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In the traditional view of reinforcement learning, the agent's goal is to find an optimal policy that maximizes its expected sum of rewards. Once the agent finds this policy, the learning ends. This view contrasts with \emph{continual reinforcement learning}, where learning does not end, and agents are expected to continually learn and adapt indefinitely. Despite the clear distinction between these two paradigms of learning, much of the progress in continual reinforcement learning has been shaped by foundations rooted in the traditional view of reinforcement learning. In this paper, we first examine whether the foundations of traditional reinforcement learning are suitable for the continual reinforcement learning paradigm. We identify four key pillars of the traditional reinforcement learning foundations that are antithetical to the goals of continual learning: the Markov decision process formalism, the focus on atemporal artifacts, the expected sum of rewards as an evaluation metric, and episodic benchmark environments that embrace the other three foundations. We then propose a new formalism that sheds the first and the third foundations and replaces them with the history process

Large language models (LLMs) have taken centre stage in debates on Artificial Intelligence. Yet there remains a gap in how to assess LLMs' conformity to important human values. In this paper, we investigate whether state-of-the-art LLMs, GPT-4 and Claude 2.1 (Gemini Pro and LLAMA 2 did not generate valid results) are moral hypocrites. We employ two research instruments based on the Moral Foundations Theory: (i) the Moral Foundations Questionnaire (MFQ), which investigates which values are considered morally relevant in abstract moral judgements; and (ii) the Moral Foundations Vignettes (MFVs), which evaluate moral cognition in concrete scenarios related to each moral foundation. We characterise conflicts in values between these different abstractions of moral evaluation as hypocrisy. We found that both models displayed reasonable consistency within each instrument compared to humans, but they displayed contradictory and hypocritical behaviour when we compared the abstract values present in the MFQ to the evaluation of concrete moral violations of the MFV.

This chapter introduces a conceptual framework for qualitative risk assessment of AI, particularly in the context of the EU AI Act. The framework addresses the complexities of legal compliance and fundamental rights protection by itegrating definitional balancing and defeasible reasoning. Definitional balancing employs proportionality analysis to resolve conflicts between competing rights, while defeasible reasoning accommodates the dynamic nature of legal decision-making. Our approach stresses the need for an analysis of AI deployment scenarios and for identifying potential legal violations and multi-layered impacts on fundamental rights. On the basis of this analysis, we provide philosophical foundations for a logical account of AI risk analysis. In particular, we consider the basic building blocks for conceptually grasping the interaction between AI deployment scenarios and fundamental rights, incorporating in defeasible reasoning definitional balancing and arguments about the contextual promotion or demotion of rights. This layered approach allows for more operative models of assessment of both high-risk AI systems and General Purpose AI (GPAI) systems, emphasizing the broader

This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In this paper, we lay the foundations for this study by introducing the notion of formal manifolds in the context of differential geometry, inspired by the notion of formal schemes in algebraic geometry. We develop the basic theory for formal manifolds, and establish a fully faithful contravariant functor from the category of formal manifolds to the category of topological $\mathbb{C}$-algebras. We also prove the existence of finite products in the category of formal manifolds by studying vector-valued formal functions.

A number of software foundations have been created as legal instruments to better articulate the structure, collaboration and financial model of Open Source Software (OSS) projects. Some examples are the Apache, Linux, or Mozilla foundations. However, the mission and support provided by these foundations largely differ among them. In this paper we perform a study on the role of foundations in OSS development. We analyze the nature, activities, role and governance of 101 software foundations and then go deeper on the 27 having as concrete goal the development and evolution of specific open source projects (and not just generic actions to promote the free software movement or similar). Our results reveal the existence of a significant number of foundations with the sole purpose of promoting the free software movement and/or that limit themselves to core legal aspects but do not play any role in the day-to-day operations of the project (e.g., umbrella organizations for a large variety of projects). Therefore, while useful, foundations do not remove the need for specific projects to develop their own specific governance, contribution and development policies. A website to help projects

The so-called Geometric Trinity of Gravity includes General Relativity (GR), based on spacetime curvature; the Teleparallel Equivalent of GR (TEGR), which relies on spacetime torsion; and the Symmetric Teleparallel Equivalent of GR (STEGR), grounded in nonmetricity. Recent studies demonstrate that GR, TEGR, and STEGR are dynamically equivalent, raising questions about the fundamental structure of spacetime, the under-determination of these theories, and whether empirical distinctions among them are possible. The aim of this work is to show that they are equivalent in many features but not exactly in everything. In particular, their relationship with the Equivalence Principle (EP) is different. The EP is a deeply theory-laden assumption, which is assumed as fundamental in constructing GR, with significant implications for our understanding of spacetime. However, it introduces unresolved conceptual issues, including its impact on the nature of the metric and connection, its meaning at the quantum level, tensions with other fundamental interactions and new physics, and its role in dark matter and dark energy problems. In contrast, TEGR and STEGR recover the EP, in particular in its st

