Deciding Conditional Termination

We address the problem of conditional termination, which is that of defining the set of initial configurations from which a given program always terminates. First we define the dual set, of initial configurations from which a non-terminating execution exists, as the greatest fixpoint of the function that maps a set of states into its pre-image with respect to the transition relation. This definition allows to compute the weakest non-termination precondition if at least one of the following holds: (i) the transition relation is deterministic, (ii) the descending Kleene sequence overapproximating the greatest fixpoint converges in finitely many steps, or (iii) the transition relation is well founded. We show that this is the case for two classes of relations, namely octagonal and finite monoid affine relations. Moreover, since the closed forms of these relations can be defined in Presburger arithmetic, we obtain the decidability of the termination problem for such loops.Comment: 61 pages, 6 figures, 2 table

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

Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems

A new class of non-homogeneous state-affine systems is introduced for use in reservoir computing. Sufficient conditions are identified that guarantee first, that the associated reservoir computers with linear readouts are causal, time-invariant, and satisfy the fading memory property and second, that a subset of this class is universal in the category of fading memory filters with stochastic almost surely uniformly bounded inputs. This means that any discrete-time filter that satisfies the fading memory property with random inputs of that type can be uniformly approximated by elements in the non-homogeneous state-affine family.Comment: 41 page

A modular architecture for transparent computation in recurrent neural networks

publisher: Elsevier articletitle: A modular architecture for transparent computation in recurrent neural networks journaltitle: Neural Networks articlelink: http://dx.doi.org/10.1016/j.neunet.2016.09.001 content_type: article copyright: © 2016 Elsevier Ltd. All rights reserved

Dynamical Systems Theory for Transparent Symbolic Computation in Neuronal Networks

In this thesis, we explore the interface between symbolic and dynamical system computation, with particular regard to dynamical system models of neuronal networks. In doing so, we adhere to a definition of computation as the physical realization of a formal system, where we say that a dynamical system performs a computation if a correspondence can be found between its dynamics on a vectorial space and the formal system’s dynamics on a symbolic space. Guided by this definition, we characterize computation in a range of neuronal network models. We first present a constructive mapping between a range of formal systems and Recurrent Neural Networks (RNNs), through the introduction of a Versatile Shift and a modular network architecture supporting its real-time simulation. We then move on to more detailed models of neural dynamics, characterizing the computation performed by networks of delay-pulse-coupled oscillators supporting the emergence of heteroclinic dynamics. We show that a correspondence can be found between these networks and Finite-State Transducers, and use the derived abstraction to investigate how noise affects computation in this class of systems, unveiling a surprising facilitatory effect on information transmission. Finally, we present a new dynamical framework for computation in neuronal networks based on the slow-fast dynamics paradigm, and discuss the consequences of our results for future work, specifically for what concerns the fields of interactive computation and Artificial Intelligence

Computation and Learning in High Dimensions (hybrid meeting)

The most challenging problems in science often involve the learning and accurate computation of high dimensional functions. High-dimensionality is a typical feature for a multitude of problems in various areas of science. The so-called curse of dimensionality typically negates the use of traditional numerical techniques for the solution of high-dimensional problems. Instead, novel theoretical and computational approaches need to be developed to make them tractable and to capture fine resolutions and relevant features. Paradoxically, increasing computational power may even serve to heighten this demand, since the wealth of new computational data itself becomes a major obstruction. Extracting essential information from complex problem-inherent structures and developing rigorous models to quantify the quality of information in a high-dimensional setting pose challenging tasks from both theoretical and numerical perspective. This has led to the emergence of several new computational methodologies, accounting for the fact that by now well understood methods drawing on spatial localization and mesh-refinement are in their original form no longer viable. Common to these approaches is the nonlinearity of the solution method. For certain problem classes, these methods have drastically advanced the frontiers of computability. The most visible of these new methods is deep learning. Although the use of deep neural networks has been extremely successful in certain application areas, their mathematical understanding is far from complete. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computational methods and to promote the exchange of ideas emerging in various disciplines about how to treat multiscale and high-dimensional problems