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Chris Pollett
>Old Classes
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CS154 Spring 2006Practice FinalThe practice exam will appear one week before the exam.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 three times. Second and third time try to see how huch 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) The final will be in class Friday, May 19 from 9:45am to 12:00pm for Sec3 and Monday, May 22 from 12:15pm to 2:30pm for Sec5. (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) If your cell-phone or beeper goes off you will be excused from the test at that point and graded on what you have done till your excusal. (f) One problem (less typos) on the actual test will be from the practice test. [Practice Final Student Solutions-PDF] 1. Prove that the power set of a set S is not of the same cardinality as S. 2. Assume the alphabet for ATM contains the same symbols as used to typeset the textbook. Assume that the TMs of ATM are all encoded over this alphabet using seven tuples as in the book. Give an explicit string which is in ATM and an explicit string which is not in ATM. 3. Consider the language {<M> | M halts on all inputs beginning with a 1}. Prove this language is undecidable by giving a mapping reduction of the halting problem to it. Then prove it is undecidable by applying Rice's Theorem. 4. Consider the language {<M> | M is a TM such that L(M) is context free}. Prove this language is undecidable without appealing to Rice's Theorem. 5. Let |x| denote the length of a string x. Consider the language EPTIME={ <M, p> | M is TM and p is an encoding of a polynomial such that the set of strings M accepts within p(|x|) steps is empty. Here x is supposed to be an input to M and |x| is the length of x. } Show that this language is undecidable. 6. Give a decidable language H such that HALTTM = {< M, x> | ∃ w, < w, M, x > ∈ H }. 7. Show that the variant of PCP where we require that each tile be played at most once is decidable. 8. For each of the following give an example as well a proof of the correctness of your example. (a) a language which is Turing-recognizable but not co-Turing Recognizable, (b) a language which is co-Turing recognizable but not Turing recognizable, and (c) a language which is neither Turing Recognizable nor co-Turing Recognizable. 9. Prove using the recursion theorem that HALTTM is undecidable. 10. Prove incompressible strings of every length exist. |