By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a "quantum probability theory". As such, it is regarded a generalization of the "classical probability theory" due to Kolmogorov. Outside of quantum physics, however, Kolmogorov's axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann's axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic $N$-body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov's axioms and physically natural random variables. Though this theory is provably distinct from its quantum mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical

Foundation models have enormous potential in advancing Earth and climate sciences, however, current approaches may not be optimal as they focus on a few basic features of a desirable Earth and climate foundation model. Crafting the ideal Earth foundation model, we define eleven features which would allow such a foundation model to be beneficial for any geoscientific downstream application in an environmental- and human-centric manner.We further shed light on the way forward to achieve the ideal model and to evaluate Earth foundation models. What comes after foundation models? Energy efficient adaptation, adversarial defenses, and interpretability are among the emerging directions.

Criticality is hypothesized as a physical mechanism underlying efficient transitions between cortical states and remarkable information processing capacities in the brain. While considerable evidence generally supports this hypothesis, non-negligible controversies persist regarding the ubiquity of criticality in neural dynamics and its role in information processing. Validity issues frequently arise during identifying potential brain criticality from empirical data. Moreover, the functional benefits implied by brain criticality are frequently misconceived or unduly generalized. These problems stem from the non-triviality and immaturity of the physical theories that analytically derive brain criticality and the statistic techniques that estimate brain criticality from empirical data. To help solve these problems, we present a systematic review and reformulate the foundations of studying brain criticality, i.e., ordinary criticality (OC), quasi-criticality (qC), self-organized criticality (SOC), and self-organized quasi-criticality (SOqC), using the terminology of neuroscience. We offer accessible explanations of the physical theories and statistic techniques of brain criticality, pr

In this paper we give a contribution to the taxonomy of physical theories. We provide here a thorough description of the axiomatic foundations of the most relevant physical theories, Mechanics, Special Relativity, General Relativity, Quantum Mechanics. The corresponding interactions will be dealt with as well, i.e. Gravity in the Minkowskian limit, Electricity without quantized energy, Gravity without quantized energy, Electricity with quantized energy. We pose the problem if the extension of the principle of solidarity to all interactions can impose to consider all variables as dynamic.

In this paper, we explore the 'equivalence principle' (EP): roughly, statements about mathematical objects should be invariant under an appropriate notion of equivalence for the kinds of objects under consideration. In set theoretic foundations, EP may not always hold: for instance, the statement '1 \in N' is not invariant under isomorphism of sets. In univalent foundations, on the other hand, EP has been proven for many mathematical structures. We first give an overview of earlier attempts at designing foundations that satisfy EP. We then describe how univalent foundations validates EP.

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes significantly upon the replacement of 'second-order representations' to 'third-order functions'. We discuss the implications and connections to the Reverse Mathematics program and its foundational claims regarding predicativist mathematics and Hilbert's program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.

We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander.

This is a transcript of the round table that took place during the conference Quantum Theory: Reconsideration of Foundations - 3, June 2005, Vaxjo, Sweden. There are presented opinions of leading experts in quantum foundations on such fundamental problems as the origin of quantum fluctuations and completeness of quantum mechanics.

Due to its longevity and enormous information density, DNA is an attractive medium for archival data storage. Thanks to rapid technological advances, DNA storage is becoming practically feasible, as demonstrated by a number of experimental storage systems, making it a promising solution for our society's increasing need of data storage. While in living things, DNA molecules can consist of millions of nucleotides, due to technological constraints, in practice, data is stored on many short DNA molecules, which are preserved in a DNA pool and cannot be spatially ordered. Moreover, imperfections in sequencing, synthesis, and handling, as well as DNA decay during storage, introduce random noise into the system, making the task of reliably storing and retrieving information in DNA challenging. This unique setup raises a natural information-theoretic question: how much information can be reliably stored on and reconstructed from millions of short noisy sequences? The goal of this monograph is to address this question by discussing the fundamental limits of storing information on DNA. Motivated by current technological constraints on DNA synthesis and sequencing, we propose a probabilistic

Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof ass

This paper is a programmatic article presenting an outline of a new view of the foundations of quantum mechanics and quantum field theory. In short, the proposed foundations are given by the following statements: * Coherent quantum physics is physics in terms of a coherent space consisting of a line bundle over a classical phase space and an appropriate coherent product. * The kinematical structure of quantum physics and the meaning of the fundamental quantum observables are given by the symmetries of this coherent space, their infinitesimal generators, and associated operators on the quantum space of the coherent space. * The connection of quantum physics to experiment is given through the thermal interpretation. The dynamics of quantum physics is given (for isolated systems) by the Ehrenfest equations for q-expectations.