6.841 Advanced Complexity Theory
Feb 17, 2009
Lecture 4
Lecturer: Madhu Sudan
Scribe: Adam Spanbauer
1
Introduction
Today’s lecture will focus on the proof of Barrington’s Theorem:
Barrington’s Theorem: All polynomialsize formulas have an O(1) width braching program.
Barrington proved, in particular, that all polynomialsize formulae have a width 5 branching program. His
proof used a lot of advanced algebra, so the proof presented here will show that all polynomialsize formulae
have a width 8 branching program. The proof was discovered by BenOr and Cleve.
2
Summary of concepts covered in previous lectures
2.1
Nonuniform complexity
For each input length, we are allowed to do something different – the mechanism for doing this is assigning
an
advice
string to each integer, one for each input length, and giving it to the TM along with the input.
The languages that can be decided by a polynomial time TM with polynomial length advice is called P/POLY
and is equivalent to polynomial sized circuits.
An interesting unresolved question in complexity theory is whether
NP
*
P/POLY
.
2.2
Circuits
Given by a Directed Acyclic Graph (DAG). Has source nodes for inputs, a collection of gates, connections,
and an output. Circuit gates can have arbitrary fanout, which means that the output from a gate can be
used arbitrarily many times as an input elsewhere.
2.3
Branching programs
Branching programs are a weaker model of computation than circuits.
BPs roughly correspond to space complexity, that is, they can be thought of as trying to measure non
uniformly what the spacecomplexity is.
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 Spring '09
 MadhuSudan
 Computational complexity theory, Jaguar Racing

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