Neuromorphic Twins for Networked Control and Decision-Making

We consider the problem of remotely tracking the state of and unstable linear time-invariant plant by means of data transmitted through a noisy communication channel from an algorithmic point of view. Assuming the dynamics of the plant are known, does there exist an algorithm that accepts a description of the channel's characteristics as input, and returns 'Yes' if the transmission capabilities permit the remote tracking of the plant's state, 'No' otherwise? Does there exist an algorithm that, in case of a positive answer, computes a suitable encoder/decoder-pair for the channel? Questions of this kind are becoming increasingly important with regards to future communication technologies that aim to solve control engineering tasks in a distributed manner. In particular, they play an essential role in digital twinning, an emerging information processing approach originally considered in the context of Industry 4.0. Yet, the abovementioned questions have been answered in the negative with respect to algorithms that can be implemented on idealized digital hardware, i.e., Turing machines. In this article, we investigate the remote state estimation problem in view of the Blum-Shub-Smale computability framework. In the broadest sense, the latter can be interpreted as a model for idealized analog computation. Especially in the context of neuromorphic computing, analog hardware has experienced a revival in the past view years. Hence, the contribution of this work may serve as a motivation for a theory of neuromorphic twins as a counterpart to digital twins for analog hardware

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Three models of ordinal computability

In this thesis we expand the scope of ordinal computability, i.e., the study of models of computation that are generalized to infinite domains. The discipline sets itself apart from classical work on generalized recursion theory by focusing strongly on the computational paradigm and an analysis in elementary computational steps. In the present work, two models of classical computability of which no previous generalizations to ordinals are known to the author are lifted to the ordinal domain, namely λ-calculus and Blum-Shub-Smale machines. One of the multiple generalizations of a third model relevant to this thesis, the Turing machine, is employed to further study classical descriptive set theory. The main results are: An ordinal λ-calculus is defined and confluency properties in the form of a weak Church-Rosser theorem are established. The calculus is proved to be strongly related to the constructible hierarchy of sets, a feature typical for an entire subfamily of models of ordinal computation. Ordinal Turing machines with input restricted to subsets of ω are shown to compute the Δ12 sets of reals. Conversely, the machines can be employed to reprove the absoluteness of Σ12 sets (Shoenfield absoluteness) and basic properties of Σ12 sets. New tree representations and new pointclasses defined by the means of ordinal Turing computations are introduced and studied. The Blum-Shub-Smale model for computation on the real numbers is lifted to transfinite running times. The supremum of possible runtimes is determined and an upper bound on the computational strength is given. Summarizing, this thesis both expands the field of ordinal computability by enlarging its palette of computational models and also connects the field further by tying in the new models into the existing framework. Questions that have been raised in the community, e.g. on the possibility of generalizations of λ-calculus and Blum-Shub-Smale machines, are addressed and answered

The complexity and geometry of numerically solving polynomial systems

These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system--solving, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker

Computability of Julia sets

In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable.Comment: Revised. To appear in Moscow Math. Journa