2026-07-22T13:50:12Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17145 2024-03-06T10:59:05Z ddc:004 open_access

Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time Arvind, V. Chatterjee, Abhranil Mukhopadhyay, Partha Rational Identity Testing Black-box Derandomization Cyclic Division Algebra Matrix coefficient realization theory Hrubeš and Wigderson [Hrubeš and Wigderson, 2015] initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, there are deterministic polynomial-time algorithms due to Garg, Gurvits, Oliveira, and Wigderson [Ankit Garg et al., 2016] and Ivanyos, Qiao, and Subrahmanyam [Ivanyos et al., 2018]. A central open problem in this area is to design an efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including concepts from matrix coefficient realization theory [Volčič, 2018] and properties of cyclic division algebras [T.Y. Lam, 2001]. En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas [Michael A. Forbes and Amir Shpilka, 2013] inside a cyclic division algebra of small index. Schloss Dagstuhl – Leibniz-Zentrum für Informatik V. Arvind and Abhranil Chatterjee and Partha Mukhopadhyay 2022 Is Part Of LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2022.23 urn:nbn:de:0030-drops-171451 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.23 eng https://creativecommons.org/licenses/by/4.0/legalcode