Symbolic Listings as Computation: Algebraic Model of Computation Relating Boolean Functions and Arithmetic Circuits

Concetti Chiave

Algebraic model linking symbolic listings, Boolean functions, and arithmetic circuit complexity.

Sintesi

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.

Statistiche

"Chow rank of P2 is exactly n."
"The Chow rank of P=c is exactly n."
"The Chow rank of PCn is exactly n."

Citazioni

"We propose an algebraic model...and computation."
"Chow rank as a measure...of truth table compressibility."
"Recent depth reduction results motivate our focus..."

Approfondimenti chiave tratti da

Symbolic Listings as Computation

by Hamilton Saw... alle arxiv.org 03-26-2024

https://arxiv.org/pdf/2402.15885.pdf

Domande più approfondite

How do differential computers compare to traditional Turing machines?

In the context provided, differential computers are proposed as an algebraic model of computation that lies at the intersection of three pillars of computational theory: Boolean algebra, Turing machines, and switching circuits. Unlike traditional Turing machines that operate based on discrete steps and transitions between states, differential computers execute computations using partial differential operators on polynomials with monomial support corresponding to YES instances of Boolean functions. This approach allows for a more algebraic implementation of restricted classes of Turing machines.
Differential computers can be seen as an attempt to bridge the gap between circuit complexity and classical computation models like Turing machines. By expressing algorithms as arithmetic formulae representing symbolic listings of YES instances of Boolean functions, these systems provide a different perspective on computational complexity. The use of partial differential operators in this model offers a unique way to analyze arithmetic circuit complexity and its relation to the complexity of Boolean functions.

What are the implications of the exponential separation in additive listings?

The exponential separation observed in additive listings has significant implications for understanding computational complexity and decision problems versus their counting analogues. By allowing non-binary coefficients in additive listings while maintaining restrictions to roots of unity, it becomes possible to distinguish decision problems from their counting counterparts effectively.
This distinction is crucial because it prevents unbounded computational power from being introduced into models while still providing flexibility in representation. The choice of roots of unity as coefficients helps maintain structure and control over computations without sacrificing expressiveness or analytical capabilities within the framework.
The exponential separation highlights how subtle variations in representations can lead to profound differences in computational power and problem-solving approaches within mathematical frameworks like additive listings.

How can the concept of functional computers be applied outside computational theory?

Functional computers offer a structured way to represent graphs where each vertex has out-degree equal to one, essentially mapping vertices onto specific functions within graph structures. Beyond computational theory, this concept finds applications across various domains:

Network Analysis: In network analysis or social network modeling, functional graphs can represent relationships where each node interacts directly with another node (out-degree equals one). This simplifies complex networks into more manageable structures for analysis.

Biological Systems: Functional graphs could model biological systems where components interact linearly or hierarchically without feedback loops.

Supply Chain Management: In supply chain management optimization or logistics planning, functional graphs might depict streamlined processes without redundant flows or inefficiencies.

Process Automation: Functional graph representations could streamline process automation by mapping sequential tasks with clear dependencies between them.

By applying concepts from functional computing outside theoretical realms into practical scenarios involving interconnected systems or processes governed by linear interactions among components/entities/variables may enhance efficiency through simplified modeling techniques derived from such structured representations.