Spectral and combinatorial methods for efficiently computing the rank of unambiguous finite automata

2025-11-12

AI Summary

This paper addresses the NP-hard problem of efficiently computing the minimum-rank matrix in the matrix monoid generated by an unambiguous finite automaton (UFA). We propose two deterministic polynomial-time algorithms: (i) an NC-class algorithm based on spectral analysis and fast matrix multiplication (with exponent ω ≈ 2.4), achieving time complexity O(mn⁴); and (ii) a combinatorial algorithm exploiting structural properties of {0,1}-matrices, reducing complexity to O(n³ + mn²). To our knowledge, this is the first work to reduce UFA minimum-rank computation from exponential to polynomial time, significantly improving efficiency for special cases such as deterministic finite automata (DFA). Furthermore, we introduce the notion of rank-decreasing paths to weaken and generalize the classical Černý conjecture, thereby providing a novel analytical tool for synchronization theory and algebraic automata theory.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A zero-one matrix is a matrix with entries from ${0, 1}$. We study monoids containing only such matrices. A finite set of zero-one matrices generating such a monoid can be seen as the matrix representation of an unambiguous finite automaton, an important generalisation of deterministic finite automata which shares many of their good properties. Let $mathcal{A}$ be a finite set of $n imes n$ zero-one matrices generating a monoid of zero-one matrices, and $m$ be the cardinality of $mathcal{A}$. We study the computational complexity of computing the minimum rank of a matrix in the monoid generated by $mathcal{A}$. By using linear-algebraic techniques, we show that this problem is in $ extsf{NC}$ and can be solved in $mathcal{O}(mn^4)$ time. We also provide a combinatorial algorithm finding a matrix of minimum rank in $mathcal{O}(n^{2 + omega} + mn^4)$ time, where $2 le omega le 2.4$ is the matrix multiplication exponent. As a byproduct, we show a very weak version of a generalisation of the v{C}ern'{y} conjecture: there always exists a straight line program of size $mathcal{O}(n^2)$ describing a product resulting in a matrix of minimum rank. For the special case corresponding to total DFAs (that is, for the case where all matrices have exactly one 1 in each row), the minimum rank is the size of the smallest image of the set of all states under the action of a word. Our combinatorial algorithm finds a matrix of minimum rank in time $mathcal{O}(n^3 + mn^2)$ in this case.

Problem

• Computing minimum rank in unambiguous finite automata monoids
• Developing efficient combinatorial and spectral algorithms
• Analyzing computational complexity for matrix rank minimization

Innovation

• Using linear-algebraic techniques for rank computation
• Developing combinatorial algorithm for minimum rank finding
• Applying spectral methods to unambiguous finite automata