🤖 AI Summary
Rational Identity Testing (RIT) is the fundamental problem of determining whether a noncommutative rational formula identically vanishes over the free skew field; its black-box variant lacks efficient deterministic algorithms, and its parallel complexity has remained open for decades. This work constructs, for the first time, a quasipolynomial-size hitting set for arbitrary polynomial-sized noncommutative rational formulas, leveraging tools from algebraic complexity theory, singularity analysis of matrices over the free skew field, and structured hitting set design. This yields a deterministic quasipolynomial-time algorithm for black-box RIT—the first such result. Furthermore, it implies a deterministic quasipolynomial-depth uniform circuit family (i.e., a quasipolynomial-time (mathsf{NC}) upper bound) for white-box RIT. These advances significantly extend the theoretical frontier of noncommutative identity testing and clarify its parallel feasibility.
📝 Abstract
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.
