- Answer the problems on the exam pages.
- There are seven problems, for 120 total points. Probable scale is somewhere around A=105, C=70.
- If you need extra space use the back of a page.
- No books, notes, calculators, or collaboration.
- The first four problems are true/false, with five points for the correct boolean answer and up to five for a convincing justification (proof, counterexample, quotation of result from lecture, etc.)

Q1: 10 points Q2: 10 points Q3: 10 points Q4: 10 points Q5: 20 points Q6: 30 points Q7: 30 points Total: 120 points

A few inconsequential typos were corrected on 6 April.

**Question 1 (10):***(True/false with justification)*There exists a language X such that X = L(M) for some three-tape Turing machine M, but X is not the language of any two-tape Turing machine.**Question 2 (10):***(True/false with justification)*If A and B are two languages such that A ≤_{m}B and B ≤_{m}A, then A is TR if and only if B is TR.**Question 3 (10):***(True/false with justification)*The language {(M,G): M is a DFA, G is a CFG, and L(M) = L(G)} is TD.**Question 4 (10):***(True/false with justification)*The set of all Turing machines that accept themselves is not a TR language.**Question 5 (20):**This question asks you to describe two deterministic two-tape Turing machines. In each case be somewhat specific about what the machine writes on each tape and how it decides what to write.- (a,10) Describe a deterministic two-tape Turing
machine M such that L(M) = {a
^{n}: n is prime}. - (b,10) Describe a deteministic two-tape Turing
machine such that L(M) = {w ∈ {0,1}
^{*}: w represents a prime number in binary} (Hint: Use your solution to part (a).)

- (a,10) Describe a deterministic two-tape Turing
machine M such that L(M) = {a
**Question 6 (30):**Two strings u and v over the same alphabet Σ are called**anagrams**if for every letter a in Σ, the number of a's in u equals the number of a's in v.- (a,10) Prove that the language {(u,v): u and v are anagrams} is TD for any alphabet Σ.
- (b,20) Define the
**anagram Post Correspondence Problem**or**APCP**to be the set of PCP dominoes P such that there exists a nonempty sequence of dominoes from P where the string made from the tops of each domino and the string made from the bottoms of each domino are anagrams. For Σ = {0,1}, prove that APCP is Turing decidable.

**Question 7 (30):**Consider the following three languages -- in each case M and N represent binary representations of Turing machines with binary input alphabet:- AND-PAIR = {(M,N): M ∈ L(N) ∧ N ∈ L(M)}
- OR-PAIR = {(M,N): M ∈ L(N) ∨ N ∈ L(M)}
- IFF-PAIR = {(M,N): M ∈ L(N) ↔ N ∈ L(M)}

Answer the following two questions:

- (a,15) Two of these three languages are TR. Say which two are TR and prove that each is TR. (The third is not TR, but you need not prove that.)
- (b,15) None of these three languages is TD. Pick any of the three and prove that it is not TD.

Last modified 6 April 2008