Games for Functions: Baire Classes, Weihrauch Degrees, Transfinite Computations, and Ranks

2022

Preprint

Weihrauch complexity is now an established and active part of mathematical logic. It can be seen as a computability-theoretic approach to classifying the uniform computational content of mathematical problems. This theory has become an important interface between more proof-theoretic and more computability-theoretic studies in the realm of reverse mathematics. Here we present a historical account of the early developments of Weihrauch complexity by the Hagen school of computable analysis that started more than thirty years ago, and we indicate how this has influenced, informed, and anticipated more recent developments of the subject.

Section: Discussionmentioning

confidence: 99%

Weihrauch Complexity and the Hagen School of Computable Analysis

2022

Preprint

“…In Section 4 we prove that the axiom of choice is actually required for such a construction. This can be achieved with the help of a variant of Wadge games for problems f :⊆ X ⇒ Y , originally considered by Nobrega and Pauly [20,21]. In fact, we can prove the following result (see Theorem 4.3), part (1) of which is already due to Nobrega and Pauly.…”

mentioning

confidence: 83%

The Discontinuity Problem

2022

J. symb. log.

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.

“…In section 4 we prove that the axiom of choice is actually required for such a construction. This can be achieved with the help of a variant of Wadge games for problems f :⊆ X ⇒ Y , originally considered by Nobrega and Pauly [20,21]. In fact, we can prove the following result (see Theorem 27), part (1) of which is already due to Nobrega and Pauly.…”

mentioning

confidence: 84%

“…Wadge [27] introduced games on subsets A, B ⊆ N N to characterize the reducibility that is named after him. Nobrega and Pauly [20,21] have used a modification of Wadge games for problems f :⊆ X ⇒ Y in order to characterize lower cones in the Weihrauch lattice. We consider similar generalized 6 versions of Wadge and Lipschitz games, defined as follows.…”

Section: A Game Characterizationmentioning

confidence: 99%

The Discontinuity Problem

2020

Preprint

Matthias Schröder has asked the question whether there is a weakest discontinuous problem in the continuous 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 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 AD, but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity.