Bootstrap percolation, introduced by Chalupa, Leath, and Reich in their 1979 work [4], serves as a simplified model for ferromagnetic dynamics. Since then, it has been applied in numerous fields of physics and has become a significant topic of interest in mathematics. The process starts with an initial set of "infected" vertices in a graph G, where at each step, any vertex with at least r infected neighbors also becomes infected. A key problem in this framework is determining the minimum size of an initial set that results in the entire graph becoming infected, known as the percolating set. This minimum is denoted by m(G, r).

In the present work we use maximum entropy methods to derive several theorems in probabilistic number theory, including a version of the Hardy-Ramanujan Theorem. We also provide a theoretical argument explaining the experimental observations of Yang-Hui He about the learnability of primes, and posit that the Erd\H{o}s-Kac law would very unlikely be discovered by current machine learning techniques. Numerical experiments that we perform corroborate our theoretical findings.

Quantum artificial intelligence is a frontier of artificial intelligence research, pioneering quantum AI-powered circuits to address problems beyond the reach of deep learning with classical architectures. This work implements a large-scale quantum-activated recurrent neural network possessing more than 3 trillion hardware nodes/cm$^2$, originating from repeatable atomic-scale nucleation dynamics in an amorphous material integrated on-chip, controlled with 0.07 nW electric power per readout channel. Compared to the best-performing reservoirs currently reported, this implementation increases the scale of the network by two orders of magnitude and reduces the power consumption by six, reaching power efficiencies in the range of the human brain, dissipating 0.2 nW/neuron. When interrogated by a classical input, the chip implements a large-scale hardware security model, enabling dictionary-free authentication secure against statistical inference attacks, including AI's present and future development, even for an adversary with a copy of all the classical components available. Experimental tests report 99.6% reliability, 100% user authentication accuracy, and an ideal 50% key uniqueness. Due to its quantum nature, the chip supports a bit density per feature size area three times higher than the best technology available, with the capacity to store more than $2^{1104}$ keys in a footprint of 1 cm$^2$. Such a quantum-powered platform could help counteract the emerging form of warfare led by the cybercrime industry in breaching authentication to target small to large-scale facilities, from private users to intelligent energy grids.

Simplicity bias is an intriguing phenomenon prevalent in various input-output maps, characterized by a preference for simpler, more regular, or symmetric outputs. Notably, these maps typically feature high-probability outputs with simple patterns, whereas complex patterns are exponentially less probable. This bias has been extensively examined and attributed to principles derived from algorithmic information theory and algorithmic probability. In a significant advancement, it has been demonstrated that the renowned logistic map $x_{k+1}=\mu x_k(1-x_k)$, and other one-dimensional maps exhibit simplicity bias when conceptualized as input-output systems. Building upon this foundational work, our research delves into the manifestations of simplicity bias within the random logistic map, specifically focusing on scenarios involving additive noise. This investigation is driven by the overarching goal of formulating a comprehensive theory for the prediction and analysis of time series.Our primary contributions are multifaceted. We discover that simplicity bias is observable in the random logistic map for specific ranges of $\mu$ and noise magnitudes. Additionally, we find that this bias persists even with the introduction of small measurement noise, though it diminishes as noise levels increase. Our studies also revisit the phenomenon of noise-induced chaos, particularly when $\mu=3.83$, revealing its characteristics through complexity-probability plots. Intriguingly, we employ the logistic map to underscore a paradoxical aspect of data analysis: more data adhering to a consistent trend can occasionally lead to reduced confidence in extrapolation predictions, challenging conventional wisdom.We propose that adopting a probability-complexity perspective in analyzing dynamical systems could significantly enrich statistical learning theories related to series prediction.

