CMPSCI 501: Theory of Computation

Second Midterm Exam, Spring 2016

David Mix Barrington

24 March 2016

Directions:

• Answer the problems on the exam pages.
• There are eight problems (some with multiple parts) for 120 total points plus 10 extra credit. Actual scale was A = 105, C = 60.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• 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: 20+10 points
  Q8: 40 points
 Total: 120+10 points

The set N of natural numbers is {0, 1, 2, 3,...}, not quite as defined in Sipser.

For this exam, the input alphabet of all machines will be Σ = {0, 1}.

A palindrome is a string that is equal to its own reversal. Let PAL = {w: w is a palindrome}.

If C is any class of computers, such as DFA's, CFG'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^*}

• ANYPAL_C = {(M): M is a computer in C and L(M) includes at least one palindrome} (That is, L(M) ∩ PAL ≠ ∅.)

• ALLPAL_C = {(M): M is a computer in C and L(M) includes every palindrome} (That is, PAL ⊆ L(M).)

• ONLYPAL_C = {(M): M is a computer in C and L(M) includes only palindromes} (That is, L(M) ⊆ PAL.)

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 is co-TR if and only if its complement is TR.

A function f from strings to strings 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 Turing computable function f: Σ^* → Σ^* such that for any string w, w ∈ A ↔ f(w) ∈ B.

A Volatile Memory Turing Machine or VMTM is a multitape deterministic Turing machine where Tape 1 is a read-only input tape and the other tapes have the following property: Any cell that has not been written to in the last ten time steps changes its content to the blank symbol, without the intervention of the head.

A Write-Once Turing Machine or WOTM is a deterministic one-tape Turing machine with the property that each cell may have its contents changed only once during a computation. That is, once a new character replaces the original contents of the cell, that new character remains there forever. There are no restrictions on the machine's reading or moving.

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

• Question 2 (10): True or false with justification: The language ONLYPAL_TM is Turing recognizable.

• Question 3 (10): True or false with justification: The language ONLYPAL_DFA is Turing decidable.

• Question 4 (10): True or false with justification: The language ANYPAL_CFG is Turing recognizable.

• Question 5 (10): True or false with justification: The language A_WOTM is Turing recognizable but not Turing decidable.

• Question 6 (10): True or false with justification: There exists a Turing machine R with the following property. For every string w such that R halts on input w with output n (a natural number), R also halts on input w with output n + 1.

• Question 7 (20+10): These three questions all deal with the model of Volatile Memory Turing Machines (or VMTM's) defined above.

  • (a, 10) Prove that every regular language is the language of some VMTM. You may quote Kleene's Theorem without proof.

  • (b, 10) Prove that the language A_VMTM is Turing decidable.

  • (c, 10XC) If M is a VMTM, must the language L(M) be regular? Prove your answer. You may quote results stated but not proved in lecture.

• Question 8 (40): These four questions all deal with the languages ALL_CFG, ALLTM, and ALLPAL_TM, as defined above.
  • (a, 10) Prove that ALLTM ≤_m ALLPAL_TM, using the definition of ≤_m above.

  • (b, 10) Prove that ALL_CFG ≤_m ALL_TM.

  • (c, 10) Prove that A_TM ≤_m ALL_TM. (Hint: At least one suitable function f was used more than once in lecture.)

  • (d, 10) Explain how the results of parts (a), (b), and (c), together with results from the course, imply that the language ALLPAL_TM is neither TR nor co-TR. (You may use those results even if you were unable to prove them.) (Hint: What exactly does the proof from class say about a mapping reduction involving A_TM and ALL_CFG?)

  Last modified 6 April 2016