Midterm Computational Complexity

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Midterm Computational Complexity March 23, 2009 You are not allowed to collaborate or refer to any book or online source (except possibly to look at the definitions). Due on Monday, March 30 in the class. 1. A language L is called unary if L ⊆ 1∗. Show that if a unary language is NP-complete, then P=NP. Hint: If there is a polytime reduction from SAT to a unary language, then this reduction will map a boolean formula of size n to a string 1i with i ≤ poly(n). Use this and downward self-reducibility of SAT to efficiently find a satisfying assignment. Start out with a recursive brute-force (i.e. depth-first-pruning) algorithm for finding a satisfying assignment. Modify it so that the algorithm runs in polytime. 2. For any positive integer k, show that there is a language in PH with circuit-complexity Ω(nk). In fact you can exhibit such a language in some fixed finite level of PH, say Σ5. 3. Let BPL be the class of languages accepted by a polytime randomized logspace machine with two sided error. Show that BPL ⊆ P. Hint: Construct the configuration graph for a BPL machine, with two outgoing edges from each node, representing the action taken by the machine depending upon the random bit read. The acceptance probability is then the probability that a random walk from the start vertex ends in the accept vertex. How do you calculate this probability efficiently? Note: A randomized logspace machine has workspace of size O(log(n)) and access to poly(n) random bits. The random bits are available on a read-once random tape. This means that once the algorithm reads 1 a random bit, it cannot go back and read the same bit again from the random tape. A language L is accepted by such a machine if strings in L are accpetd with probability at least 0:9 and strings not in L are accpeted with probability at most 0:1. 2.

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The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that deﬁnes commands to typeset (almost) all of the complexity classes you’ll ﬁnd here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/.
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Solutions to Exercises
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