Languages Accepted by Weighted Restarting Automata*

Weighted restarting automata have been introduced to study quantitative aspects of computations of restarting automata. In earlier works we studied the classes of functions and relations that are computed by weighted restarting automata. Here we use them to define classes of formal languages by restricting the weight associated to a given input word through an additional requirement. In this way, weighted restarting automata can be used as language acceptors. First, we show that by using the notion of acceptance relative to the tropical semiring, we can avoid the use of auxiliary symbols. Furthermore, a certain type of word-weighted restarting automata turns out to be equivalent to non-forgetting restarting automata, and another class of languages accepted by word-weighted restarting automata is shown to be closed under the operation of intersection. This is the first result that shows that a class of languages defined in terms of a quite general class of restarting automata is closed under intersection. Finally, we prove that the restarting automata that are allowed to use auxiliary symbols in a rewrite step, and to keep on reading after performing a rewrite step can be simulated by regular-weighted restarting automata that cannot do this.

Sparse data embedding and prediction by tropical matrix factorization

Background Matrix factorization methods are linear models, with limited capability to model complex relations. In our work, we use tropical semiring to introduce non-linearity into matrix factorization models. We propose a method called Sparse Tropical Matrix Factorization () for the estimation of missing (unknown) values in sparse data. Results We evaluate the efficiency of the method on both synthetic data and biological data in the form of gene expression measurements downloaded from The Cancer Genome Atlas (TCGA) database. Tests on unique synthetic data showed that approximation achieves a higher correlation than non-negative matrix factorization (), which is unable to recover patterns effectively. On real data, outperforms on six out of nine gene expression datasets. While assumes normal distribution and tends toward the mean value, can better fit to extreme values and distributions. Conclusion is the first work that uses tropical semiring on sparse data. We show that in certain cases semirings are useful because they consider the structure, which is different and simpler to understand than it is with standard linear algebra.

Remarks on a Tropical Key Exchange System

We consider a key-exchange protocol based on matrices over a tropical semiring which was recently proposed in [2]. We show that a particular private parameter of that protocol can be recovered with a simple binary search, rendering it insecure.

Weak Cost Register Automata are Still Powerful

We consider one of the weakest variants of cost register automata over a tropical semiring, namely copyless cost register automata over [Formula: see text] with updates using [Formula: see text] and increments. We show that this model can simulate, in some sense, the runs of counter machines with zero-tests. We deduce that a number of problems pertaining to that model are undecidable, namely equivalence, upperboundedness, and semilinearity. In particular, the undecidability of equivalence disproves a conjecture of Alur et al. from 2012. To emphasize how weak these machines are, we also show that they can be expressed as a restricted form of linearly-ambiguous weighted automata.

Bounds for the Completely Positive Rank of a Symmetric Matrix over a Tropical Semiring

In this paper, an upper bound for the CP-rank of a matrix over a tropical semiring is obtained, according to the vertex clique cover of the graph prescribed by the positions of zero entries in the matrix. The graphs that beget the matrices with the lowest possible CP-ranks are studied, and it is proved that any such graph must have its diameter equal to $2$.

THE FINITE BASIS PROBLEM FOR THE MONOID OF TWO-BY-TWO UPPER TRIANGULAR TROPICAL MATRICES

For each positive $n$, let $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ denote the identity obtained from the Adjan identity $(xy)(yx)(xy)(xy)(yx)\approx (xy)(yx)(yx)(xy)(yx)$ by substituting $(xy)\rightarrow (x_{1}x_{2}\ldots x_{n})$ and $(yx)\rightarrow (x_{n}\ldots x_{2}x_{1})$. We show that every monoid which satisfies $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ for each positive $n$ and generates a variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid $U_{2}(\mathbb{T})$ (respectively, $U_{2}(\overline{\mathbb{Z}})$) of two-by-two upper triangular tropical matrices over the tropical semiring $\mathbb{T}=\mathbb{R}\cup \{-\infty \}$ (respectively, $\overline{\mathbb{Z}}=\mathbb{Z}\cup \{-\infty \}$) is nonfinitely based.