David Fernandez

Dynamical systems are abstract models of interaction between space and time. They are often used in fields such as physics and engineering to understand complex processes, but due to their general nature, they have found applications for studying computational processes, interaction in multi-agent systems, machine learning algorithms and other computer science related phenomena. In the vast majority of applications, a dynamical system consists of the action of a continuous `transition function' on a metric space. In this work, we consider decidable formal systems for reasoning about such structures. Spatial logics can be traced back to the 1940's, but our work follows a more dynamic turn that these logics have taken due to two recent developments: the study of the topological mu-calculus, and the the integration of linear temporal logic with logics based on the Cantor derivative. In this paper, we combine dynamic topological logics based on the Cantor derivative and the `next point in time' operators with an expressively complete fixed point operator to produce a combination of the topological mu-calculus with linear temporal logic. We show that the resulting logics are decidable and have a natural axiomatisation. Moreover, we prove that these logics are complete for interpretations on the Cantor space, the rational numbers, and subspaces thereof.

Dynamical systems are general models of change or movement over time with a broad area of applicability to many branches of science, including computer science and AI. Dynamic topological logic (DTL) is a formal framework for symbolic reasoning about dynamical systems. DTL can express various liveness and reachability conditions on such systems, but has the drawback that the only known axiomatisation requires an extended language. In this paper, we consider dynamic topological logic restricted to the class of scattered spaces. Scattered spaces appear in the context of computational logic as they provide semantics for provability and enjoy definable fixed points. We exhibit the first sound and complete dynamic topological logic in the original language of DTL. In particular, we show that the version of DTL based on the class of scattered spaces is finitely axiomatisable, and that the natural axiomatisation is sound and complete.

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. Unlike logics based on the topological closure operator, d-logics have not previously been studied in the framework of dynamical systems, which are pairs (X,f) consisting of a topological space X equipped with a continuous function f:X->X. We introduce the logics wK4C, K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d-semantics in this dynamical setting. In particular, we prove that wK4C is the d-logic of all dynamic topological systems, K4C is the d-logic of all T_D dynamic topological systems, and GLC is the d-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where f is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems wK4H, K4H and GLH. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d-logics. Furthermore, our result for GLC constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -- something known to be impossible over the class of all spaces.

The musical repertoire of processions of the Latin liturgy has had a singular treat- ment in the Western plainchant scenario. Scholars have focused their studies on liturgical music genres (antiphons, responsories, letanies, hymns,...) on issues related with their definitions and usages, based on a certain liturgical context; i.e., Mass or Divine Office. For this reason, this paper aims to summarize the relevant descriptive and functional aspects of these forms of liturgical music in its processional praxis, in the context of a research of a corpus, which could be inferred to the medieval processional books.

En este artículo se realiza una indagación hermenéutica en la que se articulará el concepto lacaniano de campo del Otro con la noción kelseniana de sistema jurídico. Se sostendrá que así como el campo del Otro tiene un carácter paradójico, como un lugar en el que se supone el lenguaje y como una falta inherente al mismo, el sistema jurídico puede entenderse de manera analógica, como un campo en el que se suponen las respuestas jurídicas, y como un lugar fundado en sus propias contradicciones. Como resultado, se obtuvo que el concepto de campo del Otro se puede observar en tres escenarios donde Kelsen identifica las inconsistencias del derecho, a saber, la dinámica jurídica, la ausencia de un metalenguaje en la interpretación y la Norma Fundamental. Al final, se concluirá que son precisamente estas inconsistencias las que permiten el funcionamiento del derecho.

The European Union’s Artificial Intelligence Act (AI Act) is a groundbreaking regulatory framework that integrates technical concepts and terminology from the rapidly evolving ecosystems of AI research and innovation into the legal domain. Precise definitions accessible to both AI experts and lawyers are crucial for the legislation to be effective. This paper provides an interdisciplinary analysis of the concepts of AI system, general purpose AI system, foundation model and generative AI across the different versions of the legal text (Commission proposal, Parliament position and Council General Approach) before the final political agreement. The goal is to help bridge the understanding of these key terms between the technical and legal communities and contribute to a proper implementation of the AI Act. We provide an analysis of the concept of AI system considering its scientific foundation and the crucial role that it plays in the regulation, which requires a sound definition both from legal and technical standpoints. We connect the outcomes of this discussion with the analysis of the concept of general purpose AI system and its evolution during the negotiations. We also address the distinct conceptual meanings of AI system vs AI model and explore the technical nuances of the term foundation model. We conclude that rooting the definition of foundation model to its general purpose capabilities following standardised evaluation methodologies appears to be most appropriate approach. Lastly, we tackle the concept of generative AI, arguing that definitions of AI system that include “content” as one of the system’s outputs already captures it, and concluding that not all generative AI is based on foundation models.

