Approximate decidability in euclidean spaces

Abstract

We study concepts of decidability (recursivity) for subsets of Euclidean spaces Ê within the framework of approximate computability (type two theory of effectivity). A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff's concept of resolvable sets, and it modifies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of sets can equivalently be expressed by computability of the characteristic functions by means of appropriately working oracle Turing machines. The notion fulfills some natural requirements and is hereditary under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidability can be applied to investigate concepts of reducibility for subsets of Euclidean spaces.

Slutsky Matrix Norms and Revealed Preference Tests of Consumer Behaviour

“…When the null is false the numerator will increase, thus we conclude that the true SMN is positive when the test quantity is large enough. This is in line with δ−decidability (Gao et al, 2012) or approximate decidability (Hemmerling, 2003).…”

Characterizations of the class Δta2 over Euclidean spaces

“…Nevertheless, the class ∆ ta 2 plays a crucial role in this context, since it establishes a frame general enough to include all reasonable concepts of decidability. So the notions of decidability introduced and discussed [9] yield (proper) subclasses of ∆ ta 2 . Weihrauch [22] is said to be r. e. closed iff A = ∅ or there is a total recursive function ϕ :…”

“…So it gives merely a necessary condition of decidability. For a more detailed treatment, the reader is referred to [9]. Nevertheless, the class ∆ ta 2 plays a crucial role in this context, since it establishes a frame general enough to include all reasonable concepts of decidability.…”

“…[9]. In this paper, notions of recursivity related to point sets of Euclidean spaces and defined on the basis of rational open balls will be denoted as topologically recursive, briefly: t. r..…”

“…In this paper, we concentrate ourselves to the case of Euclidean spaces R k as universes, which is fundamental for computable analysis, even if some of our results and techniques also apply to more general effective metric or topological spaces as specified, e. g., in [21,8], The class ∆ ta 2 of the TAH over Euclidean spaces is already large enough to include most classes of sets of points which are important in computable analysis. In particular, the several notions of decidability (recursivity, computability) for subsets of Euclidean spaces form subclasses of ∆ ta 2 , as it was shown and discussed in detail in [9]. So it is highly desirable to develop classifications for the members of that class, i. e., hierarchies of subclasses.…”