Neural networks have in recent years shown promise for helping software engineers write programs and even formally verify them. While semantic information plays a crucial part in these processes, it remains unclear to what degree popular neural architectures like transformers are capable of modeling that information. This paper examines the behavior of neural networks learning algorithms relevant to programs and formal verification proofs through the lens of mechanistic interpretability, focusing in particular on structural recursion. Structural recursion is at the heart of tasks on which symbolic tools currently outperform neural models, like inferring semantic relations between datatypes and emulating program behavior. We evaluate the ability of transformer models to learn to emulate the behavior of structurally recursive functions from input-output examples. Our evaluation includes empirical and conceptual analyses of the limitations and capabilities of transformer models in approximating these functions, as well as reconstructions of the ``shortcut" algorithms the model learns. By reconstructing these algorithms, we are able to correctly predict 91 percent of failure cases for one of the approximated functions. Our work provides a new foundation for understanding the behavior of neural networks that fail to solve the very tasks they are trained for.

The Game Developer's Conference, the largest event in the games industry, has hosted over 50 talks in the last decade about procedural generation, from small-scale independent speakers to large AAA companies, covering disciplines from programming to art to writing. Correspondingly, procedural generation has been an increasingly hot topic among game AI researchers in the last two decades. The Procedural Generation Workshop at FDG, now in its twelfth year, is one of the longest-running workshops in the field of game AI, and dedicated paper tracks at conferences are a regular occurrence. Despite the huge importance of content generation, and the wealth of time invested into developing practical techniques, the analysis of procedural generators is a relatively underdeveloped area of study. A few notable techniques have emerged over the last two decades of research [7, 8], as well as studies of efficacy [4, 9], but many of the techniques used by game researchers have changed little in that time. As a result, a lot of procedural generation work is done by'feel', with postmortems shared at events such as the Roguelike Celebration

Genetic algorithms (GA) are optimisation, search and learning algorithms famous for their ability to solve problems with large number parameters and complex mathematical representations and they are sometimes used to tune models parameters in machine learning . NASA used them to create optimised antennas for their spacecrafts . The initial population is a set of valid candidates (individuals) with a specified population size chosen randomly .

According to the No Free Lunch (NFL) theorems all black-box algorithms perform equally well when compared over the entire set of optimization problems. An important problem related to NFL is finding a test problem for which a given algorithm is better than another given algorithm. Of high interest is finding a function for which Random Search is better than another standard evolutionary algorithm. In this paper, we propose an evolutionary approach for solving this problem: we will evolve test functions for which a given algorithm A is better than another given algorithm B. Two ways for representing the evolved functions are employed: as GP trees and as binary strings. Several numerical experiments involving NFL-style Evolutionary Algorithms for function optimization are performed. The results show the effectiveness of the proposed approach. Several test functions for which Random Search performs better than all other considered algorithms have been evolved.

The complexity of a string can be measured by the richness of its substrings. For example in genetics a region of DNA is considered to be highly informative if many of the possible substrings of a certain length actually occur. Abstractly this kind of complexity is captured by the standard string complexity function. When dealing with binary strings, we have the additional feature that substrings can be viewed as subsets of an index set. This allows us to apply measures of subset complexity such as VC dimension. In this paper we define a notion of VC dimension for binary strings and investigate the structure of strings of finite VC dimension.

We show how complexity theory can be introduced in machine learning to help bring together apparently disparate areas of current research. We show that this new approach requires less training data and is more generalizable as it shows greater resilience to random attacks. We investigate the shape of the discrete algorithmic space when performing regression or classification using a loss function parametrized by algorithmic complexity, demonstrating that the property of differentiation is not necessary to achieve results similar to those obtained using differentiable programming approaches such as deep learning. In doing so we use examples which enable the two approaches to be compared (small, given the computational power required for estimations of algorithmic complexity). We find and report that (i) machine learning can successfully be performed on a non-smooth surface using algorithmic complexity; (ii) that parameter solutions can be found using an algorithmic-probability classifier, establishing a bridge between a fundamentally discrete theory of computability and a fundamentally continuous mathematical theory of optimization methods; (iii) a formulation of an algorithmically directed search technique in non-smooth manifolds can be defined and conducted; (iv) exploitation techniques and numerical methods for algorithmic search to navigate these discrete non-differentiable spaces can be performed; in application of the (a) identification of generative rules from data observations; (b) solutions to image classification problems more resilient against pixel attacks compared to neural networks; (c) identification of equation parameters from a small data-set in the presence of noise in continuous ODE system problem, (d) classification of Boolean NK networks by (1) network topology, (2) underlying Boolean function, and (3) number of incoming edges.