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A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results

Vikraman Arvind, Pushkar S Joglekar

TL;DR

This work establishes that key noncommutative computation problems, namely noncommutative Rational Identity Testing ($\mathrm{RIT}$) and noncommutative rank ($\mathrm{ncRANK}$), admit deterministic NC reductions from multivariate to bivariate instances, leveraging Cohn’s embedding and Higman linearization. It proves an NC-Turing reduction from bivariate $\mathrm{RIT}$ to bivariate $\mathrm{ncRANK}$, implying that if bivariate $\mathrm{ncRANK}$ lies in deterministic NC, then both multivariate $\mathrm{ncRANK}$ and multivariate $\mathrm{RIT}$ do as well. The paper develops NC-depth reduction for noncommutative formulas (with and without inverses), and provides parallelized algorithms to perform Higman linearization, enabling NC reductions from general multivariate to bivariate settings. Together, these results reveal a close, parallel-complexity bridge between noncommutative identity testing and rank problems, while highlighting open questions about unconditional NC-depth reduction and explicit NC algorithms for bivariate cases. The contributions illuminate a path toward fully NC solutions contingent on resolving remaining depth and embedding technicalities for noncommutative rational formulas.

Abstract

We study the noncommutative rank problem, ncRANK, of computing the rank of matrices with linear entries in $n$ noncommuting variables and the problem of noncommutative Rational Identity Testing, RIT, which is to decide if a given rational formula in $n$ noncommuting variables is zero on its domain of definition. Motivated by the question whether these problems have deterministic NC algorithms, we revisit their interrelationship from a parallel complexity point of view. We show the following results: 1. Based on Cohn's embedding theorem \cite{Co90,Cohnfir} we show deterministic NC reductions from multivariate ncRANK to bivariate ncRANK and from multivariate RIT to bivariate RIT. 2. We obtain a deterministic NC-Turing reduction from bivariate $\RIT$ to bivariate ncRANK, thereby proving that a deterministic NC algorithm for bivariate ncRANK would imply that both multivariate RIT and multivariate ncRANK are in deterministic NC.

A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results

TL;DR

This work establishes that key noncommutative computation problems, namely noncommutative Rational Identity Testing () and noncommutative rank (), admit deterministic NC reductions from multivariate to bivariate instances, leveraging Cohn’s embedding and Higman linearization. It proves an NC-Turing reduction from bivariate to bivariate , implying that if bivariate lies in deterministic NC, then both multivariate and multivariate do as well. The paper develops NC-depth reduction for noncommutative formulas (with and without inverses), and provides parallelized algorithms to perform Higman linearization, enabling NC reductions from general multivariate to bivariate settings. Together, these results reveal a close, parallel-complexity bridge between noncommutative identity testing and rank problems, while highlighting open questions about unconditional NC-depth reduction and explicit NC algorithms for bivariate cases. The contributions illuminate a path toward fully NC solutions contingent on resolving remaining depth and embedding technicalities for noncommutative rational formulas.

Abstract

We study the noncommutative rank problem, ncRANK, of computing the rank of matrices with linear entries in noncommuting variables and the problem of noncommutative Rational Identity Testing, RIT, which is to decide if a given rational formula in noncommuting variables is zero on its domain of definition. Motivated by the question whether these problems have deterministic NC algorithms, we revisit their interrelationship from a parallel complexity point of view. We show the following results: 1. Based on Cohn's embedding theorem \cite{Co90,Cohnfir} we show deterministic NC reductions from multivariate ncRANK to bivariate ncRANK and from multivariate RIT to bivariate RIT. 2. We obtain a deterministic NC-Turing reduction from bivariate to bivariate ncRANK, thereby proving that a deterministic NC algorithm for bivariate ncRANK would imply that both multivariate RIT and multivariate ncRANK are in deterministic NC.
Paper Structure (25 sections, 19 theorems, 25 equations)

This paper contains 25 sections, 19 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. The Edmonds' Problem and $\textrm{ncRANK}$
  3. This paper: overview of results and proofs
  4. Preliminaries
  5. Noncommutative Rational Formulas
  6. The Free Skew Field
  7. Noncommutative Rank
  8. The Algorithmic Problems of Interest
  9. The complexity class $\mathsf{NC}$ and $\mathsf{NC}$ reductions
  10. Cohn's Embedding Theorem
  11. The Reduction from multivariate $\textrm{RIT}$ to bivariate $\textrm{RIT}$
  12. Reduction from $n$-variate $\textrm{ncRANK}_{poly}$ to $2$-variate $\textrm{ncRANK}$
  13. Depth reduction for noncommutative formulas without divisions
  14. $\mathsf{NC}$ Reduction from $\textrm{RIT}$ to bivariate $\textrm{ncRANK}$
  15. Depth reduction for noncommutative formulas with inversion gates
  16. ...and 10 more sections

Key Result

Proposition 2.5

Cohnfir Let $M$ be an $n\times n$ matrix over the ring $\mathbb{F}\langle X \rangle$. Then

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: inner rank
  • Definition 2.4: full matrices
  • Proposition 2.5
  • Theorem 2.7: Cohn's embedding theorem
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 25 more