ps6 - CS 151 Complexity Theory Spring 2011 Problem Set 6...

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CS 151 Complexity Theory Spring 2011 Problem Set 6 Out: May 12 Due: May 19 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. The following problem comes from Learning Theory, where the VC-dimension gives impor- tant information about the difficulty of learning a given concept. Given a collection S = { S 1 , S 2 , . . . , S m } of subsets of a finite set U , the VC dimension of S is the size of the largest set X U such that for every X X , there is an i for which S i X = X (we say that X is shattered by S ). A Boolean circuit C that computes a function f : { 0 , 1 } m ×{ 0 , 1 } n → { 0 , 1 } succinctly represents a collection S of 2 m subsets of U = { 0 , 1 } n as follows: the set S i consists of exactly those elements x for which
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ps6 - CS 151 Complexity Theory Spring 2011 Problem Set 6...

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