arXiv subjects
Publications and source records attributed to Alexandre Blondin Massé.
Given a discrete rectangle R of dimensions h x w, let W be the set of snake-like polyominoes contained in R represented as binary matrices, i.e. polyominoes whose underlying simple graph is a chain with respect to the 4-adjacency relation. We present an algorithm that generates W for any h and w. Also, let a be the maximal area that can be realized by an element of W. We provide exact formulas of a for h <= 5 and any w.
We present new results about fully leafed induced subtrees in Penrose P2 tilings, also known as kites and darts tilings. We first determine the graph structure of these subtrees and show that they are caterpillars up to an appendix of at most six tiles. In other words, if we remove their degree one vertices, then they are path graphs up to an additional connected path of at most two tiles. We then study bi-infinite fully leafed induced caterpillars in P2 tilings and their geometric properties. In particular, we refute the conjecture proposed by C. Porrier, A. Goupil and A. Blondin Massé that there is a unique bi-infinite fully leafed caterpillar in Penrose P2 tilings.
In a recent article by C. Porrier, A. Blondin Massé and A. Goupil, a first bi-infinite fully leafed induced subcaterpillar of Penrose P2 tilings is presented. In this paper, we formally construct this caterpillar for the first time. We then prove that every fully leafed induced subtree in Penrose P2 tilings is a caterpillar with at most one appendix of at most two internal tiles, and we characterize fully leafed induced subtrees that have the property of saturation. We also refute the conjecture that there is a unique bi-infinite fully leafed induced subcaterpillar by constructing a new one. Finally, we present progress on the construction of all bi-infinite fully leafed induced subcaterpillars in Penrose P2 tilings.
Time series forecasting is a critical task across domains such as energy, finance, and meteorology, where accurate predictions enable informed decision-making. While transformer-based and large-parameter models have recently achieved state-of-the-art results, their complexity can lead to overfitting and unstable forecasts, especially when older data points become less relevant. In this paper, we propose a lightweight multi-view approach to short-term load forecasting that leverages single-value embeddings and a scaled time-range input to capture temporally relevant features efficiently. We introduce an embedding dropout mechanism to prevent over-reliance on specific features and enhance interpretability. Our method achieves competitive performance with significantly fewer parameters, demonstrating robustness across multiple datasets, including scenarios with noisy or sparse data, and provides insights into the contributions of individual features to the forecast.
Reinforcement learning (RL) is a powerful framework for optimizing decision-making in complex systems under uncertainty, an essential challenge in real-world settings, particularly in the context of the energy transition. A representative example is remote microgrids that supply power to communities disconnected from the main grid. Enabling the energy transition in such systems requires coordinated control of renewable sources like wind turbines, alongside fuel generators and batteries, to meet demand while minimizing fuel consumption and battery degradation under exogenous and intermittent load and wind conditions. These systems must often conform to extensive regulations and complex operational constraints. To ensure that RL agents respect these constraints, it is crucial to provide interpretable guarantees. In this paper, we introduce Shielded Controller Units (SCUs), a systematic and interpretable approach that leverages prior knowledge of system dynamics to ensure constraint satisfaction. Our shield synthesis methodology, designed for real-world deployment, decomposes the environment into a hierarchical structure where each SCU explicitly manages a subset of constraints. We demonstrate the effectiveness of SCUs on a remote microgrid optimization task with strict operational requirements. The RL agent, equipped with SCUs, achieves a 24% reduction in fuel consumption without increasing battery degradation, outperforming other baselines while satisfying all constraints. We hope SCUs contribute to the safe application of RL to the many decision-making challenges linked to the energy transition.
