CMPSCI 401: Theory of Computation

Second Midterm Exam, Spring 2010

David Mix Barrington

29 March 2010

Directions:

• Answer the problems on the exam pages.
• There are seven problems for 125 total points. Actual scale was A=105, 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: 35 points
  Q6: 30 points
  Q7: 20 points

  Total: 125 points

If X is a computational model, such as DFA's, CFG's, or TM's, we follow Sipser in defining the following languages:

• A_X = {(M, w): M is a computer in X and w ∈ L(M)}
• E_X = {(M): M is a computer in X and L(M) = ∅}
• ALL_X = {(M): M is a computer in X and L(M) = Σ^*}
• EQ_X = {(M, N): M and N are computers in X and L(M) = L(N)}
• (not in Sipser) COMMON_X = {(M, N): M and N are computers in X and L(M) ∩ L(N) ≠ ∅} (that is, L(M) and L(N) have a string in common)

Recall also that the Post Correspondence Problem takes as input a finite set of dominoes {D₁,..., D_n}, such that each domino D_i has a top string t_i and a bottom string b_i, each a nonempty string in &Sigma'^* for some alphabet Σ. A set of dominoes is in the language PCP if there exists a nonempty sequence of dominoes that is a match, meaning that the concatenation of the top strings and the concatenation of the bottom strings form the same word.

• Question 1 (10): True or false with justification: A dedicated tape Turing machine (or DTTM) is a multitape deterministic TM, each of whose tapes is either read-only or write-only. (The input tape is read-only, and the others are all work tapes that start out with all blanks.) That is, the transition function takes as input the current state and the letter seen on each of the read-only tapes, and gives as output the new state, the letter written to each write-only tape, and the head movement of each tape. (It may help to recall that a tape head ordered to move left from the leftmost cell of a tapr just stays there, without indicating to the TM that this has happened.) Then the language A_DTTM is undecidable.

• Question 2 (10): True or false with justification: A ten-state binary-tape Turing machine (or 10BTTM) is a deterministic single-tape TM with exactly ten states, whose tape alphabet is {0, 1}. (0 is the blank character, so the input alphabet is {1} and the only valid inputs are strings in 1^*.) Then the language E_10BTTM is decidable.

• Question 3 (10): True or false with justification: A ten-state Turing machine (or 10TM) is a deterministic single-tape TM with exactly ten states and some finite tape alphabet Γ and some finite input alphabet Σ ⊆ Γ. Then the language A_10TM is decidable.

• Question 4 (10): True or false with justification: Let Σ be a finite alphabet. A codebook over Σ is a finite set of pairs of strings (p_i, c_i), where in each pair p_i is called the plaintext and c_i is called the ciphertext. A string w in Σ^* is called a plaintext string if it is in the regular language (p₁ ∪ p₂ ∪ ... ∪ p_n), and its corresponding ciphertext string is obtained by replacing each string p_i by the corresponding string c_i. (Note that a given plaintext string could have more than one ciphertext string, if there are two pairs with the same p_i or if there is more than one way to break the string into p_i's.)

  The codebook fixed-point problem (or CFPP) is the language consisting of all codebooks for which there is some nonempty plaintext string that has a ciphertext string that is identical to the palintext. Then this language CFPP is undecidable.

• Question 5 (35): Let M and N be arbitrary Turing machines.

  • (a,10) Argue carefully that the following language is Turing recognizable: L_M,N consists of al triples of strings (w, x, y) such that either (a) is true or both (b) and (c) are true (that is, "a ∨ (b ∧ c)"):
    • (a) w is in L(M) ∩ L(N)
    • (b) x is not an accepting computation history of M on w
    • (c) y is not an accepting computation history of N on w

  • (b,15) Argue carefully that the pair (M, N) is in the language EQ_TM if and only if the Turing machine recognizing L_M,N is in the language ALL_TM. (Here we assume that every string represents some triple of strings, so to be in ALL_TM means accepting all possible triples.)

  • (c,10) Rephrase the result of part (b) as a mapping reduction. What does this mapping reduction (and the results from lecture) imply about the language ALL_TM?

• Question 6 (30): This problem uses the definition of the language COMMON_X for a computing model X, given at the start of the exam above.

  • (a,10) Indicate how a multitape Turing machine can decide the language COMMON_DFA.

  • (b,10) Prove that the language COMMON_CFG is Turing recognizable. (I would ask you whether it's decidable, but I'm not sure...)

  • (c,10) Prove that the language COMMON_TM is Turing recognizable but not Turing decidable. You may use any method and any facts known from lecture about the other languages defined for TM's at the start of the exam.

• Question 7 (20): Here are two problems involving variants of the Post Correspondence Problem, which is defined above at the start of the exam:

  • (a,10) Define UNARY-PCP to be the subset of PCP where the instance's alphabet Σ has one letter (and there is a match). Prove that UNARY-PCP is decidable. (This is Problem 5.17 in Sipser, which we didn't do this term -- obviously you can't just quote the fact that it's an exercise as your answer.)

  • (b,10) Let SINGLE-PCP be the subset of PCP where the instance has a match in which no domino is used more than once. Prove that SINGLE-PCP is decidable.

Last modified 2 April 2010