## Theory of Computations, Final Exam for the courseCS 5315, Spring 2020

Name: _____________________________________________________

You can use up to 10 handwritten pages of notes.

1. Primitive recursive and mu-recursive functions:

1a. Why do we need to study primitive-recursive and mu-recursive functions in the first place? What programming concepts do they correspond to?

1b. Translate, step-by-step, the following double loop into a mu-recursive expression:

``` int s = 2;
for(int i = 1; i <= b; i++){
while(s < a)
{s = s + i + 1;}
}
```
You can use the addition function is this description. What is the result of this program when
a = 5 and b = 3?

1c. Prove, from scratch, that the function (a % b) + (a / b) is primitive recursive.

2. Computable sets, computable functions, and Turing machines:

2a. Why do we need to study decidable and recursively enumerable (r.e.) sets?

2b. Is the union of two r.e. sets always r.e.? Is the difference of two r.e. sets always r.e.? In both cases, if yes, prove it, if no, provide a counterexample.

2c. Prove that it is not possible, given a program that always halts, to check whether this program always computes n + 8.

2d. Design a Turing machine that, given n in unary code, computes 2n + 1 in binary code. For example:

• for n = 11 = 110, it should return 21 + 1 = 11;
• for n = 111 = 210, it should return 22 + 1 = 101;
• for n = 1111 = 310, it should return 23 + 1 = 1001, etc.
Trace this machine on the example of n = 111. Explain why in a Turing machine (and in most computers) binary numbers are represented starting with the least significant digit.

3. NP-hardness:

3a-d. Reduce the satisfiability problem for the formula (~a \/ ~c) & (b \/ ~c) & (~a \/ b \/ ~c) to:

a) 3-coloring,
b) clique,
c) subset sum problem, and
d) interval computations.

3e. Reduce the satisfiability problem for the formula (~a \/ ~b \/ c \/ ~d) & (a \/ b \/ c) to 3-SAT.

3f. In proofs of what results are these reductions used? What do we gain by proving that a problem is NP-hard?

3g. Use the definitions of the classes P, NP, NP-hard, and NP-complete to describe, for each of these four classes, whether this class contains search, 3-coloring, 2-SAT, and interval computations. Explain your answers.

3h. Give two examples of why the current definition of a feasible algorithm is not perfect:

• an example when an algorithm is feasible according to this definition but not practically feasible, and
• an example when an algorithm is not feasible according to this definition but is practically feasible.

3i. Use the general algorithm to find a satisfying vector for the 2-SAT formula

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

4. Parallelization:

4a. Matrices (2-D arrays) are subtracted component-wise. Does the problem of subtracting two n-by-n matrices belong to the class NC? Explain your answer.

4b. If we parallelize and take into account communication time, what is the fastest that we can compute the difference between the two matrices? Explain your answer.

4c. Where are similar arguments used in the proof that satisfiability is NP-hard?

4d. What are the physical assumptions behind these arguments? Give two examples of how non-standard physics -- in which these assumptions are not satisfied -- can be used to solve NP-hard problems in polynomial time.

5. How to solve NP-hard problems?

5a. Suppose that a probabilistic algorithm for checking the program correctness misses the program's mistake 2% of the time. How many times do we need to repeat this algorithm to achieve reliability of 99.9%? And why do we need probabilistic algorithms in the first place?

5b. Apply both greedy algorithms to solve the following particular case of the knapsack problem: we have 4 objects with weights 20, 30, 40, and 50, values 10, 12, 12, and 15, and the overall weight of 50. Why do we need to use these algorithms, if they do not produce optimal results?

5c. Give an example of how quantum computing can speed up computations.

6. Beyond NP: polynomial hierarchy:

6a. Which class of the polynomial hierarchy contains optimization problems? The problem of winning in 2 moves? Explain your answers.

6b. Which class of polynomial hierarchy contains Σ2Π3?

6c. (For extra credit) Give an example of an oracle A for which PA = NPA. Explain your answer.

7. Projects:

7a. Briefly describe your project for this class.

7b. Briefly describe someone else's project for this class.