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Maria Francis

Publications and source records attributed to Maria Francis.

9 recordsLinked to original sources

Challenging the Abilities of Large Language Models in Italian: a Community Initiative

The rapid progress of Large Language Models (LLMs) has transformed natural language processing and broadened its impact across research and society. Yet, systematic evaluation of these models, especially for languages beyond English, remains limited. "Challenging the Abilities of LAnguage Models in ITAlian" (CALAMITA) is a large-scale collaborative benchmarking initiative for Italian, coordinated under the Italian Association for Computational Linguistics. Unlike existing efforts that focus on leaderboards, CALAMITA foregrounds methodology: it federates more than 80 contributors from academia, industry, and the public sector to design, document, and evaluate a diverse collection of tasks, covering linguistic competence, commonsense reasoning, factual consistency, fairness, summarization, translation, and code generation. Through this process, we not only assembled a benchmark of over 20 tasks and almost 100 subtasks, but also established a centralized evaluation pipeline that supports heterogeneous datasets and metrics. We report results for four open-weight LLMs, highlighting systematic strengths and weaknesses across abilities, as well as challenges in task-specific evaluation. Beyond quantitative results, CALAMITA exposes methodological lessons: the necessity of fine-grained, task-representative metrics, the importance of harmonized pipelines, and the benefits and limitations of broad community engagement. CALAMITA is conceived as a rolling benchmark, enabling continuous integration of new tasks and models. This makes it both a resource -- the most comprehensive and diverse benchmark for Italian to date -- and a framework for sustainable, community-driven evaluation. We argue that this combination offers a blueprint for other languages and communities seeking inclusive and rigorous LLM evaluation practices.

cs.CL

On Two Signature Variants Of Buchberger's Algorithm Over Principal Ideal Domains

Signature-based algorithms have brought large improvements in the performances of Gr\"obner bases algorithms for polynomial systems over fields. Furthermore, they yield additional data which can be used, for example, to compute the module of syzygies of an ideal or to compute coefficients in terms of the input generators. In this paper, we examine two variants of Buchberger's algorithm to compute Gr\"obner bases over principal ideal domains, with the addition of signatures. The first one is adapted from Kandri-Rody and Kapur's algorithm, whereas the second one uses the ideas developed in the algorithms by L. Pan (1989) and D. Lichtblau (2012). The differences in constructions between the algorithms entail differences in the operations which are compatible with the signatures, and in the criteria which can be used to discard elements. We prove that both algorithms are correct and discuss their relative performances in a prototype implementation in Magma.

cs.SC

Signature-based M\"oller's algorithm for strong Gr\"obner bases over PIDs

Signature-based algorithms are the latest and most efficient approach as of today to compute Gr\"obner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention of several authors. In this paper, we present a signature-based version of M\"oller's classical variant of Buchberger's algorithm for computing strong Gr\"obner bases over Principal Ideal Domains (or PIDs). It ensures that the signatures do not decrease during the algorithm, which makes it possible to apply classical signature criteria for further optimization. In particular, with the F5 criterion, the signature version of M\"oller's algorithm computes a Gr\"obner basis without reductions to zero for a polynomial system given by a regular sequence. We also show how Buchberger's chain criterion can be implemented so as to be compatible with the signatures. We prove correctness and termination of the algorithm. Furthermore, we have written a toy implementation in Magma, allowing us to quantitatively compare the efficiency of the various criteria for eliminating S-pairs.

cs.SC

A Signature-based Algorithm for computing Computing Gr\"obner Bases over Principal Ideal Domains

Signature-based algorithms have become a standard approach for Gr\"obner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gr\"obner bases over commutative integral domains. It is adapted from a general version of M\"oller's algorithm (1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

cs.SC

On Ideal Lattices and Gröbner Bases

In this paper, we draw a connection between ideal lattices and Gröbner bases in the multivariate polynomial rings over integers. We study extension of ideal lattices in $\mathbb{Z}[x]/\langle f \rangle$ (Lyubashevsky \& Micciancio, 2006) to ideal lattices in $\mathbb{Z}[x_1,\ldots,x_n]/\mathfrak{a}$, the multivariate case, where $f$ is a polynomial in $\mathbb{Z}[X]$ and $\mathfrak{a}$ is an ideal in $\mathbb{Z}[x_1,\ldots,x_n]$. Ideal lattices in univariate case are interpreted as generalizations of cyclic lattices. We introduce a notion of multivariate cyclic lattices and we show that multivariate ideal lattices are indeed a generalization of them. We show that the fact that existence of ideal lattice in univariate case if and only if $f$ is monic translates to short reduced Gröbner basis (Francis \& Dukkipati, 2014) of $\mathfrak{a}$ is monic in multivariate case. We, thereby, give a necessary and sufficient condition for residue class polynomial rings over $\mathbb{Z}$ to have ideal lattices. We also characterize ideals in $\mathbb{Z}[x_1,\ldots,x_n]$ that give rise to full rank lattices.

