Theory of Computation
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:

Solution to Problem 1

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

if P(n) then f(n) elseif Q(n) then g(n) else h(n)
is also primitive recursive.

Solution to Problem 2

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.

Solution to Problem 3

4. (Due February 2) Write a Java program corresponding to the following primitive recursive function f = σ(PR(π22, σ(add(π41, π42, π43)))). For this function f, what is the value of f(0, 2, 2)?

Solution to Problem 4

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, πki, 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.

Solution to Problem 5

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).

Solution to Problem 6

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

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

Solution to Problem 7

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.

Solution to Problem 8

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.

Solution to Problem 9

10. (Due February 11) In class, we designed a Turing machine for computing π21. Use this design as a sample to design a Turing machine for computing π31. 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 π21 is based on the following idea:

This Turing machine has the following rules:

Solution to Problem 10

11. (Due February 11 for extra credit, due February 16 for the regular credit) In class, we designed a Turing machine for computing π22. Use this design as a sample to design a Turing machine for computing π32. 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 π22 is based on the following idea:

A special care needs to be taken for a special case when the second component of the original pair is number 0. In this case, once we erase the 1st number, there is nothing left to erase, so we simply go back (and replace 1 back to blank when we reach the starting cell).

This Turing machine has the following main rules:

The following three additional rules take care of the case when the second number is 0:

Solution to Problem 11

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:

This machine has the following rules: The Turing machine for computing f(n) is based on the following idea: We need to take special care of the case when n = 0.

Solution to Problem 12

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.

Solution to Problem 13

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.

Solution to Problem 14

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.

Solution to Problem 15

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

This program needs to keep track of a current location of the head. Initially, this location is 0.

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:

In this case: Here:

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.

Solution to Problem 17

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?

Solution to Problem 18

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?

Solution to Problem 19

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?

Solution to Problem 20

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:

Solution to Problem 21

22. (Due March 9) Give:

These examples must be different from the ones we had in class.

Solution to Problem 22

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

by using the formulas ln(x1 * x2) = ln(x1) + ln(x2) and ln(x1 / x2) = ln(x1) − ln(x2). Specifically: Describe each of these two reductions in general terms: what is C(x, y), what is C'(x', y'), what is U1, U2, and U3.

Solution to Problem 23

24. (Due March 11) Use the general algorithm to come up with DNF form and CNF form of the formula 0.5 * x1 + 0.3 * x2 ≥ 0.7 * x3.

Solution to Problem 24

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:

Solution to Problem 25

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.

Solution to Problem 26

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.

Solution to Problem 27

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.

Solution to Problem 28

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.

Solution to Problem 29

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.

Solution to Problem 30

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?

Solution to Problem 31

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.

For the sum problem, Tsequential = n.

Solution to Problem 32

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?

Solution to Problem 33

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).

Solution to Problem 35

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.

Solution to Problem 36

37. (Due April 20) What class of polynomial hierarchy contains Σ4PΠ1P? Explain your answer.

Solution to Problem 37

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.

Solution to Problem 38

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?

Solution to Problem 39