Revision #5 Authors: Vikraman Arvind, Abhranil Chatterjee, Partha Mukhopadhyay

Accepted on: 25th February 2024 12:22

Downloads: 94

Keywords:

Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen and Makam (2017)] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result of Derksen and Makam (2017).

Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson (2016), these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind, Chatterjee, and Mukhopadhyay (2022)].

In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.

Revision #4 Authors: Vikraman Arvind, Abhranil Chatterjee, Partha Mukhopadhyay

Accepted on: 25th February 2024 12:21

Downloads: 81

Keywords:

Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen and Makam (2017)] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result of Derksen and Makam (2017).

Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson (2016), these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind, Chatterjee, and Mukhopadhyay (2022)].

In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.

Revision #3 Authors: Vikraman Arvind, Abhranil Chatterjee, Partha Mukhopadhyay

Accepted on: 25th February 2024 12:21

Downloads: 64

Keywords:

Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen and Makam (2017)] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result of Derksen and Makam (2017).

Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson (2016), these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind, Chatterjee, and Mukhopadhyay (2022)].

In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.

Revision #2 Authors: Vikraman Arvind, Abhranil Chatterjee, Partha Mukhopadhyay

Accepted on: 25th February 2024 11:51

Downloads: 105

Keywords:

Rational Identity Testing (RIT) is the decision problem of determining whether or not a given noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg et al., 2016; Ivanyos et al., 2018; Hamada and Hirai, 2021], and a randomized polynomial-time black-box algorithm [Derksen and Makam, 2017] via singularity testing of linear matrices over the free skew field.

Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson, this question has seen very limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind et al., 2022].

In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for *all* rational formulas of polynomial size.

Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.

*Changes in abstract

*Comment on the use of division algebra

*Fixed some typos

Revision #1 Authors: Vikraman Arvind, Abhranil Chatterjee, Partha Mukhopadhyay

Accepted on: 15th November 2023 15:40

Downloads: 144

Keywords:

A white-box quasi-NC RIT algorithm has been added.

TR23-147 Authors: Vikraman Arvind, Abhranil Chatterjee, Partha Mukhopadhyay

Publication: 1st October 2023 14:56

Downloads: 225

Keywords:

Rational Identity Testing (RIT) is the decision problem of determining whether or not a given noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg et al., 2016; Ivanyos et al., 2018; Hamada and Hirai, 2021], and a randomized polynomial-time black-box algorithm [Derksen and Makam, 2017] via singularity testing of linear matrices over the free skew field.

Designing a subexponential-time deterministic RIT algorithm in black-box is a major open problem in this area. Despite being open for several years, this question has seen very limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind et al., 2022].

In this paper, we settle this problem and obtain a deterministic quasipolynomial-time RIT algorithm for the general case in the black-box setting. Our algorithm uses ideas from the theory of finite dimensional division algebras, algebraic complexity theory, and the theory of generalized formal power series.

Duplicate copies have been created for our submission. Consider Revision #5 as the final revised version. We have added a comment on the use of division algebra and corrected a few typos over Revision #1.