CMPSCI 501: Theory of Computation

Second Midterm Exam, Spring 2014

David Mix Barrington

27 March 2014

Directions:

• Answer the problems on the exam pages.
• There are nine problems (some with multiple parts) for 120 total points plus 10 extra credit. Actual scale was A = 96, C = 56.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• Many useful definitions are given below.
• The first six 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: 10 points
  Q7: 30 points
  Q8: 20 points
  Q9: 10+10 points
 Total: 120+10 points

The set N of natural numbers is {1, 2, 3,...} as defined in Sipser.

If C is any class of computers, such as DFA's, CFG's, LBA's, TM's, 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^*}
• SEP_C = {<X, Y, Z>: X, Y, and Z are computers in C ∧ (L(X) ⊆ L(Z)) ∧ (L(Y) ⊆ (the complement of L(Z)))}

A language is Turing recognizable (TR) if it is equal to L(M) for some Turing machine M.

A language is Turing decidable (TD) if it is equal to L(M) for some Turing machine M that halts on every input.

A language X is co-TR if and only if its complement is TR.

A function f is Turing computable if there exists a TM 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 Turing computable function f from Σ^* to Σ^* such that for any string w, w ∈ A ⇔ f(w) ∈ B.

A reversible stack pushdown automaton (RSPDA) is a nondeterministic machine with a stack that stores a string over its stack alphabet. On any step, the machine may push a new first letter onto the stack, pop the current first letter, or reverse the enture stack. It reads input letters and accepts or rejects just like an ordinary PDA.

A marker Turing machine (MTM) is like an ordinary deterministic one-tape Turing machine except that (1) it has a special marker symbol $ that is in its tape alphabet and not in its input alphabet, (2) its initial configuration on input w is q₀w$, (3) it must always maintain the property that there is exactly one $ on its tape, which is the rightmost non-blank symbol (unless it is in the process of moving the $ to another location), and (4) if the initial input is length n, and the head is ever more than 2n tape cells to the left of the $, the machine halts and rejects.

Sipser defines an LBA to be a deterministic one-tape Turing machine that never accesses any tape cells to the right of where the input initially resides. Here we define a nondeterministic linear bounded automaton (NLBA) to be a nondeterministic one-tape Turing machine with the same property.

• Question 1 (10): True or false with justification: Assuming that the function of Question 9b exists, the languages E_NLBA and ALL_NLBA are both undecidable (that is, neither is TD).

• Question 2 (10): True or false with justification: The language SEP_TM is TR.

• Question 3 (10): True or false with justification: The language {<X, Y>: ∃<Z>: <X, Y, Z> ∈ SEP_DFA} is TD.

• Question 4 (10): True or false with justification: The language E_RSPDA is co-TR but not TD.

• Question 5 (10): True or false with justification: If the languages A and B are not both TR, A ≤_m C, and B ⊆ C, then it is possible that C is TR.

• Question 6 (10): True or false with justification: Let PCP_k be the variant of the Post Correspondence Problem where every string on the top or bottom of any domino has at most k letters. Then for every fixed positive integer k, the language PCP_k is TD.

• Question 7 (40): Let f be any function from N × N to {0, 1}. For any number i ∈ N, we define the function f_i from N to {0, 1} by the rule f_i(x) = f(i, x).
  • (a, 5) Show that if f is any such function, computable or not, there exists a function g from N to {0, 1} such that for all i ∈ N, g ≠ f_i. (That is, there exists an x such that g(x) ≠ f_i(x).)

  • (b, 5) Explain why, if f is a Turing computable function, we can find such a g that is also a Turing computable function.

  • (c, 5) Let {S_i: i ∈ N} be any collection of subsets of N. (That is, for any i, S_i ⊆ N.) Prove that there exists a subset T of N such that for all i ∈ N, T ≠ S_i.

  • (d, 5) Explain why part (c) implies that the power set of N is uncountable, making clear that you understand the definition of "uncountable".

  • (e, 10) Professor Colbert argues as follows: "Suppose each of the sets S_i in part (c) is TD. Then we can apply the reasoning of part (b) to the construction of the set in part (c), so that the set T constructed there has a Turing computable characteristic function and is therefore TD. Then, reasoning as in part (d), there are uncountably many TD sets." What is wrong with this argument?

• Question 8 (20): Marker Turing machines (MTM's) are defined above.
  • (a, 10) Explain how any MTM can be simulated by an ordinary Turing machine. (You may use any constant number of tapes in your simulation.)

  • (b, 10) Can every ordinary one-tape TM be simulated by an NTM? That is, for every ordinary one-tape TM, does there exist an MTM with the same language? Prove your answer.

• Question 9 (10 + 10 extra credit): Nondeterministic linear bounded automata (NLBA's) are defined above.
  • (a, 10) Prove that the language A_NLBA is TD.

  • (b, 10) Show that there exists a Turing computable function f such that if G is any context-free grammar, then f(<G>) = <N>, where N is an NLBA such that L(G) = L(N).

  Last modified 10 April 2014