Andrew Ryzhikov

Stefan Kiefer, Andrew Ryzhikov

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 and $\mathcal{O}(n^2)$ space. We also provide a combinatorial algorithm finding a matrix of minimum rank in $\mathcal{O}(mn^4)$ time and $\mathcal{O}(n^3)$ space. 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.

Stefan Kiefer, Andrew Ryzhikov

In this paper we investigate the maximum size of finite semigroups of rational $n \times n$ matrices, with the goal of shedding more light on their structure. Such semigroups provide a rich generalisation of transition monoids of unambiguous (and, in particular, deterministic) finite automata. While in general such semigroups can be arbitrarily large in terms of $n$, a classical result of Schützenberger from 1962 implies an upper bound of $2^{O(n^2 \log n)}$ for irreducible semigroups. A semigroup of rational matrices is called irreducible if the only subspaces of $\mathbb{Q}^n$ that are invariant for all matrices in the semigroup are $\mathbb{Q}^n$ and the subspace consisting only of the zero vector. Irreducible matrix semigroups can be viewed as the building blocks of general matrix semigroups, and as such play an important role in mathematics and computer science. From the point of view of automata theory, they can be seen as a generalisation of strongly connected weighted automata. Using a very different technique from that of Schützenberger, we improve the upper bound on the cardinality to $3^{n^2}$. This is the main result of the paper. The bound is in some sense tight, as we show that there exists, for every $n$, a finite irreducible semigroup with $3^{\lfloor n^2/4 \rfloor}$ rational matrices. Our main result also leads to an improvement of a bound, due to Almeida and Steinberg, on the mortality threshold of finite semigroups of rational matrices. The mortality threshold is a number $\ell$ such that if the zero matrix is in the semigroup, then the zero matrix can be written as a product of at most $\ell$ matrices from any subset that generates the semigroup.