- Answer the problems on the exam pages.
- There are seven problems for 120 total points. Probable scale is around A=100, C=70 but will be determined after I grade the exam.
- If you need extra space use the back of a page.
- No books, notes, calculators, or collaboration.
- The first five 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: 40 points Total: 120 points

**Question 1 (10):***True or false with justification:*Define a**bizarro Turing machine**to be like an ordinary deterministic one-tape Turing machine except that the tape alphabet Γ is countably infinite, so that Γ = {a_{1}, a_{2}, a_{3},...}. The input alphabet Σ is still finite. The transition function δ from (Q × Γ) to (Q × Γ × {L, R}) can no longer be given by a finite lookup table, but it is computable by an ordinary Turing machine (that always halts).Given this definition, there exists a bizarro Turing machine B whose language L(B) = {w ∈ Σ

^{*}: B halts in its accepting state on input w} is not TR.**Question 2 (10):***True or false with justification:*It is possible to have a countably infinite family S_{1}, S_{2}, S_{3},... of countably infinite sets, such that for any positive integers i and j where i ≠ j, the intersection S_{i}∩ S_{j}is a nonempty finite set. (Recall that a set is countably infinite if and only if it can be put into one-to-one correspondence with the positive integers.)**Question 3 (10):***True or false with justification:*For every ordinary Turing machine M with input alphabet Σ and every string w in Σ^{*}, the set of accepting computation histories of M on w is a regular language.**Question 4 (10):***True or false with justification:*If f is a one-to-one function from {0, 1}^{*}to {0, 1}^{*}, and n is a positive integer, then there must exist a string w of length n such that the length of f(w) is at least n.**Question 5 (10):***True or false with justification:*Recall that an LBA is a deterministic one-tape Turing machine that never leaves the portion of its tape that originally contained its input. Define ALL_{LBA}to be the set {<M>: M is an LBA with input alphabet Σ and L(M) = Σ^{*}}. Then the language ALL_{LBA}is not TD.**Question 6 (30):**A**Latin square**is a d by d matrix of numbers, each in the set {1, 2, 3,..., d}, such that each row and each column has each possible number exactly once. (For example, a solution to a standard Sudoku puzzle is a 9 by 9 Latin square that also has an additional property.) Let LS be the following language: A string in {a, b}^{*}is in LS if it is of the form a^{i1}ba^{i2}ba^{i3}b... a^{in}b where n is equal to d^{2}for some positive integer d, each i_{j}is in the set {1, 2, 3,..., d}, and the numbers i_{1}, i_{2},..., i_{n}form a Latin square if arranged into a d by d matrix in row-major order. Describe a deterministic k-tape Turing machine deciding the language LS, where k is a fixed positive integer of your choice.**Question 7 (40):**Let Z be the language {<M_{1}, M_{2}>: M_{1}and M_{2}are Turing machines and there exists a string w such that M_{1}and M_{2}both accept w and M_{1}does so in fewer steps than does M_{2}}. Here are four questions concerning Z. Your answers to some may imply answers to others --- it is fine to quote your solutions to other problems, but beware of circular reasoning!- (a, 10) Is the language Z a TR language? Prove your answer.
- (b, 10) Is the language Z a TD language? Prove your answer.
- (c, 10) Is it true that A
_{TM}≤_{m}Z? Prove your answer by either describing a mapping reduction or proving that it cannot exist. - (d, 10) Is it true that
A
_{TM}≤_{m}Z? Prove your answer by either describing a mapping reduction or proving that it cannot exist.

Last modified 15 April 2009