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

TL;DR

This paper introduces efficient spectral and combinatorial algorithms for computing the minimum rank of matrices generated by unambiguous finite automata, improving computational complexity and providing new theoretical insights.

Contribution

It presents novel algorithms with improved complexity for determining the minimum rank in monoids of zero-one matrices representing unambiguous finite automata.

Findings

The problem is in NC and solvable in O(mn^4) time.

A combinatorial algorithm finds minimum rank matrices in O(n^{2 + } + mn^4) time.

A weak version of a generalized ernfd conjecture is demonstrated.

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 $n \times n$ 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 $\textsf{NC}$ 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.