Keskeiset käsitteet
Algebraic model linking symbolic listings, Boolean functions, and arithmetic circuit complexity.
Tiivistelmä
This article proposes an algebraic model that connects symbolic listings, Boolean functions, and low-depth arithmetic circuit complexity. It discusses the Chow rank as a measure of complexity and introduces differential computers. The content covers functional graphs, additive listings, and their relation to computational complexity.
Introduction
Boole's contribution to Boolean algebra.
Equivalence of computability theories by Godel, Church, and Turing.
Shannon's implementation of Boolean algebra via switching circuits.
Differential Computers
Computation through differential operators on polynomials.
Importance of Chow rank in measuring complexity.
Relationship between arithmetic circuit complexity and Boolean functions.
A Symbolic Model of Computation
Definition of symbolic adjacency matrix AG.
Functional graphs representation using monomial edge listing MG.
The Chow Rank
Definition and significance of the Chow rank in polynomial decomposition.
Functional Computers
Simplification by considering functional graphs.
Totally Non-Overlapping Polynomials
Monomial non-overlapping lemma for polynomial analysis.
Applications to Totally Non-Overlapping Polynomials
Lower bounds on the Chow ranks of specific polynomials.
Acknowledgments
Support from the United States Office of Naval Research.
Gratitude to individuals for insightful discussions.
Tilastot
"Chow rank of P2 is exactly n."
"The Chow rank of P=c is exactly n."
"The Chow rank of PCn is exactly n."
Lainaukset
"We propose an algebraic model...and computation."
"Chow rank as a measure...of truth table compressibility."
"Recent depth reduction results motivate our focus..."