CMPSCI 401: Theory of Computation

Final Exam, Spring 2013

David Mix Barrington

9 May 2013

Directions:

• Answer the problems on the exam pages.
• There are eight problems for 125 total points plus 10 extra credit. Actual scale was A = 105, C = 70.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first five questions are statements -- in each case say whether the statement is true or false and give a convincing justification of your answer -- a proof, counterexample, quotation from the book or from lecture, etc. You get five points for the correct boolean answer (so there is no reason not to guess if you don't know) and up to five for the justification.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 10 points
  Q6: 25 points
  Q7: 30+10 points
  Q8: 20 points
 Total: 125+10 points

If C is any class of computers, such as DFA's, CFG's, LBA's TM, strange variant TM's, etc.:

• A_C = {(M, w): M is a computer in C and w ∈ L(M)}
• E_C = {(M): M is a computer in C and L(M) = ∅}
• ALL_C = {(M): M is a computer in C and L(M) = Σ^*}
• ALLSTAR_C = {(M): M is a computer in C and L(M)^* = Σ^*}
• INF_C = {(M): M is a computer in C and L(M) contains infinitely many strings}

A is Turing decidable if A = L(M) for some Turing machine M that always halts.

A function f is Turing computable if there exists a Turing machine M such that for any string w, M when started on w halts with f(w) on its tape.

Recall that if A and B are two languages, A is mapping reducible to B, written A ≤_m B, if there exists a function f: Σ^* → Σ^* such that for any string w, w ∈ A ↔ f(w) ∈ B.

Although we did not prove this in lecture, you may assume both:

• There exists a Turing decidable language that is not in PSPACE.

• There exists a language in PSPACE that is not in L = DPSACE(log n).

Define a weighted graph to be an undirected graph where each vertex has a positive integer weight -- you may assume that each weight is at most n, the number of vertices. The language WC (weighted clique) is the set {(G, S, k): S is a clique in the weighted graph G and the sum of the weights of the vertices in S is at least k}. The language MWC (maximum weighted clique) is the set {(G, k): there exists a clique in the weighted graph G such that the sum of the vertex weights in the clique is at least k}.

A directed graph is strongly connected if for every pair of vertices u and v, there is a (directed) path from u to v. The language STRCONN is the set of all strongly connected directed graphs.

• Question 1 (10): True or false with justification: The language INF_CFG is Turing decidable.

• Question 2 (10): True or false with justification: There exist languages X and Y such that X ≤_m Y is true but X ≤_p Y is false.

• Question 3 (10): True or false with justification: Every language in the class L = DSPACE(log n) is context-free.

• Question 4 (10): True or false with justification: For any language X, X is Turing decidable if and only if X* is Turing decidable.

• Question 5 (10): True or false with justification: The language of the grammar with rules S → Ta, T → bU, T → Ub, U → c, and U → S is regular.

• Question 6 (25): These questions use the definition of the languages MWC and WC above.

  • (a, 10) Prove that the language WC is in the class L = DSPACE(log n).

  • (b, 15) Prove that the language MWC is NP-complete.

• Question 7 (30+10): These questions all involve the language STRCONN of strongly connected directed graphs, defined above.
  • (a, 10): Prove that STRCONN is in the class P.

  • (b, 10): Prove that STRCONN is in the class DSPACE(log² n).

  • (c, 10): Prove that STRCONN is in the class NL.

  • (d, 10XC): Prove that STRCONN is complete for the class NL under ≤_L reductions.

• Question 8 (20): These questions both use the definition of the language ALLSTAR_C from above.
  • (a, 10): Prove that the language ALLSTAR_DFA is Turing decidable.

  • (b, 10): Prove that the language ALLSTAR_TM is not Turing decidable.

Last modified 29 May 2013