Weihrauch Complexity in Computable Analysis

Reverse mathematics and Weihrauch analysis motivated by finite complexity theory

“…The results extend the currently small catalog of Weihrauch problems at this level, for example like those in [7]. Weihrauch analysis at the level of Π 1 1 -CA 0 appears to parallel familiar reverse mathematics closely, in contrast to recent results related to ATR 0 by Kihara, Marcone, and Pauly [8].…”

“…and note that LPO is a total problem of the form ∀p∃n(LPO(p) = n). This version of LPO differs from that presented in the survey by Brattka, Gherardi, and Pauly [1]. It is Weihrauch equivalent to their version, but not strongly Weihrauch equivalent, as the range of their version includes only {0, 1} and the range of this version includes all of N. However, for our purposes it is desirable to have the underlying predicate be ∃-free.…”

“…An early article of Hirst and Lempp [4] was motivated by the following example. The problem of determining if a finite graph has a Hamiltonian path is NP complete, while the problem for Eulerian paths is in the class P. In reverse mathematics, the problem of selecting which infinite graphs in an infinite sequence have Hamiltonian paths is equivalent to Π 1 1 -CA 0 , while the corresponding problem for Eulerian graphs is equivalent to ACA 0 . While very evocative, Hirst and Lempp showed that this parallel does not generally hold.…”

“…The final section addresses Weihrauch analysis of stronger results, with parallelizations at the Π 1 1 -CA 0 level. The formulas describing the problems are too complicated for formal analysis techniques, so we turn to traditional Weihrauch analysis.…”

Effective aspects of Hausdorff and Fourier dimension

“…We say that f is Weihrauch reducible to g (f ≤ W g) iff there are two computable maps Φ, Ψ :⊆ N N → N N s.t., for every realizer G of g, the map p → Ψ( p, GΦ(p) ) is a realizer for f . A thorough presentation on Weihrauch reducibility is out of the scope of this paper, and the reader is referred to [8].…”