Effective local compactness and the hyperspace of located sets

Pauly, Arno

doi:10.48550/arxiv.1903.05490

Cited by 3 publications

(7 citation statements)

References 8 publications

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“…The price to pay for this generalization in the classic setting is that we no longer obtain a unique probability measure, but merely a locally finite measure identified up to a constant scaling factor. A notion of effective local compactness is available (see [20]), but any such generalization seems to require new proof techniques beyond those employed in this article.…”

Section: Discussionmentioning

confidence: 99%

Computing Haar Measures

Pauly¹,

Seon²,

Ziegler³

2019

Preprint

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According to Haar's Theorem, every compact group G admits a unique left-invariant Borel probability measure µG. Let the Haar integral (of G) denote the functional G : C(G) f → f dµG integrating any continuous function f : G → R with respect to µG. This generalizes, and recovers for the additive group G = [0; 1) mod 1, the usual Riemann integral: computable (cmp. Weihrauch 2000, Theorem 6.4.1), and of computational cost characterizing complexity class #P1 (cmp. Ko 1991, Theorem 5.32).We establish that in fact every computably compact computable metric group renders the Haar integral computable: once using an elegant synthetic argument; and once presenting and analyzing an explicit, imperative algorithm. Regarding computational complexity, for the groups SO(3) and SU(2) we reduce the Haar integral to and from Euclidean/Riemann integration. In particular both also characterize #P1. ACM Subject ClassificationTheory of computation → Constructive mathematics; Mathematics of computing → Continuous mathematics Keywords and phrases computable analysis, topological groups, exact real arithmetic, Haar measure Digital Object Identifier 10.4230/LIPIcs.CSL.2020.00 European Union's Horizon 2020 MSCA IRSES project 731143. Motivation and OverviewComplementing empirical approaches, heuristics, and recipes [23,25], Computable Analysis[30] provides a rigorous algorithmic foundation to Numerics, as well as a way of formally measuring the constructive contents of theorems in classical Calculus. Haar's Theorem is such an example, of particular beauty combining three categories: compact metric spaces, algebraic groups, and measure spaces:Fact 1. Let (G, e, •, • −1 ) denote a group and (G, d) a compact metric space such that the group operation • and inverse operation • −1 are continuous with respect to d (that is, a topological group). There exists a unique left-invariant Borel probability measure µ G , called Haar measure, on G.We refrain from expanding on generalizations to locally-compact Hausdorff spaces. Recall that a left-invariant measure satisfies µ(U ) = µ(g • U ) for every g ∈ G and every measurable

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Section: Discussionmentioning

confidence: 99%

Computing Haar Measures

Pauly¹,

Seon²,

Ziegler³

2019

Preprint

Self Cite

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Section: Basic Verification Questionsmentioning

confidence: 99%

“…Subsequently, we can obtain every closed ball as a compact set given a radius less than E x . If every closed ball is compact, we can even obtain them computably as elements of K(X) by [27,Proposition 10].…”

Section: Basic Verification Questionsmentioning

confidence: 99%

A Computability Perspective on (Verified) Machine Learning

Crook¹,

Morgan²,

Pauly³

et al. 2021

Preprint

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There is a strong consensus that combining the versatility of machine learning with the assurances given by formal verification is highly desirable. It is much less clear what verified machine learning should mean exactly. We consider this question from the (unexpected?) perspective of computable analysis. This allows us to define the computational tasks underlying verified ML in a model-agnostic way, and show that they are in principle computable.

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“…To this end, we need to obtain closed balls B(x, r) as elements of (V ∧ K)(X). The property that for every x ∈ X we can find an R > 0 such that for every r < R we can compute B(x, r) ∈ K(X) is a characterisation of effective local compactness of a computable metric space X [128]. We generally get clB(x, r), the closure of the open ball, as elements of V(X).…”

Section: A Theory Of Treating Adversarial Examplesmentioning

confidence: 99%

“…We generally get clB(x, r), the closure of the open ball, as elements of V(X). For all but countably many radii r we have that B(x, r) = clB(x, r), and we can effectively compute suitable radii within any interval [128]. Definition 7.10 Let (X, d) be a computable metric space, and C(X, k ⊥ ) the space of classifiers.…”

Section: A Theory Of Treating Adversarial Examplesmentioning

confidence: 99%

Computable Analysis and Game Theory: From Foundations to Applications

Crook

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This body of research showcases several facets of the intersection between computer science and game theory. On the foundational side, we explore the obstructions to the computability of Nash equilibria in the setting of computable analysis. In particular, we study the Weihrauch degree of the problem of ﬁnding a Nash equilibrium for a multiplayer game in normal form. We conclude that the Weihrauch degree Nash for multiplayer games lies between AoUC∗[0,1] and AoUC⋄[0,1] (Theorem 5.3). As a slight detour, we also explore the demarcation between computable and non-computable computational problems pertaining to the veriﬁcation of machine learning. We demonstrate that many veriﬁcation questions are computable without the need to specify a machine learning framework (Section 7.2). As well as looking into the theory of learners, robustness and sparisty of training data. On the application side, we study the use of Hypergames in Cybersecurity. We look into cybersecurity AND/OR attack graphs and how we could turn them into a hypergame (8.1). Hyper Nash equilibria is not an ideal solution for these games, however, we propose a regret-minimisation based solution concept. In Section 8.2, we survey the area of Hypergames and their connection to cybersecurity, showing that even if there is a small overlap, the reach is limited. We suggest new research directions such as adaptive games, generalisation and transferability (Section 8.3).

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Effective local compactness and the hyperspace of located sets

Cited by 3 publications

References 8 publications

Computing Haar Measures

Computing Haar Measures

A Computability Perspective on (Verified) Machine Learning

Computable Analysis and Game Theory: From Foundations to Applications

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