Goodstein’s theorem states that certain sequences based on exponential notation for the natural numbers are always finite. The result is independent of Peano arithmetic and is a prototypical example of a proof of termination by transfinite induction. A variant based instead on the Ackermann function has more recently been proposed by Arai, Fernández-Duque, Wainer, and Weiermann, and instead is independent of the more powerful theory ATR0. However, this result is contingent on rather elaborate normal forms for natural numbers based on a “sandwiching” procedure. This leaves open both the question of whether the sandwiching procedure can be eliminated while retaining the full strength of the Ackermannian Goodstein principle, and whether other normal forms can lead to nontermination. In this article we settle these questions by showing that any Goodstein process based on the Ackermann function is terminating, and indeed the sandwiching procedure gives rise to Goodstein principles of maximal length. We thus obtain an equivalent principle which does not involve normal forms at all and immediately implies all Ackermannian Goodstein principles that have been considered. Our techniques provide a new approach to termination proofs, where terms in a sequence do not necessarily decrease in complexity, but instead are majorized by some “master” process, already known to be terminating.

David Fernández-Quijada’s contribution addresses the influence of the technical network as ideology in today’s world. He argues that the modern doctrine of retiology was strengthened and intensified with the advent of the digital era and that, despite the dematerialization of networks, the physical is still very much present in the digital universe.In the ethos associated with the Internet he perceives a materialization of Saint-Simonian utopianism, highlighting peer to peer networks in this context as free sharing communities based on the premise that all members contribute to the operation and maintenance of shared information flows. Fernández-Quijada argues that this model was not, however, created by the new media, nor does it produce egalitarian outcomes, because digital networks themselves produce new hierarchies.An example of the way in which the determinism of retiological rhetoric has been reinforced is to be found in the liberalization of the broadcasting industry, with the aim of reusing its frequencies for profit-making mobile telecommunications services. Increasingly, with the changeover to digital television, commercial interests have been favoured over public service interests, and this illustrates the way in which digitalization has accentuated these problems and dynamics.Fernández-Quijada concurs with the territorial nature of networks, following Musso’s suggestion that territory, in addition to its physical aspect, also has a cultural and political existence. He underlines the significance of physical space in the geographical grouping practices (clusters) of industrial firms and of industrial actors associated with production and distribution, and its impact on the geography and characteristics of regions in terms of customers and suppliers, infrastructure and natural and human resources, and low transaction costs as a result of the short distances involved. Rather than arguing that modern retiology can be traced to the earliest stages of the media, Fernández-Quijada demonstrates that retiological determinism is a mirror of capitalist dynamics, just as the threat hanging over public broadcasting companies is a derivative of the neoliberal ethos.

Despite recent efforts by the Artificial Intelligence (AI) community to move towards standardised procedures for documenting models, methods, systems or datasets, there is currently no methodology focused on use cases aligned with the risk-based approach of the European AI Act (AI Act). In this paper, we propose a new framework for the documentation of use cases that we call use case cards, based on the use case modelling included in the Unified Markup Language (UML) standard. Unlike other documentation methodologies, we focus on the intended purpose and operational use of an AI system. It consists of two main parts: firstly, a UML-based template, tailored to allow implicitly assessing the risk level of the AI system and defining relevant requirements, and secondly, a supporting UML diagram designed to provide information about the system-user interactions and relationships. The proposed framework is the result of a co-design process involving a relevant team of EU policy experts and scientists. We have validated our proposal with 11 experts with different backgrounds and a reasonable knowledge of the AI Act as a prerequisite. We provide the 5 use case cards used in the co-design and validation process. Use case cards allows framing and contextualising use cases in an effective way, and we hope this methodology can be a useful tool for policy makers and providers for documenting use cases, assessing the risk level, adapting the different requirements and building a catalogue of existing usages of AI.

The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote $\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $\mathsf {S5}$, and prove soundness and strong completeness for the cases where X is crowded and either completely metrizable and continuum-sized or locally compact Hausdorff. We also show that every extension of $\mathsf {S5}$ is the modal logic of a subalgebra of $\mathbf {Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality.

Goodstein’s principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to diverge, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we provide a self-contained and modern analysis of Goodstein’s principle, obtaining some variations and improvements. We explore notions of optimality for notation systems and apply them to the classical Goodstein process and to a weaker variant based on multiplication rather than exponentiation. In particular, we introduce the notion of base-change maximality, and show how it leads to far-reaching extensions of Goodstein’s result. We moreover show that by varying the initial base of the Goodstein process, one readily obtains independence results for each of the fragments $\mathsf {I}\Sigma _n$ of Peano arithmetic.

Clinical practices are considered one of the cornerstones in nurses' education. This study provides a framework to determine how factors in the academic environment, influence nursing student's satisfaction with their practices. A cross-sectional analytical study was conducted in a convenience sample of 574 nursing students at a private university in Valencia, during the 2016/2017 academic year, 79% were women. Two statistical methodologies were used for data analysis: hierarchical regression models and fuzzy sets qualitative comparative analysis. The HRM indicate that the students' mean score influences all dimensions of satisfaction. Furthermore, in the fsQCA, the type of service and center, as well as the type of management, the preference in the choice of the practice center and the number of students per period per clinical educator influence satisfaction with clinical practices. These results could be used to understand how academic factors influence nursing students' satisfaction with their clinical practices and to create intervention programmes that improve it. This will help prepare students to be the future nursing workforce.

Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel’s provability predicate and its natural extensions. If Λ is any ordinal, the Gödel-Löb calculus GLPΛ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLPω has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In this paper we generalize Beklemishev and Gabelaia’s result to GLPΛ for countable Λ. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLPΛ for the class of provability ambiances based on Icard polytopologies.