Thermal convection in 2D soap film

1
Toward Optimal Tree Construction of Monotone
Functions
Miao Chen Department of Mathematics and Computer
Science Duquesne University
2
Outline

• Introduction
• Primary Conjecture
• Influence
• Secondary Conjecture
• Algorithm
• Results
• Acknowledgments

3
Introduction

• Boolean function f 0,1n?0,1
• Representations of f

• Disjunctive Normal Form (DNF)
• x1x2x2x3, x1x2x1
• Boolean Decision Tree

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Introduction

• Monotone function
• f(x)f(x ei), for all i and x

i.e. x00110101010011
ei00000000001000
x ei00110101011011

• Why is Monotone function important?

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Introduction

• An example of monotone function

x1x2x2x3
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Primary Conjecture

• In a decision tree of a monotone function f
• If vb is relevant in only one sub-tree w/
  rootvr,
• vr is relevant in both sub-trees w/ rootvb
  
• Then Opt_Tree_size(rootvr)
• Opt_Tree_size(rootvb)

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Influence
Influence of the ith variable vi Probability
of change of functions value when vi 0?1 or
vice versa.
Inf(x1)1/4 Inf(x2)3/4 Inf(x3)1/4
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Secondary Conjecture

• A monotone function f
• Infi( f )Infj( f ) for all vj?vi,
• Then an optimal-sized tree for f
• can be constructed with viroot.

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Algorithm
Max Inf.
Influence-based OPT size calculation
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Relation between primary secondary conjectures

• vr and vb have the relation in Primary
• ? ( f ) ( f )
• Primary conjecture fails
• ? secondary conjecture fails

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Algorithm
fi U fj fj?Fkfi fj 00 U 01 01 ? 0001
Generating monotone functions of n variables.
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Algorithm
Brute force OPT size calculation
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Results

• All monotone functions up to 6
• variables? no counterexamples
• Some interesting monotone
• functions of 7 and 8 variables?
• no counterexamples

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Acknowledgments

• Prof. Jeffrey Jackson
• Prof. Donald L. Simon
• Prof. Patrick Juola
• Prof. Kathleen Taylor