lect03

lect03 - 6.841 Advanced Complexity Theory Feb 11, 2009...

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6.841 Advanced Complexity Theory Feb 11, 2009 Lecture 3 Lecturer: Madhu Sudan Scribe: Debmalya Panigrahi In today’s lecture, we will focus on non-uniform models of computation . In non-uniform computation, we have a different gadget/program/machine for each input size for a given problem. Specifically, in this lecture, we will focus on circuit families (which are equivalent to deterministic Turing Machines (DTMs) with advice strings), branching programs and formulas , and compare them. Finally, we will study Neciporuk’s lower bound on the succinctness of branching programs. 1 Circuits and Circuit Families A circuit is a directed acyclic graph (DAG), where the nodes are of three kinds: input nodes , logic gates and an output node . The goal of a circuit with n input nodes is to compute a boolean function f : { 0 , 1 } n → { 0 , 1 } 1 in the following way: if a boolean vector ¯ x = ( x 1 ,x 2 ,...,x n ) is fed to the n input nodes, then the logic gates compute the function f x ) and output it at the output node. A basis is a set of logic gates that can be used to compute any boolean function. A circuit C has two main complexity measures: The size of circuit C , denoted by SIZE( C ) is the number of edges in C . 2 For a function f , C - SIZE( f ) = min C computes f SIZE( C ). C - SIZE( f ) depends on the basis of gates used in the circuit, but only upto a constant factor. The depth of circuit C , denoted by DEPTH( C ) is the longest path in C . The following is a set of simple facts about circuits: Given a circuit C of size S , evaluating C x ) for an input vector ¯ x takes O ( S 2 ) time. To simulate a DTIME ( t ( n )) TM on an input of length n , we need a circuit of size O ( t ( n ) 2 ). For any function
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This note was uploaded on 04/02/2010 for the course CS 6.841 taught by Professor Madhusudan during the Spring '09 term at MIT.

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lect03 - 6.841 Advanced Complexity Theory Feb 11, 2009...

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