Computability on subsets of Euclidean space I: closed and compact subsets

Brattka, Vasco; Weihrauch, Klaus

doi:10.1016/s0304-3975(98)00284-9

Cited by 112 publications

(105 citation statements)

References 17 publications

Supporting

Mentioning

105

Contrasting

Order By: Relevance

“…We recall that the co-c.e. closed sets A ⊆ [0, 1] are exactly those for which there exists a computable function f : [0, 1] → R with f −1 {0} = A (see [14,52]). Somewhat surprisingly, we can attach an arbitrary computable point to an arbitrary co-c.e.…”

Section: Proof If a ⊆ R Is Non-empty Then There Is A Rational Intermentioning

confidence: 99%

Effective Choice and Boundedness Principles in Computable Analysis

2011

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn-Banach Theorem and Weak Kőnig's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example. §1. Introduction. The purpose of this paper is to propose a new approach to classify mathematical theorems according to their computational content and according to their logical complexity.2000 Mathematics Subject Classification. 03F60,03D30,03B30,03E15.

show abstract

Section: Proof If a ⊆ R Is Non-empty Then There Is A Rational Intermentioning

confidence: 99%

Effective Choice and Boundedness Principles in Computable Analysis

2011

Self Cite

View full text Add to dashboard Cite

show abstract

“…By properties of computable compact sets, the distance function d K is computable [2], and, as a corollary, the set {(y , )|d K (y ) < } is Σ-definable. By Proposition 4 and Theorem 3, there exists a required sequence of quantifier free formulas {ψ} i∈ω .…”

Section: Bounded Existential Quantifier Case Suppose ϕ(X)mentioning

confidence: 99%

“…Let K be co-semicomputable compact set and time bounded by N and f a computable disturbance. The unreachability problem can be formalised as follows: ϕ ∀a ∈ Init X ∀t ∈ [0, N]H(f, a, t) ∈ K. By properties of cosemicomputable compact sets, the distance function d K is lower semicomputable [2], and, as a corollary, {x|x ∈ K} = {x|d K (x) > 0} is Σ-definable. By Theorem 6 and Proposition 2, ϕ is equivalent to a Σ-formula.…”

Section: Shs = T S X U D Init F Conv1 a Conv2mentioning

confidence: 99%

See 1 more Smart Citation

Σ K –constraints for Hybrid Systems

Korovina

Kudinov

2010

Perspectives of Systems Informatics

View full text Add to dashboard Cite

show abstract

“…The precise condition for this is that the complement set Ω c is poly-time computable as a subset of R d in the sense of Computable Analysis. See for example [2,17,3] for more details on poly-time computability of real sets. The vast majority of domains in applications satisfy this condition.…”

Section: The Walk On Spheres Algorithmmentioning

confidence: 99%

The complexity of simulating Brownian Motion

Binder¹,

Braverman²

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

We analyze the complexity of the Walk on Spheres algorithm for simulating Brownian Motion in a domain Ω ⊂ R d . The algorithm, which was first proposed in the 1950s, produces samples from the hitting probability distribution of the Brownian Motion process on ∂Ω within an error of ε. The algorithm is used as a building block for solving a variety of differential equations, including the Dirichlet Problem. The WoS algorithm simulates a BM starting at a point X 0 = x in a given bounded domain Ω until it gets ε-close to the boundary ∂Ω. At every step, the algorithm measures the distance d k from its current position X k to ∂Ω and jumps a distance of d k /2 in a uniformly random direction from X k to obtain X k+1 . The algorithm terminates when it reaches X n that is ε-close to ∂Ω. It is not hard to see that the algorithm requires at least Ω(log 1/ε) steps to converge. Only partial results with respect to the upper bound existed. In 1959 M. Motoo established an O(log 1/ε) bound on the running time for convex domains. The results were later generalized for a wider, but still very restricted, class of planar and 3-dimensional domains by G.A. Mikhailov (1979). In our earlier work (2007), we established an upper bound of O(log 2 1/ε) on the rate of convergence of WoS for arbitrary planar domains. In this paper we introduce energy functions using Newton potentials to obtain very general upper bounds on the convergence of the algorithm. Special instances of the upper bounds yield the following results for bounded domains Ω:• if Ω is a planar domain with connected exterior, the WoS converges in O(log 1/ε) steps;• if Ω is a domain in R 3 with connected exterior, the WoS converges in O(log 2 1/ε) steps;• for d > 2, if Ω is a domain in R d , the WoS converges in O ((1/ε) 2−4/d ) steps;

show abstract

Computability on subsets of Euclidean space I: closed and compact subsets

Cited by 112 publications

References 17 publications

Effective Choice and Boundedness Principles in Computable Analysis

Effective Choice and Boundedness Principles in Computable Analysis

Σ K –constraints for Hybrid Systems

The complexity of simulating Brownian Motion

Contact Info

Product

Resources

About