Trading Determinism for Noncommutativity in Edmonds' Problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913310600855552 |
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| author | Arvind, V. Chatterjee, Abhranil Mukhopadhyay, Partha |
| author_facet | Arvind, V. Chatterjee, Abhranil Mukhopadhyay, Partha |
| contents | Let $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ be a partitioned set of variables such that the variables in each part $X_i$ are noncommuting but for any $i\neq j$, the variables $x\in X_i$ commute with the variables $x'\in X_j$. Given as input a square matrix $T$ whose entries are linear forms over $\mathbb{Q}\langle{X}\rangle$, 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 $\mathbb{Q}\langle{X}\rangle$ [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 \emph{weighted} automata (for constant $k$) resolving a long-standing open problem [Harju and Karhum"{a}ki(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 \emph{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)]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trading Determinism for Noncommutativity in Edmonds' Problem Arvind, V. Chatterjee, Abhranil Mukhopadhyay, Partha Computational Complexity Formal Languages and Automata Theory Let $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ be a partitioned set of variables such that the variables in each part $X_i$ are noncommuting but for any $i\neq j$, the variables $x\in X_i$ commute with the variables $x'\in X_j$. Given as input a square matrix $T$ whose entries are linear forms over $\mathbb{Q}\langle{X}\rangle$, 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 $\mathbb{Q}\langle{X}\rangle$ [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 \emph{weighted} automata (for constant $k$) resolving a long-standing open problem [Harju and Karhum"{a}ki(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 \emph{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)]. |
| title | Trading Determinism for Noncommutativity in Edmonds' Problem |
| topic | Computational Complexity Formal Languages and Automata Theory |
| url | https://arxiv.org/abs/2404.07986 |