Related papers: On Turedo Hierarchies and Intrinsic Universality
Wolfram [2, p. 707] and Cook [1, p. 3] claim to prove that a (2,5) Turing machine (2 states, 5 symbols) is universal, via a universal cellular automaton known as Rule 110. The first part of this paper points out a critical gap in their…
We demonstrate the existence of transient two-dimensional surfaces where a random-walking particle escapes to infinity in contrast to localization in standard flat 2D space. We first prove that any rotationally symmetric 2D membrane…
Neural surface representation has demonstrated remarkable success in the areas of novel view synthesis and 3D reconstruction. However, assessing the geometric quality of 3D reconstructions in the absence of ground truth mesh remains a…
Motivated by Wheeler's bottom up pregeometry, we introduce a pregeometric approach that does not assume Wheeler's probability amplitudes for establishing spacetime neighborhoods. Rather, a non-trivial metric is produced via the concept of a…
Using tools from topology and functional analysis, we provide a framework where artificial neural networks, and their architectures, can be formally described. We define the notion of machine in a general topological context and show how…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
We study an elementary problem of the topological robotics: collective motion of a set of $n$ distinct particles which one has to move from an initial configuration to a final configuration, with the requirement that no collisions occur in…
We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long-term behavior. It is shown that a very general class of decision problems regarding…
Spatial reasoning requires both location-bound computation and location-invariant structure: agents must make local moves while preserving route, object, or constraint-level plans. We propose interaction locality, a task-geometry-aware…
In a variety of studies of dynamical systems, the edge of order and chaos has been singled out as a region of complexity. It was suggested by Wolfram, on the basis of qualitative behaviour of cellular automata, that the computational basis…
Laguerre geometry of surfaces in $\R^3$ is given in the book of Blaschke [1], and have been studied by E.Musso and L.Nicolodi [5], [6], [7], B. Palmer [8] and other authors. In this paper we study Laguerre differential geometry of…
This paper introduces a class of objects called decision rules that map infinite sequences of alternatives to a decision space. These objects can be used to model situations where a decision maker encounters alternatives in a sequence such…
Manipulating an articulated object requires perceiving itskinematic hierarchy: its parts, how each can move, and howthose motions are coupled. Previous work has explored per-ception for kinematics, but none infers a complete…
External neural memory structures have recently become a popular tool for algorithmic deep learning (Graves et al. 2014, Weston et al. 2014). These models generally utilize differentiable versions of traditional discrete memory-access…
Spatial curvature is one of the fundamental cosmological parameters that is routinely constrained from observations. The forward modelling of observations, in particular of large-scale structure, often relies on large cosmological…
We examine different generalizations of checking stack automata by allowing multiple input heads and multiple stacks, and characterize their computing power in terms of two-way multi-head finite automata and space-bounded Turing machines.…
The main result of this article is a geometric interpretation of magnitude, a real-valued invariant of metric spaces. We introduce a Euclidean embedding of a (suitable) finite metric space $X$ such that the magnitude of $X$ can be expressed…
The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of…
Reverse engineering the brain is proving difficult, perhaps impossible. While many believe that this is just a matter of time and effort, a different approach might help. Here, we describe a very simple idea which explains the power of the…
What kinds of symmetry-protected topologically ordered (SPTO) ground states can be used for universal measurement-based quantum computation in a similar fashion to the 2D cluster state? 2D SPTO states are classified not only by global…