CS 5315, Final Exam, Thursday, May 11, 2006

Name

(Please do not forget to put your name on all extra sheets of paper)

1. Primitive recursive functions
(a) Prove, from scratch, that the functions a^b, 3 - a (funny minus that never goes below 0), "a and b", and "not a" are primitive recursive.

(b) Start by giving a definition of a primitive recursive function, and explain how your procedure for "not a" fits this general definition.

(c) The notion of a primitive recursive function is a formalization of the for-loop. Translate your description of "not a" into a program that uses for-loop(s) to compute the product.

2. Mu-recursive functions
(a) Prove that the functions a^b, 3 - a, "a and b", "not a", and a nowhere defined function are mu-recursive.

(b) Start by giving a definition of a mu-recursive function, and explain how your procedure for computing a nowhere defined function fits this general definition.

(c) Since division is an inverse operation to multiplication, it may seem reasonable to define a div b as the following expression: mu m.(b*m=a). Describe this expression in English and as a while loop, and show, step by step, what happens when we try to compute this expression for a=2 and b=3.

3. Turing machines (TM)
(a) Describe a TM that computes a + b.

(b) Trace your TM for a=2 and b=1.

(c) Use your TM to design a new TM for computing composition (n1 + n2) + n3.

4. Computability in general
(a) Is every computable function primitive recursive? Explain your answer (no need to produce a detailed proof).

(b) Is every computable function mu-recursive? explain your answer. Formulate Church-Turing thesis. Is it a mathematical theorem? a statement about the physical world? can it turn out to be false?

(c) Use the known fact that halting problem is undecidable to prove that zero-checker is not possible.

5. Decidable and r.e. sets
(a) Prove that the union and intersection of two decidable sets and a complement to a decidable set are decidable.

(b) Prove that the union and intersection of two r.e. sets are r.e.

(c) Is the complement to a r.e. set always r.e.? Explain your answer.

6. Kolmogorov complexity
(a) What is Kolmogorov complexity? Give an exact definition.

(b) Estimate the Kolmogorov complexity of a string 011011...01011 (011 repeated 2,006 times). For extra credit: prove that Kolmogorov complexity is not computable.

7. NP
(a) Define what it means for a problem to be from the class P, from the class NP, to be NP-hard.

(b) Use a general algorithm to transform a Boolean expression x1+0.3*x2<0.2+5*x1 into DNF and CNF forms.

(c)-(f) Reduce the satisfiability problem for the formula (x1 \/ x2) & (x1 \/ -x2 \/ -x3) to:
(c) 3-coloring
(d) clique
(e) subset problem
(f) interval computations.

8. Parallelism
(a) Describe, step by step, how to compute the dot product a1*b1+...+an*bn of two given vectors a1,...,an and b1,...,bn in parallel.

(b) Does this problem belong to the class NC? explain your answer.