cs.SC

On Gröbner Bases and Krull Dimension of Residue Class Rings of Polynomial Rings over Integral Domains

Given an ideal $\mathfrak{a}$ in $A[x_1, \ldots, x_n]$, where $A$ is a Noetherian integral domain, we propose an approach to compute the Krull dimension of $A[x_1,\ldots,x_n]/\mathfrak{a}$, when the residue class polynomial ring is a free $A$-module. When $A$ is a field, the Krull dimension of $A[x_1,\ldots,x_n]/\mathfrak{a}$ has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. For a Noetherian integral domain, $A$ we introduce the notion of combinatorial dimension of $A[x_1, \ldots,x_n]/\mathfrak{a}$ and give a Gröbner basis method to compute it for residue class polynomial rings that have a free $A$-module representation w.r.t. a lexicographic ordering. For such $A$-algebras, we derive a relation between Krull dimension and combinatorial dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$. An immediate application of this relation is that it gives a uniform method, the first of its kind, to compute the dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$ without having to consider individual properties of the ideal. For $A$-algebras that have a free $A$-module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Gröbner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such $A$-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to $A$-algebras with a free $A$-module representation w.r.t. a degree compatible ordering as well.

cs.SC

Border Bases for Polynomial Rings over Noetherian Rings

The theory of border bases for zero-dimensional ideals has attracted several researchers in symbolic computation due to their numerical stability and mathematical elegance. As shown in (Francis & Dukkipati, J. Symb. Comp., 2014), one can extend the concept of border bases over Noetherian rings whenever the corresponding residue class ring is finitely generated and free. In this paper we address the following problem: Can the concept of border basis over Noetherian rings exists for ideals when the corresponding residue class rings are finitely generated but need not necessarily be free modules? We present a border division algorithm and prove the termination of the algorithm for a special class of border bases. We show the existence of such border bases over Noetherian rings and present some characterizations in this regard. We also show that certain reduced Gröbner bases over Noetherian rings are contained in this class of border bases.

cs.SC

On Ideal Lattices, Gröbner Bases and Generalized Hash Functions

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Ideal lattices are ideals in the residue class ring, $\mathbb{Z}[x]/\langle f \rangle$ (here $f$ is a monic polynomial), and cryptographic primitives have been built based on these objects. As ideal lattices in the univariate case are generalizations of cyclic lattices, we introduce the notion of multivariate cyclic lattices and show that multivariate ideal lattices are indeed a generalization of them. Based on multivariate ideal lattices, we establish the existence of collision resistant hash functions using Gröbner basis techniques. For the construction of hash functions, we define a worst case problem, shortest substitution problem w.r.t. an ideal in $\mathbb{Z}[x_1,\ldots, x_n]$, and establish hardness results using functional fields.

cs.SC

Reduced Gröbner Bases and Macaulay-Buchberger Basis Theorem over Noetherian Rings

In this paper, we extend the characterization of $\mathbb{Z}[x]/\ < f \ >$, where $f \in \mathbb{Z}[x]$ to be a free $\mathbb{Z}$-module to multivariate polynomial rings over any commutative Noetherian ring, $A$. The characterization allows us to extend the Gröbner basis method of computing a $\Bbbk$-vector space basis of residue class polynomial rings over a field $\Bbbk$ (Macaulay-Buchberger Basis Theorem) to rings, i.e. $A[x_1,\ldots,x_n]/\mathfrak{a}$, where $\mathfrak{a} \subseteq A[x_1,\ldots,x_n]$ is an ideal. We give some insights into the characterization for two special cases, when $A = \mathbb{Z}$ and $A = \Bbbk[θ_1,\ldots,θ_m]$. As an application of this characterization, we show that the concept of border bases can be extended to rings when the corresponding residue class ring is a finitely generated, free $A$-module.

cs.SC