We study a graph-theoretic problem in the Penrose P2-graphs which are the dual graphs of Penrose tilings by kites and darts. Using substitutions, local isomorphism and other properties of Penrose tilings, we construct a family of arbitrarily large induced subtrees of Penrose graphs with the largest possible number of leaves for a given number $n$ of vertices. These subtrees are called fully leafed induced subtrees. We denote their number of leaves $L_{P2}(n)$ for any non-negative integer $n$, and the sequence $\left(L_{P2}(n)\right)_{n\in\mathbb{N}}$ is called the leaf function of Penrose P2-graphs. We present exact and recursive formulae for $L_{P2}(n)$, as well as an infinite sequence of fully leafed induced subtrees, which are caterpillar graphs. In particular, our proof relies on the construction of a finite graded poset of 3-internal-regular subtrees.
Abstract meaning representation (AMR) is a semantic formalism used to represent the meaning of sentences as directed acyclic graphs. In this paper, we describe how real digital dictionaries can be embedded into AMR directed graphs (digraphs), using state-of-the-art pre-trained large language models. Then, we reduce those graphs in a confluent manner, i.e. with transformations that preserve their circuit space. Finally, the properties of these reduces digraphs are analyzed and discussed in relation to the symbol grounding problem.
Let d be an integer between 0 and 4, and W be a 2-dimensional word of dimensions h x w on the binary alphabet {0, 1}, where h, w in Z > 0. Assume that each occurrence of the letter 1 in W is adjacent to at most d letters 1. We provide an exact formula for the maximum number of letters 1 that can occur in W for fixed (h, w). As a byproduct, we deduce an upper bound on the length of maximum snake polyominoes contained in a h x w rectangle.
Quasi-static time series (QSTS) simulations have great potential for evaluating the grid's ability to accommodate the large-scale integration of distributed energy resources. However, as grids expand and operate closer to their limits, iterative power flow solvers, central to QSTS simulations, become computationally prohibitive and face increasing convergence issues. Neural power flow solvers provide a promising alternative, speeding up power flow computations by 3 to 4 orders of magnitude, though they are costly to train. In this paper, we envision how recently introduced grid foundation models could improve the economic viability of neural power flow solvers. Conceptually, these models amortize training costs by serving as a foundation for a range of grid operation and planning tasks beyond power flow solving, with only minimal fine-tuning required. We call for collaboration between the AI and power grid communities to develop and open-source these models, enabling all operators, even those with limited resources, to benefit from AI without building solutions from scratch.
Foundation models (FMs) currently dominate news headlines. They employ advanced deep learning architectures to extract structural information autonomously from vast datasets through self-supervision. The resulting rich representations of complex systems and dynamics can be applied to many downstream applications. Therefore, FMs can find uses in electric power grids, challenged by the energy transition and climate change. In this paper, we call for the development of, and state why we believe in, the potential of FMs for electric grids. We highlight their strengths and weaknesses amidst the challenges of a changing grid. We argue that an FM learning from diverse grid data and topologies could unlock transformative capabilities, pioneering a new approach in leveraging AI to redefine how we manage complexity and uncertainty in the electric grid. Finally, we discuss a power grid FM concept, namely GridFM, based on graph neural networks and show how different downstream tasks benefit.
Hydro-Quebec (HQ) is a vertically integrated utility that produces, transmits, and distributes most of the electricity in the province of Quebec. The power grid it operates has a particular architecture created by large hydroelectric dams located far north and the extensive 735kV transmission grid that allows the generated power to reach the majority of the load located thousands of kilometers away in the southern region of Quebec. The specificity of the grid has led HQ to develop monitoring tools responsible for generating so-called stability limits. Those stability limits take into account several nonlinear phenomena such as angular stability, frequency stability, or voltage stability. Since generation planning and operation tools rely mostly on mixed integer linear programming formulation, HQ had to adapt its tools to integrate stability limits into them. This paper presents the challenges it faced, especially considering its reserve monitoring tool and unit commitment tool.
