hwk1 - CSE 200 Computability and Complexity Homework 1...

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Unformatted text preview: CSE 200 Computability and Complexity Homework 1 Models of Computation, Efficient Computation, and Undecidability Due Wednesday, April 21 April 9, 2010 1. Consider a machine model with a single two-dimensional tape, where locations are indexed by pairs (x, y) of positive integers, where each location can store a symbol from a finite set , and the input is written on positions (1, 1), ..(n, 1). The tape head starts at position (1, 1), and in one step, the machine can move the tape head to the left one (decrementing x), to the right one (incrementing x), up one (incrementing y) or down 1, (decrementing y). Show that the set of languages accepted in polynomial-time on such a 2D-TM is the same as for a normal TM. 2. Remember that for multi-tape TM's, the input tape is usually considered read-only, and we only count the amount of the other tapes used to measure the amount of memory S(n) an algorithm uses on inputs of length n. (Thus, it makes sense for S(n) < n.) For the language L = {x0 n x||x| = n} from class, prove that any k-tape TM algorithm that decides membership in L in time T (n) and memory S(n) must satisfy the time-space tradeoff T (n)S(n) (n 2 ). 3. Show that a function f (x) from {0, 1} to {0, 1} is in F P (functions computable in polynomial time) if and only if there is a k so that |f (x)| O(|x| k ) and the language {(x, i, b)|i |f (x)| and the i'th bit of f (x) is b} is in P . 4. Let L = {< M, w > |M is a Turing Machine program so that there is an input y {0, 1} so that M halts on y in at most 100|y| steps and M (y) = w } Is L recursive? Is L recursively enumerable? Is L co-r.e.? 5. Give an example of a language L that is neither R.E. nor co-R.E. Prove your answer correct. 1 ...
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