Related papers: Linearizing the Word Problem in (some) Free Fields
We study the language-theoretic aspects of the word problem, in the sense of Duncan & Gilman, of free products of semigroups and monoids. First, we provide algebraic tools for studying classes of languages known as super-AFLs, which…
Gr\"obner bases, in their noncommutative version, and word reversing are methods for solving the word problem of a presented monoid, and both rely on iteratively completing the initial list of relations. Simple examples may suggest to…
This paper focuses on the noiseless complete dictionary learning problem, where the goal is to represent a set of given signals as linear combinations of a small number of atoms from a learned dictionary. There are two main challenges faced…
This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear…
We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…
We consider commutative regular and context-free grammars, or, in other words, Parikh images of regular and context-free languages. By using linear algebra and a branching analog of the classic Euler theorem, we show that, under an…
We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be…
Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
We investigate the properties of formal languages expressible in terms of formulas over quantifier-free theories of word equations, arithmetic over length constraints, and language membership predicates for the classes of regular, visibly…
We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus…
The question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words. Recently, Aleksi Saarela has introduced a new approach to the problem that is based on linear-algebraic…
Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an…
Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. Based on the Gr\"obner basis theory for $A$ and for free modules over $A$, an elimination theory…
These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following…
We argue that reducing nonlinear programming problems to a simple canonical form is an effective way to analyze them, specially when the problem is degenerate and the usual linear independence hypothesis does not hold. To illustrate this…
We present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms. Beginning from basic results developed by Riehm, our solution to this problem hinges on the classification of…
Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…
Automatic Word problem solving has always posed a great challenge for the NLP community. Usually a word problem is a narrative comprising of a few sentences and a question is asked about a quantity referred in the sentences. Solving word…
A new algorithm is presented for computing a direct solution to a system of consistent linear equations. It produces a minimum norm particular solution, a generalized inverse (of type {124}), and a null space projection operator. In…