In graph theory, the minimum directed feedback vertex set (FVS) problem consists in identifying the smallest subsets of vertices in a directed graph whose deletion renders the directed graph acyclic. Although being known as NP-hard since 1972, this problem can be solved in a reasonable time on small instances, or on instances having special combinatorial structure. In this paper we investigate graph reductions preserving all or some minimum FVS and focus on their properties, especially the Church-Rosser property, also called confluence. The Church-Rosser property implies the irrelevance of reduction order, leading to a unique directed graph. The study seeks the largest subset of reductions with the Church-Rosser property and explores the adaptability of reductions to meet this criterion. Addressing these questions is crucial, as it may impact algorithmic implications, allowing for parallelization and speeding up sequential algorithms.
In the last years, several variants of transformers have emerged. In this paper, we compare different transformer-based models for solving the reverse dictionary task and explore their use in the context of a serious game called The Dictionary Game.
Let $G$ be a simple graph on $n$ vertices. We consider the problem LIS of deciding whether there exists an induced subtree with exactly $i \leq n$ vertices and $\ell$ leaves in $G$. We study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by $L_G(i)$, realized by an induced subtree with $i$ vertices, for $0 \le i \le n$. We begin by proving that the LIS problem is NP-complete in general and then we compute the values of the map $L_G$ for some classical families of graphs and in particular for the $d$-dimensional hypercubic graphs $Q_d$, for $2 \leq d \leq 6$. We also describe a nontrivial branch and bound algorithm that computes the function $L_G$ for any simple graph $G$. In the special case where $G$ is a tree of maximum degree $Δ$, we provide a $\mathcal{O}(n^3Δ)$ time and $\mathcal{O}(n^2)$ space algorithm to compute the function $L_G$.
Given a simple graph $G$ with $n$ vertices and a natural number $i \leq n$, let $L_G(i)$ be the maximum number of leaves that can be realized by an induced subtree $T$ of $G$ with $i$ vertices. We introduce a problem that we call the \emph{leaf realization problem}, which consists in deciding whether, for a given sequence of $n+1$ natural numbers $(\ell_0, \ell_1, \ldots, \ell_n)$, there exists a simple graph $G$ with $n$ vertices such that $\ell_i = L_G(i)$ for $i = 0, 1, \ldots, n$. We present basic observations on the structure of these sequences for general graphs and trees. In the particular case where $G$ is a caterpillar graph, we exhibit a bijection between the set of the discrete derivatives of the form $(ΔL_G(i))_{1 \leq i \leq n - 3}$ and the set of prefix normal words.
How many words (and which ones) are sufficient to define all other words? When dictionaries are analyzed as directed graphs with links from defining words to defined words, they reveal a latent structure. Recursively removing all words that are reachable by definition but that do not define any further words reduces the dictionary to a Kernel of about 10%. This is still not the smallest number of words that can define all the rest. About 75% of the Kernel turns out to be its Core, a Strongly Connected Subset of words with a definitional path to and from any pair of its words and no word's definition depending on a word outside the set. But the Core cannot define all the rest of the dictionary. The 25% of the Kernel surrounding the Core consists of small strongly connected subsets of words: the Satellites. The size of the smallest set of words that can define all the rest (the graph's Minimum Feedback Vertex Set or MinSet) is about 1% of the dictionary, 15% of the Kernel, and half-Core, half-Satellite. But every dictionary has a huge number of MinSets. The Core words are learned earlier, more frequent, and less concrete than the Satellites, which in turn are learned earlier and more frequent but more concrete than the rest of the Dictionary. In principle, only one MinSet's words would need to be grounded through the sensorimotor capacity to recognize and categorize their referents. In a dual-code sensorimotor-symbolic model of the mental lexicon, the symbolic code could do all the rest via re-combinatory definition.
The object under study is a particular closed curve on the square lattice $\Z^2$ related with the Fibonacci sequence $F_n$. It belongs to a class of curves whose length is $4F_{3n+1}$, and whose interiors by translation tile the plane. The limit object, when conveniently normalized, is a fractal line for which we compute first the fractal dimension, and then give a complexity measure.