A topological view on algebraic computation models

Abstract

We investigate the topological aspects of some algebraic computation models, in particular the BSS-model. Our results can be seen as bounds on how different BSS-computability and computability in the sense of computable analysis can be. The framework for this is Weihrauch reducibility. As a consequence of our characterizations, we establish that the solvability complexity index is (mostly) independent of the computational model, and that there thus is common ground in the study of non-computability between the BSS and TTE setting. * Pauly has since moved to the Département d'Informatique, Université libre de Bruxelles, Belgium.

Finite Choice, Convex Choice and Sorting

“…We also get an alternative proof of the following, which was previously shown in [24] using the squashing principle from [15]:…”

“…In [28] it is shown that Sort n+1 ≡ W Sort n ; it follows that Sort * ≡ W Sort * d ≡ W d∈N Sort d . The degree Sort * was shown in [24] to capture the strength of the strongly analytic machines [14,16], which in turn are an extension of the BSS-machines [1]. Sort is equivalent to Thomae's function; and to the translation of the standard representation of the reals into the continued fraction representation [30].…”

How constructive is constructing measures?

“…follows from arguments in Neumann and Pauly [34]. Since lim ≡ W lim, we can semidecide whether inf U d > 0 for all rational d in parallel, and this suffices to obtain inf{d…”

Comparing Representations for Function Spaces in Computable Analysis

“…2 The Weihrauch degree corresponding to C N has received significant attention, e.g. in [3,4,6,7,[15][16][17][18][19]. In particular, as shown in [23], a function between computable Polish spaces is Weihrauch reducible to C N if and only if it is piecewise computable or equivalently is effectively 0 2 -measurable.…”

“…Now, LPO * corresponds to carrying out a fixed finite but arbitrary high number of equality tests on the real or complex numbers via the operator from [20]. The operator introduced in [17] captures using the given degree an arbitrary finite number of times (without the requirement that the number is fixed in advance), and it holds that [C N ] = LPO . Define one last operator on the Weihrauch degrees by setting [f ] := [f N ].…”