CS154 Spring 2011Practice Final

To study for the final I would suggest you: (1) Know how to do (by heart) all the practice problems. (2) Go over your notes at least three times. Second and third time try to see how much you can remember from the first time. (3) Go over the homework problems. (4) Try to create your own problems similar to the ones I have given and solve them. (5) Skim the relevant sections from the book. (6) If you want to study in groups, at this point you are ready to quiz each other. The practice final is below. Here are some facts about the actual final: (a) It is comprehensive (b) It is closed book, closed notes. Nothing will be permitted on your desk except your pen (pencil) and test. (c) You should bring photo ID. (d) There will be more than one version of the test. Each version will be of comparable difficulty. (e) It is 10 problems, 6 problems will be on material since the midterm, four problems will come from the topics covered prior to the midterm. (f) Two problems will be exactly (less typos) off of the practice final, and one will be off of practice midterm

1. Give the formal definition of a Turing Machine that converts a string over `0`'s and `1`'s to a string of the same length involving just the symbols `a`. Draw a state diagram for this machine. Write down the sequence of configurations this machine would go through on the input `1001`.
2. Suppose we had a two dimension Turing Machine. i.e., rather than use a 1D tape of it uses a 2D sheet of paper of unbounded size. Give a formal definition for such a machine, and for such a machine to recognize a language. Show that a usual TM can simulate a 2D Turing Machine.
3. Why do `mbox{TIME}(n)` and `cup _k mbox{TIME}(k cdot n))` have the same languages? Is linear time on a 1-tape machine the same as on a 2-tape machine? If not, give an example (give intuition why the example works, you do not need to give a formal proof).
4. Give a RAM that takes as its input an integer `n` and outputs `5n`.
5. For each of the following languages, either give an algorithm to show its decidable or prove that its not: `mbox{ALL}_{DFA}`, `\mbox{EQ}_{REX}`, `\mbox{EQ}_{CFG}`.
6. What is a Universal Turing Machine? Give a brief description of how one might build such a UTM.
7. Prove that `n^2` is a proper complexity function. State the deterministic time hierarchy theorem as presented in class.
8. Come up with a bounded version of the `\mbox{REGULAR}_{TM}` problem from class such that this variant is now NP complete under Karp-reductions. Prove this.
9. Prove an upper bound on the number of distinct configurations that a LBA with `q` states, and `g` symbol alpabet can go through on an input of length `n`.
10. Prove using Rice's Theorem that the set of encodings of Turing machines that only accepts strings with Kolmogorov complexity less than 10 is undecidable.