ps3 - CS 151 Complexity Theory Spring 2011 Problem Set 3...

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CS 151 Complexity Theory Spring 2011 Problem Set 3 Out: April 14 Due: April 21 Reminder: you are encouraged to work in groups of two or three; however you must turn in your own write-up and note with whom you worked. You may consult the course notes and the text (Papadimitriou). Please attempt all problems. To facilitate grading, please turn in each problem on a separate sheet of paper and put your name on each sheet. Do not staple the separate sheets. 1. Barrington’s Theorem. We begin with a few words of background before the actual problem. A branching program is a directed acyclic graph in which each node is labelled by a variable x i , one of these is designated as the start node and there are two special nodes labelled “accept” and “reject.” All of the nodes labelled with variables have exactly two outgoing edges, one labelled “0” and the other labelled “1”. An input x = x 1 x 2 . . . x n defines a path from the start node to the accept or reject node as follows: at every node labelled x i , we follow the outgoing edge whose label coincides with the value of x i in the input. If we reach the accept node, the input is accepted; if we reach the reject node, the input is rejected. Polynomial-size branching programs capture L/poly in the same way that polynomial-size circuits capture P/poly . In this problem we will consider a very restricted subclass of polynomial-size branching pro- grams. With the exception of the accept and reject nodes, all of the nodes will be divided into levels 1 , ℓ 2 , . . . ℓ m , with each level containing at most 5 nodes

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