2024/04/11 by V. Arvind, Arvind, V., Abhranil Chatterjee +3
Computer Science · Mathematics · #Algebraic structures and combinatorial models #Coding theory and cryptography #Computational Complexity (cs.CC) #FOS: Computer and information sciences #Formal Languages and Automata Theory (cs.FL) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2404.07986
openalex publication_date 2024/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X=X1\sqcup X2\sqcup…\sqcup Xk be a partitioned set of variables such that the variables in each part Xi are noncommuting but for any i≠ j, the variables x∈ Xi commute with the variables x'∈ Xj. Given as input a square matrix T whose entries are linear forms over ℚ⟨X⟩, we consider the problem of checking if T is invertible or not over the universal skew field of fractions of the partially commutative polynomial ring ℚ⟨X⟩ [Klep-Vinnikov-Volcic (2020)]. In this paper, we design a deterministic polynomial-time algorithm for this problem for constant k. The special case k=1 is the noncommutative Edmonds' problem (NSINGULAR) which has a deterministic polynomial-time algorithm by recent results [Garg-Gurvits-Oliveira-Wigderson (2016), Ivanyos-Qiao-Subrahmanyam (2018), Hamada-Hirai (2021)]. En-route, we obtain the first deterministic polynomial-time algorithm for the equivalence testing problem of k-tape weighted automata (for constant k) resolving a long-standing open problem [Harju and Karhum"aki(1991), Worrell (2013)]. Algebraically, the equivalence problem reduces to testing whether a partially commutative rational series over the partitioned set X is zero or not [Worrell (2013)]. Decidability of this problem was established by Harju and Karhumäki (1991). Prior to this work, a randomized polynomial-time algorithm for this problem was given by Worrell (2013) and, subsequently, a deterministic quasipolynomial-time algorithm was also developed [Arvind et al. (2021)].