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

TL;DR

The paper addresses the problem of computing the minimum real rank of matrices in zero-one matrix monoids generated by a finite set, which correspond to unambiguous finite automata. It introduces a weight-based linear-algebraic framework and a complementary combinatorial approach, proving the rank problem lies in NC$^2$ with a near-optimal time bound of $O(mn^4)$ and providing a combinatorial method with $O(n^{2+\omega}+mn^4)$ time to also construct a minimum-rank matrix. For the special case of total DFAs, it achieves a tighter constructive bound of $O(n^3+mn^2)$ time. A byproduct is a very weak, but informative, Černý-like bound via straight-line programs of size $O(n^2)$ describing a product achieving minimum rank. These results strengthen the links between synchronising automata, weighted automata, and matrix semigroups, and open up new directions for parallelizable algorithms in this area.

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 $\mathcal{O}(n^{2 + ω} + mn^4)$ time, where $2 \le ω\le 2.4$ is the matrix multiplication exponent. As a byproduct, we show a very weak version of a generalisation of the Černý 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.

Figures (5)

• Figure 1: The flower automaton of the code $X = \{aa, aab, aba, abab\}$ (left), two adjacent interpretations of $ababa$ over $X$ (top right), and two disjoint interpretations of $bababaaba$ over $X$ (bottom right). Note that this code is not complete, but still illustrates all the discussed properties.
• Figure 2: The configuration that is forbidden in a UFA.
• Figure 3: $[a \cdot 3] = M(a) [3] = [\{1, 2\}]$ is a column; $[2 \cdot a]^T = [2]^T M(a) = [\{1, 3\}]^T$ is a row.
• Figure 4: An example of $\mathcal{A}$ (left), and a part of the underlying digraph $G^{(2)}$ of its square automaton (right). Merging vertices are doubly circled. The edges of $T$ for $p = 7$ are represented by dotted edges, and these edges are labelled with one of the letters labelling the corresponding transition in $\mathcal{A}^{(2)}$. Furthermore, we have $\rho_1 = (7, 1) \to (8, 2) \to (1, 3) \to (4, 4)$, $\rho_2 = (5, 7) \to (6, 8) \to (3, 1) \to (4, 4)$, $\rho_3 = (7, 3) \to (8, 4) \to (1, 5) \to (4, 6) \to (5, 7)$. A set-SLP encoding the labels of these path is presented in \ref{['ex:set-slp']}, with $u_i$ labelling the path $\rho_i$, $i \in \{1, 2, 3\}$.
• Figure :

Theorems & Definitions (64)

• Theorem 4.1: Césari Cesari74
• Corollary 4.2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• proof
• Theorem 4.6
• Lemma 4.7
• proof
• Proposition 4.8