Matthew de Brecht

Closed choice and a uniform low basis theorem

Abstract We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed choice of the singleton space, the natural numbers, Cantor space and Baire space correspond to the class of computable functions, functions computable with finitely many mind changes, weakly computable functions and effectively Borel measurable functions, respectively. We also prove that all these classes correspond to classes of non-deterministically computable functions with the respective spaces as advice spaces. The class of limit computable functions can be characterized with parallelized choice of natural numbers. On top of these results we provide further insights into algebraic properties of closed choice. In particular, we prove that closed choice on Euclidean space can be considered as “locally compact choice” and it is obtained as product of closed choice on the natural numbers and on Cantor space. We also prove a Quotient Theorem for compact choice which shows that single-valued functions can be “divided” by compact choice in a certain sense. Another result is the Independent Choice Theorem, which provides a uniform proof that many choice principles are closed under composition. Finally, we also study the related class of low computable functions, which contains the class of weakly computable functions as well as the class of functions computable with finitely many mind changes. As a main result we prove a uniform version of the Low Basis Theorem that states that closed choice on Cantor space (and the Euclidean space) is low computable. We close with some related observations on the Turing jump operation and its initial topology.

Quasi-Polish spaces

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is level \Pi_2 in the Borel hierarchy. Quasi-Polish spaces can be characterized within the framework of Type-2 Theory of Effectivity as precisely the countably based spaces that have an admissible representation with a Polish domain. They can also be characterized domain theoretically as precisely the spaces that are homeomorphic to the subspace of all non-compact elements of an \omega-continuous domain. Every countably based locally compact sober space is quasi-Polish, hence every \omega-continuous domain is quasi-Polish. A metrizable space is quasi-Polish if and only if it is Polish. We show that the Borel hierarchy on an uncountable quasi-Polish space does not collapse, and that the Hausdorff-Kuratowski theorem generalizes to all quasi-Polish spaces.

A neural network implementation of a saliency map model

The saliency map model proposed by Itti and Koch [Itti, L., & Koch, C. (2000). A saliency-based search mechanism for overt and covert shifts of visual attention. Vision Research, 40, 1489-1506] has been a popular model for explaining the guidance of visual attention using only bottom-up information. In this paper we expand Itti and Kochs model and propose how it could be implemented by neural networks with biologically realistic dynamics. In particular, we show that by incorporating synaptic depression into the model, network activity can be normalized and competition within the feature maps can be regulated in a biologically plausible manner. Furthermore, the dynamical nature of our model permits further analysis of the time course of saliency computation, and also allows the model to calculate saliency for dynamic visual scenes. In addition to explaining the high saliency of pop-out targets in visual search tasks, our model explains attentional grab by sudden-onset stimuli, which was not accounted for by previous models.

Non-deterministic computation and the Jayne-Rogers Theorem

We provide a simple proof of a computable analogue to the Jayne Rogers Theorem from descriptive set theory. The difficulty of the proof is delegated to a simulation result pertaining to non-deterministic type-2 machines. Thus, we demonstrate that developments in computational models can have applications in fields thought to be far removed from it.

Inferability of closed set systems from positive data

In this paper, we generalize previous results showing connections between inductive inference from positive data and algebraic structures by using tools from universal algebra. In particular, we investigate the inferability from positive data of language classes defined by closure operators. We show that some important properties of language classes used in inductive inference correspond closely to commonly used properties of closed set systems. We also investigate the inferability of algebraic closed set systems, and show that these types of systems are inferable from positive data if and only if they contain no infinite ascending chain of closed sets. This generalizes previous results concerning the inferability of various algebraic classes such as the class of ideals of a ring. We also show the relationship with algebraic closed set systems and approximate identifiability as introduced by Kobayashi and Yokomori [11]. We propose that closure operators offer a unifying framework for various approaches to inductive inference from positive data.

Descriptive Set Theory in the Category of Represented Spaces

We propose to extend descriptive set theory (DST) beyond its traditional setting of Polish spaces to the represented spaces. There, we can reformulate DST in terms of endofunctors on the categories of represented spaces and computable or continuous functions. In particular, this approach satisfies the demand for a uniform approach to both classic and effective DST -- computability follows naturally from the setting, rather than having to be explicitly demanded. The previous endeavour to extend DST to the Quasi-Polish spaces is subsumed by this work. In several cases the category-theoretic setting enables new, very succinct proofs, and sheds a new light on why certain results are true. The framework lets us make formal some natural questions not easily approachable by traditional methods.

The effect of mood state on visual search times for detecting a target in noise:An application of smartphone technology

The study of visual perception has largely been completed without regard to the influence that an individual’s emotional status may have on their performance in visual tasks. However, there is a growing body of evidence to suggest that mood may affect not only creative abilities and interpersonal skills but also the capacity to perform low-level cognitive tasks. Here, we sought to determine whether rudimentary visual search processes are similarly affected by emotion. Specifically, we examined whether an individual’s perceived happiness level affects their ability to detect a target in noise. To do so, we employed pop-out and serial visual search paradigms, implemented using a novel smartphone application that allowed search times and self-rated levels of happiness to be recorded throughout each twenty-four-hour period for two weeks. This experience sampling protocol circumvented the need to alter mood artificially with laboratory-based induction methods. Using our smartphone application, we were able to replicate the classic visual search findings, whereby pop-out search times remained largely unaffected by the number of distractors whereas serial search times increased with increasing number of distractors. While pop-out search times were unaffected by happiness level, serial search times with the maximum numbers of distractors (n = 30) were significantly faster for high happiness levels than low happiness levels (p = 0.02). Our results demonstrate the utility of smartphone applications in assessing ecologically valid measures of human visual performance. We discuss the significance of our findings for the assessment of basic visual functions using search time measures, and for our ability to search effectively for targets in real world settings.

Noetherian Quasi-Polish spaces.

In the presence of suitable power spaces, compactness of

Combining sparseness and smoothness improves classification accuracy and interpretability

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Levels of discontinuity, limit-computability, and jump operators.

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