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.