The Discontinuity Problem

Abstract

Matthias Schröder has asked the question whether there is a weakest discontinuous problem in the topological version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a suitable way. However, in which sense this answers Schröder’s question sensitively depends on the axiomatic framework that is chosen, and it is a positive answer if we work in Zermelo–Fraenkel set theory with dependent choice and the axiom of determinacy $\mathsf {AD}$ . On the other hand, using the full axiom of choice, one can construct problems which are discontinuous, but not effectively so. Hence, the exact situation at the bottom of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose. We prove our result using a variant of Wadge games for mathematical problems. While the existence of a winning strategy for Player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for Player I characterizes effective discontinuity of the problem. By Weihrauch determinacy we understand the condition that every problem is either continuous or effectively discontinuous. This notion of determinacy is a fairly strong notion, as it is not only implied by the axiom of determinacy $\mathsf {AD}$ , but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity.

Stashing And Parallelization Pentagons

“…Then DIS ≤ W f ⇐⇒ Player I has a computable winning strategy in the Wadge game f . An analogous result holds for ≤ * W and (not necessarily computable) winning strategies [3,Theorem 27]. We are going to use Theorem 9 in the proof of Proposition 32 that establishes the pentagon of ACC N shown in Figure 1.…”

“…Now we are prepared to prove that stashing is an interior operator on the (strong) Weihrauch lattice. Properties (1) and (2) in the following result are only made possible by the usage of the completion Y , whereas the existence of a retraction according to Lemma 17 guarantees that the completion is not an obstacle for property (3). For a problem f :⊆ X ⇒ Y the completion is defined by…”

“…The problem NRNG is called range non-equality problem and is introduced here. 2 The universal function U has been defined differently in [3], but our definition is equivalent, as the function U defined here also satisfies a utm-and an smn-theorem [32]. In particular, the discontinuity problem DIS defined with one version of U is strongly Weihrauch equivalent to the one defined with the other version of U.…”

“…Nobrega and Pauly used Wadge games to characterize certain lower cones in the Weihrauch lattice [26]. In [3] we have characterized the upper cone of the discontinuity problem by Wadge games on problems. The characterization goes as follows [3, Theorem 17, Corollary 28].…”

“…In fact, they are related through NON ≡ sW ΠΣ(ACC N ). The discontinuity problem DIS was introduced in [3] and it was proved that (under the axiom of determinacy) DIS is actually the weakest discontinuous problem with respect to the topological version of Weihrauch reducibility ≤ * W . Part of the above relation between ACC N and NON is that we are going to prove that DIS parallelizes to NON (see Theorem 34).…”