Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symbolic Listings as Computation

Hamilton Sawczuk, Edinah Gnang

TL;DR

The paper develops an algebraic framework of differential computers that encode Boolean functions as symbolic listings and manipulate them with differential operators on polynomials, linking this approach to Chow rank and low-depth arithmetic circuits. It introduces additive and symbolic listings, defines Chow rank as a complexity measure, and demonstrates how Boolean computations can be implemented with differential operations; a key result is the monomial non-overlapping lemma, which establishes that totally non-overlapping polynomials of degree $m\ge2$ have Chow rank exactly $n$. The authors also show Chow-rank non-decreasing transformations from graphs to functional graphs and discuss how these ideas apply to functional inputs, Hamiltonian-cycle related polynomials, and determinant-based constructions to bypass certain hardness barriers. Overall, the work connects Boolean function complexity, graph representations, and low-depth circuit complexity within an elegant algebraic model, with potential implications for understanding arithmetic-formula complexity and matrix-related computations.

Abstract

We propose an algebraic model of computation which formally relates symbolic listings, complexity of Boolean functions, and low depth arithmetic circuit complexity. In this model algorithms are arithmetic formula expressing symbolic listings of YES instances of Boolean functions, and computation is executed via partial differential operators. We consider the Chow rank of an arithmetic formula as a measure of complexity and establish the Chow rank of multilinear polynomials with totally non-overlapping monomial support. We also provide Chow rank non-decreasing transformations from sets of graphs to sets of functional graphs.

Symbolic Listings as Computation

TL;DR

The paper develops an algebraic framework of differential computers that encode Boolean functions as symbolic listings and manipulate them with differential operators on polynomials, linking this approach to Chow rank and low-depth arithmetic circuits. It introduces additive and symbolic listings, defines Chow rank as a complexity measure, and demonstrates how Boolean computations can be implemented with differential operations; a key result is the monomial non-overlapping lemma, which establishes that totally non-overlapping polynomials of degree have Chow rank exactly . The authors also show Chow-rank non-decreasing transformations from graphs to functional graphs and discuss how these ideas apply to functional inputs, Hamiltonian-cycle related polynomials, and determinant-based constructions to bypass certain hardness barriers. Overall, the work connects Boolean function complexity, graph representations, and low-depth circuit complexity within an elegant algebraic model, with potential implications for understanding arithmetic-formula complexity and matrix-related computations.

Abstract

We propose an algebraic model of computation which formally relates symbolic listings, complexity of Boolean functions, and low depth arithmetic circuit complexity. In this model algorithms are arithmetic formula expressing symbolic listings of YES instances of Boolean functions, and computation is executed via partial differential operators. We consider the Chow rank of an arithmetic formula as a measure of complexity and establish the Chow rank of multilinear polynomials with totally non-overlapping monomial support. We also provide Chow rank non-decreasing transformations from sets of graphs to sets of functional graphs.
Paper Structure (9 sections, 10 theorems, 57 equations, 3 figures)

This paper contains 9 sections, 10 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Symbolic Model of Computation
  3. Symbolic Listings
  4. The Chow Rank
  5. Differential Computers
  6. Functional Computers
  7. Totally Non– Overlapping Polynomials
  8. Monomial Non– Overlapping Lemma
  9. Applications to Totally Non– Overlapping Polynomials

Key Result

Lemma 2.11

Let $\mathcal{G}_n$ and $\Tilde{\mathcal{G}}_n$ denote the set of all graphs and functional graphs respectively on $n$ vertices. Then there exist two efficient transformations $T_f:\raisebox{.15\baselineskip}{\Large$\wp$}(\mathcal{G}_n)\rightarrow\raisebox{.15\baselineskip}{\Large$\wp$}(\Tilde{\math

Figures (3)

  • Figure 2.1: Pictorial listing of functional graphs on two vertices.
  • Figure 2.2: $\sum \prod \sum$– formula.
  • Figure 2.3: Example of the transformations in \ref{['lem:trans']} showing $G$, $T_{a_{00}a_{10}}(G)$, and $T(G)$ from left to right.

Theorems & Definitions (34)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Remark 2.10
  • ...and 24 more