The "paradox" of computability and a recursive relative version of the Busy Beaver function

In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy Beaver function which we will call Busy Beaver Plus function. Therefore, we will prove that the computable Busy Beaver Plus function defined on any Turing submachine is not computable by any program running on this submachine. We will thereby demonstrate the existence of a "paradox" of computability a la Skolem: a function is computable when "seen from the outside" the subsystem, but uncomputable when "seen from within" the same subsystem. Finally, we will raise the possibility of defining universal submachines, and a hierarchy of negative Turing degrees.Comment: 10 pages. 0 figures. Supported by the National Council for Scientific and Technological Development (CNPq), Brazil. Book chapter published in Information and Complexity, Mark Burgin and Cristian S. Calude (Editors), World Scientific Publishing, 2016, ISBN 978-981-3109-02-5, available at http://www.worldscientific.com/worldscibooks/10.1142/10017. arXiv admin note: substantial text overlap with arXiv:1612.0522

Algorithmic information and incompressibility of families of multidimensional networks

This article presents a theoretical investigation of string-based generalized representations of families of finite networks in a multidimensional space. First, we study the recursive labeling of networks with (finite) arbitrary node dimensions (or aspects), such as time instants or layers. In particular, we study these networks that are formalized in the form of multiaspect graphs. We show that, unlike classical graphs, the algorithmic information of a multidimensional network is not in general dominated by the algorithmic information of the binary sequence that determines the presence or absence of edges. This universal algorithmic approach sets limitations and conditions for irreducible information content analysis in comparing networks with a large number of dimensions, such as multilayer networks. Nevertheless, we show that there are particular cases of infinite nesting families of finite multidimensional networks with a unified recursive labeling such that each member of these families is incompressible. From these results, we study network topological properties and equivalences in irreducible information content of multidimensional networks in comparison to their isomorphic classical graph.Comment: Extended preprint version of the pape

Nomic realism, simplicity, and the simplicity bubble effect

We offer an argument against simplicity as a sole intrinsic criterion for nomic realism. The argument is based on the simplicity bubble effect. Underdetermination in quantum foundations illustrates the case.Comment: Contributed talk for the Third Graduate Conference of the Italian Network for the Philosophy of Mathematics --- FilMat. Submitted: September 15, 2023. Approved: October 25, 202

Emergence and algorithmic information dynamics of systems and observers

Previous work has shown that perturbation analysis in software space can produce candidate computable generative models and uncover possible causal properties from the finite description of an object or system quantifying the algorithmic contribution of each of its elements relative to the whole. One of the challenges for defining emergence is that one observer's prior knowledge may cause a phenomenon to present itself to such observer as emergent while for another as reducible. When attempting to quantify emergence, we demonstrate that the methods of Algorithmic Information Dynamics can deal with the richness of such observer-object dependencies both in theory and practice. By formalising the act of observing as mutual algorithmic perturbation, the emergence of algorithmic information is rendered invariant, minimal, and robust in the face of information cost and distortion, while still observer-dependent. We demonstrate that the unbounded increase of emergent algorithmic information implies asymptotically observer-independent emergence, which eventually overcomes any formal theory that an observer might devise to finitely characterise a phenomenon. We discuss observer-dependent emergence and asymptotically observer-independent emergence solving some previous suggestions indicating a hard distinction between strong and weak emergence

Optimal Spatial Deconvolution and Message Reconstruction from a Large Generative Model of Models

We introduce a general-purpose univariate signal deconvolution method based on the principles of an approach to Artificial General Intelligence. This approach is based on a generative model that combines information theory and algorithmic probability that required a large calculation of an estimation of a `universal distribution' to build a general-purpose model of models independent of probability distributions. This was used to investigate how non-random data may encode information about the physical properties such as dimension and length scales in which a signal or message may have been originally encoded, embedded, or generated. This multidimensional space reconstruction method is based on information theory and algorithmic probability, and it is agnostic, but not independent, with respect to the chosen computable or semi-computable approximation method or encoding-decoding scheme. The results presented in this paper are useful for applications in coding theory, particularly in zero-knowledge one-way communication channels, such as in deciphering messages sent by generating sources of unknown nature for which no prior knowledge is available. We argue that this can have strong potential for cryptography, signal processing, causal deconvolution, life, and techno signature detection.Comment: 35 page

An algorithmically random family of MultiAspect Graphs and its topological properties

This article presents a theoretical investigation of incompressibility and randomness in generalized representations of graphs along with its implications on network topological properties. We extend previous studies on plain algorithmically random classical graphs to plain and prefix algorithmically random MultiAspect Graphs (MAGs). First, we show that there is an infinite recursively labeled infinite family of nested MAGs (or, as a particular case, of nested classical graphs) that behaves like (and is determined by) an algorithmically random real number. Then, we study some of their important topological properties, in particular, vertex degree, connectivity, diameter, and rigidity

The simplicity bubble effect as a zemblanitous phenomenon in learning systems

The ubiquity of Big Data and machine learning in society evinces the need of further investigation of their fundamental limitations. In this paper, we extend the ``too-much-information-tends-to-behave-like-very-little-information'' phenomenon to formal knowledge about lawlike universes and arbitrary collections of computably generated datasets. This gives rise to the simplicity bubble problem, which refers to a learning algorithm equipped with a formal theory that can be deceived by a dataset to find a locally optimal model which it deems to be the global one. However, the actual high-complexity globally optimal model unpredictably diverges from the found low-complexity local optimum. Zemblanity is defined by an undesirable but expected finding that reveals an underlying problem or negative consequence in a given model or theory, which is in principle predictable in case the formal theory contains sufficient information. Therefore, we argue that there is a ceiling above which formal knowledge cannot further decrease the probability of zemblanitous findings, should the randomly generated data made available to the learning algorithm and formal theory be sufficiently large in comparison to their joint complexity