if it was in the state q
_{n}and sees the symbol s_{m}; - an integer array symbol[n][m] that describes
what symbol should be on the tape after the Turing Machine in the
state q
_{n}sees the symbol s_{m}(it may be the same symbol as before, or it may be some other symbol written by the Turing machine); - a character array lr[n][m] that
describes, for each state q
_{n}and for each symbol s_{m}, whether the Turing machine moves to the left (L), or to the right (R), or stays in place (blank symbol); - the integer array of a large size describing the original contents of the tape, i.e., what symbols are written in each cell.

Your program should emulate the work of the Turing machine step-by-step. Return the printout of the method, the printout of the program that you used to test this method, and the printout of the result of this testing. Feel free to use Java, C, C+++, Fortran, or any programming language in which the code is understandable.

*Example:* A Turing machine for checking whether a binary
string is even (i.e., ends with 0) has the following rules:

- start, _ --> inNumber, R
- inNumber, 1 --> state1, R
- inNumber, _ --> reject
- inNumber, 0 --> state0, R
- state0, 1 --> state1, R
- state0, 0 --> state0, R
- state1, 1 --> state1, R
- state1, 0 --> state0, R
- state1, _ --> reject
- state0, _ --> accept

- we have N =
6 states: q
_{0}= start, q_{1}= inNumber, q_{2}= state1, q_{3}= state0, q_{4}= accept, and q_{5}= reject; - we have M = 3 symbols:
s
_{0}= _, s_{1}= 0, and s_{2}= 1;

- The first rule start, _ --> inNumber, R means that state[0][0] = 1, symbol[0][0] = 0, and lr[0][0] = R.
- The second rule inNumber, 1 --> state1, R means that state[1][2] = 2, symbol[1][2] = 2, and lr[1][2] = R, etc.

17. (Due March 2) Use the impossibility of zero-checker (that we proved in class) to prove that no algorithm is possible that, given a program p that always halts, checks whether this program always computes 5n + 8.

18-20. (Due March 4) Suppose that A, B are r.e. sets.

18. If a number n appears in the A-generating algorithm at moment 4, when will this number appear in the algorithm generating all elements of the union A U B?

19. If a number n appears in the A-generating algorithm at moment 3 and in the B-generating algorithm at moment 2, when will this number appear in the algorithm generating all elements of the intersection of A and B?

20. If a number n appears in the A-generating algorithm at moment 2, and the complement −A is also r.e., when will the deciding algorithm tell us that n is an element of the set A?

21. (Due March 4) Let us consider cases when the set A is decidable and the sets B and C are r.e. Give four examples of such cases:

- an example when the union A U B U C of the three sets is decidable,
- an example when the union A U B U C of the three sets is not decidable,
- an example when the intersection of the three sets is decidable, and
- an example when the intersection of the three sets is not decidable.

22. (Due March 9) Give:

- an example of computation time
t
_{A}(x) for which the algorithm is practically not feasible, but is feasible according to the existing definition, and - an example of computation time t
_{A}(x) for which the algorithm is practically feasible, but is not feasible according to the existing definition.

23. (Due March 9) Logarithms were invented to make multiplication and division faster: they reduce:

- multiplication to addition and
- division to subtraction

- instead of directly
computing the product y = x
_{1}* x_{2}, we first compute X_{1}= ln(x_{1}) and X_{2}= ln(x_{2}), then compute Y = X_{1}+ X_{2}, and then y = exp(Y); - instead of directly computing the ratio y
= x
_{1}/ x_{2}, we first compute X_{1}= ln(x_{1}) and X_{2}= ln(x_{2}), then compute Y = X_{1}− X_{2}, and then y = exp(Y).

24. (Due March 11) Use the general algorithm to come up with DNF
form and CNF form of the formula 0.5 * x_{1} + 0.3 *
x_{2} ≥ 0.7
* x_{3}.

25. (Due March 11) Similar to what we did in the class, illustrate the general algorithm of reducing NP problems to satisfiability on the example of the following problem:

- given a bit x,
- find a bit y for which the following formula is true: ~x \/ ~y (where ~y means negation).

26. (Due March 23) On the example of the formula (~a \/ ~b \/ c \/ ~d) & (~a \/ ~b \/ ~c), show how checking its satisfiability can be reduced to checking satisfiability of a 3-CNF formula.

27. (Due March 23) On the example of the formula (~a \/ b \/ ~c) & (a \/ ~b), show how checking its satisfiability can be reduced to coloring a graph in 3 colors.

28. (Due March 23) On the example of the formula (~a \/ b \/ ~c) & (a \/ ~b), show how checking its satisfiability can be reduced to an instance of the clique problem.

29. (Due March 23) On the example of the formula (~a \/ b \/ ~c) & (a \/ ~b), show how checking its satisfiability can be reduced to an instance of the subset sum problem (i.e., the problem of exact change.

30. (Due April 6) On the example of the formula (~a \/ b \/ ~c) & (a \/ ~b), show how checking its satisfiability can be reduced to an instance of the interval computation problem.

31. (Due April 8) Show how to compute the sum of 11 numbers in parallel if we have unlimited number of processors. How many processors do we need and how much time will the computation take? Why do we need parallel processing in the first place?

32. (Due April 8) If we take into account communication time, how
fast can you compute the sum of n numbers in parallel? *Hint:*
See Section 2 of the paper with Data Morgenstein on the class
website.

- There, we show that T
_{sequential}≤ (T_{parallel})^{4}. - So what can we tell about
T
_{parallel}? That T_{parallel}is bounded, from below, by the fourth order root of T_{sequential}.

33. (Due April 13) Suppose that we have a probabilistic algorithm that gives a correct answer 3/4 of the time. How many times do we need to repeat this algorithm to make sure that the probability of a false answer does not exceed 3%? Explain your answer. Give an example of a probabilistic algorithm. Why do we need probabilistic algorithms in the first place?

34. (Due April 13) Pick an example of an Ali-Baba problem, and explain, step by step, what solution two greedy algorithms will produce for this example.

35. (Due April 13) Use the variable-elimination algorithm for
checking satisfiability of 2-SAT formulas that we had in class to
find the values that satisfy the following formula:

(a \/ ~b) &
(~a \/ b) & (~a \/ ~b) & (~a \/ c) & (a \/ c) & (b \/ c).

36. (Due April 15) On the example of the function f(x) = 1, trace, step by step, how Deutsch-Josza algorithm will conclude that f(0) = f(1) while applying f only once.

37. (Due April 20) What class of polynomial hierarchy contains
Σ_{4}P^{Π1P}? Explain your
answer.

38. (Due April 20) Describe, in detail, at least two different schemes that use serious but still hypothetical physical processes to solve NP-complete problems in polynomial time.

39. (Due April 22) What can you say about the Kolmogorov complexity of the string 001001... in which the sequence 001 is repeated 2020 times?