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# lect04 - 6.841 Advanced Complexity Theory Lecture 4...

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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 polynomial-size formulas have an O(1) width braching program. Barrington proved, in particular, that all polynomial-size formulae have a width 5 branching program. His proof used a lot of advanced algebra, so the proof presented here will show that all polynomial-size formulae have a width 8 branching program. The proof was discovered by Ben-Or and Cleve. 2 Summary of concepts covered in previous lectures 2.1 Non-uniform 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 fan-out, 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 space-complexity is.

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