Discrete modelling frameworks of Biological networks can be divided in two distinct categories: Boolean and Multi-valued. Although Multi-valued networks are more expressive for qualifying the regulatory behaviours modelled by more than two values, the ability to automatically convert them to Boolean network with an equivalent behaviour breaks down the fundamental borders between the two approaches. Theoretically investigating the conversion process provides relevant insights into bridging the gap between them. Basically, the conversion aims at finding a Boolean network bisimulating a Multi-valued one. In this article, we investigate the bisimilar conversion where the Boolean integer coding is a parameter that can be freely modified. Based on this analysis, we define a computational method automatically inferring a bisimilar Boolean network from a given Multi-valued one.

In this paper, we study functions of the roots of a univariate polynomial in which the roots have a given multiplicity structure $\mu$. Traditionally, root functions are studied via the theory of symmetric polynomials; we extend this theory to $\mu$-symmetric polynomials. We were motivated by a conjecture from Becker et al.~(ISSAC 2016) about the $\mu$-symmetry of a particular root function $D^+(\mu)$, called D-plus. To investigate this conjecture, it was desirable to have fast algorithms for checking if a given root function is $\mu$-symmetric. We designed three such algorithms: one based on Gr\"{o}bner bases, another based on preprocessing and reduction, and the third based on solving linear equations. We implemented them in Maple and experiments show that the latter two algorithms are significantly faster than the first.

In this paper, we work on the notion of k-synchronizability: a system is k-synchronizable if any of its executions, up to reordering causally independent actions, can be divided into a succession of k-bounded interaction phases. We show two results (both for mailbox and peer-to-peer automata): first, the reachability problem is decidable for k-synchronizable systems; second, the membership problem (whether a given system is k-synchronizable) is decidable as well. Our proofs fix several important issues in previous attempts to prove these two results for mailbox automata.

We present $\mathtt{bimEX}$, a Mathematica package for exact computations in 3$+$1 bimetric relativity. It is based on the $\mathtt{xAct}$ bundle, which can handle computations involving both abstract tensors and their components. In this communication, we refer to the latter case as concrete computations. The package consists of two main parts. The first part involves the abstract tensors, and focuses on how to deal with multiple metrics in $\mathtt{xAct}$. The second part takes an ansatz for the primary variables in a chart as the input, and returns the covariant BSSN bimetric equations in components in that chart. Several functions are implemented to make this process as fast and user-friendly as possible. The package has been used and tested extensively in spherical symmetry and was the workhorse in obtaining the bimetric covariant BSSN equations and reproducing the bimetric 3$+$1 equations in the spherical polar chart.

The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.

We present a non-commutative algorithm for the multiplication of a block-matrix by its transpose over C or any finite field using 5 recursive products. We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions. Finally we propose space and time efficient schedules that enable us to provide fast practical implementations for higher-dimensional matrix products.

Machine Learning techniques have become pervasive across a range of different applications, and are now widely used in areas as disparate as recidivism prediction, consumer credit-risk analysis and insurance pricing. The prevalence of machine learning techniques has raised concerns about the potential for learned algorithms to become biased against certain groups. Many definitions have been proposed in the literature, but the fundamental task of reasoning about probabilistic events is a challenging one, owing to the intractability of inference. The focus of this paper is taking steps towards the application of tractable models to fairness. Tractable probabilistic models have emerged that guarantee that conditional marginal can be computed in time linear in the size of the model. In particular, we show that sum product networks (SPNs) enable an effective technique for determining the statistical relationships between protected attributes and other training variables. If a subset of these training variables are found by the SPN to be independent of the training attribute then they can be considered `safe' variables, from which we can train a classification model without concern that the resulting classifier will result in disparate outcomes for different demographic groups. Our initial experiments on the `German Credit' data set indicate that this processing technique significantly reduces disparate treatment of male and female credit applicants, with a small reduction in classification accuracy compared to state of the art. We will also motivate the concept of "fairness through percentile equivalence", a new definition predicated on the notion that individuals at the same percentile of their respective distributions should be treated equivalently, and this prevents unfair penalisation of those individuals who lie at the extremities of their respective distributions.

We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated gcd's. Furthermore, it is shown that the number of polynomials in the condition is optimal and the degree of polynomials is smaller than that in the previous condition based on repeated gcd's.

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a---possibly randomized---verification algorithm that proves the correctness of each output. The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size $N$, meaning $N$ times a factor $N^{o(1)}$, i.e., a factor $N^{\eta(N)}$ with $\lim\_{N\to \infty} \eta(N)$ $=$ $0$. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an $n\times n$ dense integer matrix $A$. Our certificates can be verified in Monte-Carlo bit complexity $(n^2 \log\|A\|)^{1+o(1)}$, where $\log\|A\|$ is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc.\ ISSAC 2011] subject to computational hardness assumptions. Second, we give algorithms that compute certificates for the rank of sparse or structured $n\times n$ matrices over an abstract field, whose Monte Carlo verification complexity is $2$ matrix-times-vector products $+$ $n^{1+o(1)}$ arithmetic operations in the field. For example, if the $n\times n$ input matrix is sparse with $n^{1+o(1)}$ non-zero entries, our rank certificate can be verified in $n^{1+o(1)}$ field operations. This extends also to integer matrices with only an extra $\|A\|^{1+o(1)}$ factor. All our certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic. The validity of our verification procedure is subject to standard computational hardness assumptions from cryptography.

This document contains the computer generated part of the proof of the main result in [12], giving a complete and general list of tree representations corresponding to exceptional modules over the path algebra of the canonically oriented Euclidean quiver $\widetilde{\mathbb{E}}_6$. The proofs (involving induction and symbolic computation with block matrices) were partially generated by a purposefully developed computer software, outputting in a detailed step-by-step fashion as if written "by hand". Tree representations are exhibited using matrices involving only the elements 0 and 1, and all representations enlisted in this document remain valid over any base field (since the symbolic block matrix computations present inhere hold in any field).

A small subset of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0, \infty)$. Such sequences are known as Stieltjes moment sequences. This article focuses on some classical sequences in enumerative combinatorics, denoted $Av(\mathcal{P})$, and counting permutations of $\{1, 2, \ldots, n \}$ that avoid some given pattern $\mathcal{P}$. For increasing patterns $\mathcal{P}=(12\ldots k)$, we recall that the corresponding sequences, $Av(123\ldots k)$, are Stieltjes moment sequences, and we explicitly find the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool. We show that the generating functions of the sequences $\, Av(1234)$ and $\, Av(12345)$ correspond, up to simple rational functions, to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian $\, _2F_1$ hypergeometric function, respectively to an order-two linear differential operator acting on the square of a classical modular form given as a pullback of a $\, _2F_1$ hypergeometric function. We demonstrate that the density function for the Stieltjes moment sequence $Av(123\ldots k)$ is closely, but non-trivially, related to the density attached to the distance traveled by a walk in the plane with $k-1$ unit steps in random directions. Finally, we study the challenging case of the $Av(1324)$ sequence and give compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, we show how rigorous lower bounds on the growth constant of this sequence can be constructed, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give an estimate of the (unknown) growth constant.