Core Concepts

Deterministic NC reductions from multivariate noncommutative rank and rational identity testing problems to their bivariate counterparts.

Abstract

The paper studies the noncommutative rank (ncRANK) problem and the noncommutative rational identity testing (RIT) problem. It presents the following key results:
Deterministic NC reduction from multivariate ncRANK to bivariate ncRANK:
Leverages Cohn's embedding theorem to obtain a reduction that preserves the noncommutative rank.
Requires additional NC algorithms for formula depth reduction and Higman linearization.
Deterministic NC reduction from multivariate RIT to bivariate RIT:
Again uses Cohn's embedding theorem to map multivariate rational formulas to bivariate ones while preserving equivalence.
The reduction is logspace computable.
Deterministic NC-Turing reduction from RIT to bivariate ncRANK:
Builds on the Hrubes-Wigderson reduction from RIT to ncRANK.
Parallelizes the depth reduction of noncommutative rational formulas using an oracle for bivariate ncRANK.
The reductions establish that if bivariate ncRANK has a deterministic NC algorithm, then both multivariate RIT and multivariate ncRANK would also be in deterministic NC.

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Key Insights Distilled From

by Vikraman Arv... at **arxiv.org** 04-26-2024

Deeper Inquiries

Establishing connections or reductions between noncommutative complexity classes and their commutative counterparts can provide valuable insights into the relationships between these classes and the computational power of noncommutative models compared to commutative ones. Some possible connections or reductions that can be explored include:
Reductions from noncommutative complexity classes to commutative ones: It would be interesting to investigate if problems in noncommutative complexity classes, such as noncommutative rank or noncommutative identity testing, can be efficiently reduced to their commutative counterparts. This could help in understanding the impact of noncommutativity on the complexity of computational problems.
Equivalence between noncommutative and commutative complexity classes: Exploring whether certain noncommutative complexity classes are equivalent to commutative complexity classes under certain conditions or restrictions could provide insights into the inherent differences in computational power between these models.
Comparative analysis of complexity hierarchies: Analyzing the hierarchy of noncommutative complexity classes and comparing it to the hierarchy of commutative complexity classes could reveal similarities or differences in the computational capabilities of these models.
By investigating these connections and reductions, researchers can gain a deeper understanding of the relationship between noncommutative and commutative complexity classes and the implications for computational complexity theory.

Achieving depth reduction of noncommutative rational formulas unconditionally in NC, without requiring an oracle for bivariate ncRANK, is a challenging problem in computational complexity theory. While existing algorithms, such as the one described in the context above, rely on oracle access to bivariate ncRANK for depth reduction, it is an open question whether such a reduction can be achieved without this oracle.
To address this challenge, researchers could explore alternative approaches or techniques for depth reduction of noncommutative rational formulas. This may involve developing novel algorithms that do not rely on oracle access to bivariate ncRANK but still guarantee efficient depth reduction. Additionally, investigating the inherent properties of noncommutative formulas and their depth complexity could provide insights into the feasibility of achieving unconditional depth reduction in NC.
Overall, achieving unconditional depth reduction of noncommutative rational formulas in NC is a significant research goal that requires innovative algorithmic techniques and a deep understanding of noncommutative complexity theory.

There are several other natural noncommutative problems that can be efficiently reduced to their bivariate versions, offering valuable insights into their complexity and computational properties. Some examples of such problems include:
Noncommutative Polynomial Identity Testing: Reducing the noncommutative polynomial identity testing problem to its bivariate version could provide insights into the impact of noncommutativity on the complexity of identity testing problems. This reduction could help in understanding the computational differences between noncommutative and commutative polynomial identities.
Noncommutative Matrix Rank Problems: Extending reductions from noncommutative matrix rank problems to their bivariate counterparts can shed light on the complexity of rank computations in noncommutative settings. By exploring these reductions, researchers can gain a better understanding of the computational challenges posed by noncommutative matrices.
Noncommutative Rational Formula Equivalence: Investigating reductions from noncommutative rational formula equivalence problems to bivariate versions can offer insights into the complexity of testing equivalence in noncommutative settings. Understanding the reductions between these problems can provide valuable information about the computational differences between noncommutative and commutative rational formulas.
By exploring reductions from various noncommutative problems to their bivariate versions, researchers can uncover fundamental insights into the computational properties and complexities of noncommutative models.

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