CMPSCI 601: Theory of Computation

Practice Final Exam, Spring 2010

David Mix Barrington

Posted 29 April 2010

Directions:

• Answer the problems on the exam pages.
• There are five short problems, for ten points each, and three long problems for 25 points each. Attempt all the short problems and only two of the long ones -- the maximum score is thus 100. If you attempt all three long problems I will take the scores of the best two. Likely scale is A = 90, B = 60 but the actual scale will be determined after I grade the exam.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 10 points
  Q6: 25 points
  Q7: 25 points
  Q8: 25 points

  Total: max 100 points

Minor corrections added on the evening of 29 April.

• Question 1 (10): A parallel Turing processor is a network of f(n) different Turing machines, each of has one worktape that is public. On a given time step it may read one cell of the public worktape of any other machine, by writing the number of the cell and the number of the machine on an address tape. One machine is designated the lead machine, and it gets the input string at the beginning of the computation and produces the output at the end of the computation. Define parallel-P to be the class of languages X such that there exists a parallel Turing processor M with f(n) processors and time bound t(n) on input of length n, where f and t are polynomials, such that L(M) = X. Prove that parallel-P = P.

• Question 2 (10): Let DCYCLE be the set of directed graphs that have a cycle, where a cycle is any path of one or more edges from a vertex to itself. Prove that DCYCLE is complete for the class NL under log-space reductions. (You may quote the fact that the language PATH is NL-complete, but for your proof you will probably want a slighty different language which may require revisiting that completeness proof.)

• Question 3 (10): Recall that PARITY is the set of binary strings with an odd number of ones. Prove that PARITY is in the class ATIME(log n), where of course the alternating Turing machine has random access to its input. (This means that it can write a number i on its address tape and then access input bit x_i.)

• Question 4 (10): Recall that a pseudorandom generator of stretch s(n) is called secure if for any probabilistic polynomial-time function A, the probability that A(x) = 1, for a string x of length s(n) generated from a uniform random seed of length n, differs from the probability that A(y) = 1 for a uniform random string of length s(n) by a negligible function. Prove that no secure pseudorandom generator can have the property that its output depends only on the first O(log n) bits of its seed.

• Question 5 (10): Recall that a MOD-3 gate takes zero or more boolean inputs and outputs 1 iff the sum of its inputs (as integers) is not congruent to 0 modulo 3. Consider circuits with only MOD-3 gates that compute the parity function on n boolean inputs. Describe such a circuit of depth O(log n), and show that any such circuit must have depth Ω(log n).

• Question 6 (25): Let E(k, p, 1^t) be the following partial encryption function. (It is a partial encryption function because for given inputs it may or may not produce a ciphertext.) E interprets the binary string k, if possible, as a Turing machine. It then runs k on the plaintext binary string p for at most t steps, and outputs the string, if any, that is the result of this computation.

  The language VALID is the set of tuples (c, k, 1^a, 1^b) such that there exists a binary string p of length a such that E(k, p, 1^b) = c.

  • (a,5) Argue that VALID is in the class NP.

  • (b,10) Prove that if A is any language in the class NP, then A ≤_p VALID.

  • (c,10) Let VALID' be the set of tuples (c, k, 1^a, b) where b is a binary number and E(k, p, 1^b) = c. What is the complexity of the language VALID'? Justify your answer.

• Question 7 (25): A boolean circuit is levelled if the non-input gates can be divided into sets L₁, L₂,..., L_d, where the inputs to any gate in the set L_i are either inputs to the circuit (i.e., variables or negated variables) or are gates in the set L_i-1. The width of the circuit is the maximum number of gates in any set L_i.

  • (a,10) Let A be any language in DSPACE(log n). Argue that A is decided by a family C₀, C₁,... of boolean circuits such that each circuit C_n has size polynomial in n and width O(log n).

  • (b,5) Give an example of a language that has a circuit family as in part (a) but is not in DSPACE(log n).

  • (c,10) Argue carefully that if A has a circuit family as in part (a) and that family is log-space uniform, then A is in DSPACE(log n).

• Question 8 (25): Let g(x₁,...,x_n) be a polynomial of degree d (where d is polynomial in n) over the field Z_p, where p is a prime of between n and 2n bits. Let h be the sum, over all 2ⁿ possible boolean settings of the variables x₁,...,x_n, of g(x₁,...,x_n). Describe a protocol whereby a computationally unbounded prover can convince a probabilistic verifier that h is equal to some number K, if this is true. Your protocol should have completeness 1 and soundness at most 1/2. Argue that your protocol is correct.

Last modified 29 April 2010