CMPSCI 401: Theory of Computation

Final Exam, Spring 2010

David Mix Barrington

18 May 2010

Directions:

• Answer the problems on the exam pages.
• There are eight problems for 125 total points. Actual scale is A = 102, C = 65.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first four questions are true/false, with five points for the correct boolean answer and up to five for a correct justification of your answer -- a proof, counterexample, quotation from the book or from lecture, etc. -- note that there is no reason not to guess if you don't know.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 10 points
  Q6: 30 points
  Q7: 20 points
  Q8: 25 points

  Total: 125 points

The following formal languages are each used in one or more problems:

• TWO-CLIQUE is the set of all pairs (G, k) where G is an undirected graph, k is a positive integer, and G includes two sets of vertices A and B such that (1) A ∩ B = ∅, (2) A and B each have exactly k vertices, and (3) both A and B are cliques. (A clique is a set of vertices where there is an edge between every pair of distinct vertices in the set.)

• NP-PATH is the set of all triples (G, s, t) where G is a directed graph, s and t are vertices in G, and it is possible for a nondeterministic Turing machine to choose (guess) a path from s to t in G in polynomial time.

• A_NFA is the set of pairs (N, w) where N is an NFA, w is a string over the alphabet of N, and w ∈ L(N).

• A_LINSPACE is the set of triples (M, x, 1^s) such that M is a one-tape deterministic Turing machine (with tape alphabet {0, 1, blank}), x is a string over the input alphabet of M, and M accepts x using at most s cells of space. Note that here "1^s" denotes a string of s ones, so that the length of this string is s.

• UNARY-PATH is the set of all strings of the form

  a¹b^i₁a^i₁b^i₂a^i₂b^i₃...a^i_k-1b^i_ka^i_kb²

  where k, i₁, i₂, ..., i_k are all positive integers.

• Question 1 (10): True or false with justification: Assume that P ≠ NP. Then the language TWO-CLIQUE, defined above, is NP-complete.

• Question 2 (10): True or false with justification: Assume that P ≠ NP. Then the language NP-PATH, defined above, is NP-complete.

• Question 3 (10): True or false with justification: There exists a regular language that is not in the class DSPACE(1). (Note added during test: The read-only input tape of a Turing machine has detectable endmarkers at each end.)

• Question 4 (10): True or false with justification: The language A_NFA, defined in Sipser and above, is in the class DSPACE(log² n).

• Question 5 (10): Prove that the language A_NFA can be decided by a family of boolean circuits with polynomial size, fan-in two, and depth O(log² n).

• Question 6 (30): Let M be a fixed (ordinary deterministic) Turing machine that always halts on any input.

  • (a,10) Let f_M be the function from positive integers to positive integers defined so that f(n) is the maximum space used by M on any input of length exactly n. Prove that f_M is a Turing computable function.

  • (b,10) Prove that the language A_LINSPACE, defined above, can be decided by a linear bounded automaton (LBA). (Recall that an LBA is a one-tape Turing machine that never moves its tape head to the right past the the space originally occupied by its input.) (Notes added during test: (1) An LBA knows where the ends of its tape are when it reaches them. (2) You may assume that s ≥ |x| for any triple (M, x, 1^s) in A_LINSPACE.)

  • (c,10) Parts (a) and (b) could be combined to show that any Turing decidable language is mapping reducible to the language of an LBA. Show that this result would not be so impressive, as follows. Prove that if X is any Turing decidable language, then there exists a regular language R such that X ≤_m R. (Recall that "≤_m" denotes the relation called "mapping reducible" in Sipser.)

• Question 7 (20): Both parts of this question involve the language UNARY-PATH defined above.

  • (a,10) Is UNARY-PATH a regular language? Prove your answer.

  • (b,10) Is UNARY-PATH a context-free language? Prove your answer.

• Question 8 (30): A boustrephedonic Turing machine is a single-tape nondeterministic TM that (1) begins its computation by placing an endmarker at the left end of its tape, (2) then travels right without changing the tape until it nondeterministically chooses to place an endmarker on a blank cell somewhere to the right of the input, and (3) then travels determinisitically between the endmarkers until or unless it halts, never moving the endmarkers and changing direction only at an endmarker. (A boustrephedon is a type of bidirectional text, common in ancient Greek inscriptions, where lines are written alternately left-to-right and right-to-left. The name comes from the Greek for "turning like an ox" as it plows a field.)

  • (a,10) Prove that every Turing-decidable language X is the language of some boustrephedonic TM that always halts once it has placed its right endmarker. (Note that the BTM is nondeterministic, so you must prove that your BTM can accept w if and only if w ∈ X.

  • (b,15) Let P_bous be the set of all languages of boustrephedonic TM's B such that for any string w of length n, if B can accept w at all it can accept w within time n^O(1). Prove that P_bous = P. Note that since the BTM's are nondeterministic, the inclusion P_bous ⊆ P is not trivial. It may help to reuse some of your argument from part (a).

  Last modified 19 May 2010