Weihrauch Complexity in Computable Analysis

Abstract

We provide a self-contained introduction into Weihrauch complexity and its applications to computable analysis. This includes a survey on some classification results and a discussion of the relation to other approaches. The Algebra of ProblemsThe Weihrauch lattice offers a framework to classify the uniform computational content of problems and theorems from analysis and other areas of mathematics. This framework can be seen as an attempt to create a calculus of mathematical problems, very much in spirit of Kolmogorov's interpretation of intuitionistic logic [69].We express mathematical problems with the help of partial multi-valued functions f :⊆ X ⇒ Y , which are just relations f ⊆ X ×Y . It has turned out to be fruitful for our approach to think of these relations as input-output oriented multi-valued functions f :⊆ X ⇒ Y . We consider dom( f ) = {x ∈ X : f (x) = / 0} as the set of admissible instances x of the problem f , and we consider the corresponding set of function values f (x) ⊆ Y as the set of possible results. In the case of single-valued f we identify f (x) with the corresponding singleton. An example of a mathematical

Lawvere-Tierney topologies for computability theorists

“…As one such class, we first deal with partial multi-valued functions (abbreviated as multifunctions) on N. In recent years, the notion of partial multifunction has received a great deal of attention in computability theory and related fields; see e.g. [6].…”

“…A similar notion for partial multifunctions on N N has been extensively studied under the name Weihrauch reducibility; see e.g. Brattka-Gherardi-Pauly [6]. Indeed, one-query Turing reducibility is exactly the restriction of Weihrauch reducibility to functions on N. This reducibility is also called many-one reducibility in [31].…”

“…There are a huge number of mathematical principles which are equivalent to LLPO; see Diener [11] and also Brattka-Gherardi-Pauly [6]. The principle LLPO may also be called de Morgan's law for Σ 0 1 formulas.…”

“…The principle LLPO 1/ is first introduced in Richman [34], and extensively studied in constructive mathematics and related areas. For the computability-theoretic study, see Brattka-Gherardi-Pauly [6]. Note that one can deduce several results on the structure of ({LLPO m/k } m<k , ≤ 1 LT ) from the work by Cenzer-Hinman [8].…”

“…Coincidentally, it turns out that our notion is heavily related to another notion called extended Weihrauch reducibility, which is introduced in Bauer [2]. By viewing topologies in this way, it is possible, for example, to position the study of the structure of Lawvere-Tierney topologies as an extension of the Weihrauch-style analogue [6] of reverse mathematics (note, however, that this is by no means an extension of the standard reverse mathematics [38,11] at all; reverse mathematics has more to do with the internal logic, and more finitary aspects). By bringing the arguments on topologies into pure computability theory in this way, we solve some problems proposed in [23,24].…”