Ezequiel López-Rubio

Ezequiel López-Rubio

We introduce Grid Programs, a novel model of computation in which programs are finite two-dimensional arrangements of instructions on an integer grid rather than linear sequences of statements. Three properties distinguish this model fundamentally from classical frameworks: (i) programs are planar structures through which an instruction pointer moves in the four cardinal directions; (ii) there are no syntax constraints, so any assignment of instructions to grid cells constitutes a valid program; and (iii) the model uses no named variables or explicit memory addresses. Program state is maintained through a data stack, an address stack, and a circularly doubly linked list accessed via three named pointers. Control flow is achieved spatially, with branching encoded as perpendicular turns of the instruction pointer. The address stack stores triplets (cell row, cell column, direction), enabling precise restoration of both position and heading after branches, loops, and function calls. We give a formal operational semantics, present a representative instruction set covering arithmetic, control flow, and linked-list manipulation, and work through several detailed examples, including an absolute-value function, a factorial computation, a linear-search algorithm, a string-reversal program, and a while-loop summation. We establish that Grid Programs are Turing-complete by simulating an arbitrary register machine, and we discuss their relationship to prior two-dimensional languages such as Befunge and Funge-98, to stack-based languages such as Forth and PostScript, and to dataflow and spatial computation models. Grid Programs offer a fresh vantage point for exploring the design space of computation, with potential applications in visual programming environments, cellular-automaton-inspired hardware, and obfuscation-resistant code.

Mario Pascual-González, Ariadna Jiménez-Partinen, R. M. Luque-Baena et al.

Focal cortical dysplasia (FCD) lesions in epilepsy FLAIR MRI are subtle and scarce, making joint image--mask generative modeling prone to instability and memorization. We propose SLIM-Diff, a compact joint diffusion model whose main contributions are (i) a single shared-bottleneck U-Net that enforces tight coupling between anatomy and lesion geometry from a 2-channel image+mask representation, and (ii) loss-geometry tuning via a tunable $L_p$ objective. As an internal baseline, we include the canonical DDPM-style objective ($ε$-prediction with $L_2$ loss) and isolate the effect of prediction parameterization and $L_p$ geometry under a matched setup. Experiments show that $x_0$-prediction is consistently the strongest choice for joint synthesis, and that fractional sub-quadratic penalties ($L_{1.5}$) improve image fidelity while $L_2$ better preserves lesion mask morphology. Our code and model weights are available in https://github.com/MarioPasc/slim-diff

Ezequiel López-Rubio

We introduce IsalProgram (Instruction Set and Language for Programming), a novel assembly-like programming language with three distinctive theoretical properties: (1) it is a regular language in the sense of formal language theory, meaning its programs are accepted by a finite automaton; (2) every finite string over the instruction alphabet is a syntactically valid program; and (3) it makes no explicit use of memory addresses or variable names, absolute or relative. Programs are finite sequences of tokens drawn from a fixed instruction set, and are executed on a virtual machine whose sole data structure is a circular doubly linked list (CDLL) navigated by three data pointers, with control flow governed by two code pointers. We give a complete formal definition of the language and its virtual machine, prove its regularity, and demonstrate its expressive power. We further discuss IsalProgram's potential advantages as a target language for neural program synthesis, the amenability of its program space to metric-based exploration via the Levenshtein edit distance, and directions for analyzing computability and complexity within this framework.

Nivar Anwer, Ezequiel López-Rubio, David Elizondo et al.

The analysis and control of stochastic dynamical systems rely on probabilistic models such as (continuous-space) Markov decision processes, but large or continuous state spaces make exact analysis intractable and call for principled quantitative abstraction. This work develops a unified theory of such abstraction by integrating category theory, coalgebra, quantitative logic, and optimal transport, centred on a canonical $\varepsilon$-quotient of the behavioral pseudo-metric with a universal property: among all abstractions that collapse behavioral differences below $\varepsilon$, it is the most detailed, and every other abstraction achieving the same discounted value-loss guarantee factors uniquely through it. Categorically, a quotient functor $Q_\varepsilon$ from a category of probabilistic systems to a category of metric specifications admits, via the Special Adjoint Functor Theorem, a right adjoint $R_\varepsilon$, yielding an adjunction $Q_\varepsilon \dashv R_\varepsilon$ that formalizes a duality between abstraction and realization; logically, a quantitative modal $μ$-calculus with separate reward and transition modalities is shown, for a broad class of systems, to be expressively complete for the behavioral pseudo-metric, with a countable fully abstract fragment suitable for computation. The theory is developed coalgebraically over Polish spaces and the Giry monad and validated on finite-state models using optimal-transport solvers, with experiments corroborating the predicted contraction properties and structural stability and aligning with the theoretical value-loss bounds, thereby providing a rigorous foundation for quantitative state abstraction and representation learning in probabilistic domains.

Karl Thurnhofer-Hemsi, Ezequiel López-Rubio, Miguel A. Molina-Cabello et al.

Support Vector Machines (SVMs) are still one of the most popular and precise classifiers. The Radial Basis Function (RBF) kernel has been used in SVMs to separate among classes with considerable success. However, there is an intrinsic dependence on the initial value of the kernel hyperparameter. In this work, we propose OKSVM, an algorithm that automatically learns the RBF kernel hyperparameter and adjusts the SVM weights simultaneously. The proposed optimization technique is based on a gradient descent method. We analyze the performance of our approach with respect to the classical SVM for classification on synthetic and real data. Experimental results show that OKSVM performs better irrespective of the initial values of the RBF hyperparameter.