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The Discontinuity Problem

Vasco Brattka

TL;DR

The paper introduces the discontinuity problem $DIS$ within the continuous Weihrauch lattice and proves it characterizes effective discontinuity via Weihrauch reducibility. It develops a robust set of tools, including the universal function $U$, the Smn theorem, and a uniform recursion framework, to relate discontinuity to game-theoretic notions. A generalized Wadge game characterization shows that continuity corresponds to II-winning strategies while effective discontinuity corresponds to I-winning strategies, with determinacy results tying these concepts to $AD$ and broader set-theoretic assumptions. The work also connects computable discontinuity to productivity, analyzes the impact of the axiom of choice, and demonstrates the precise bottom structure (under $ZF+DC+AD$) with $DIS$ occupying a minimal discontinuous degree, while highlighting rich interplays with $NRNG$, $ Bernstein$ sets, and parallelization to $NON$.

Abstract

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.

The Discontinuity Problem

TL;DR

The paper introduces the discontinuity problem within the continuous Weihrauch lattice and proves it characterizes effective discontinuity via Weihrauch reducibility. It develops a robust set of tools, including the universal function , the Smn theorem, and a uniform recursion framework, to relate discontinuity to game-theoretic notions. A generalized Wadge game characterization shows that continuity corresponds to II-winning strategies while effective discontinuity corresponds to I-winning strategies, with determinacy results tying these concepts to and broader set-theoretic assumptions. The work also connects computable discontinuity to productivity, analyzes the impact of the axiom of choice, and demonstrates the precise bottom structure (under ) with occupying a minimal discontinuous degree, while highlighting rich interplays with , sets, and parallelization to .

Abstract

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.

Paper Structure

This paper contains 6 sections, 32 theorems, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Universal Function and the Recursion Theorem
  3. Effectively Discontinuous Problems
  4. A Game Characterization
  5. Computable Discontinuity and Productivity
  6. Conclusions

Key Result

Theorem 4

Let $f:\subseteq X\rightrightarrows Y$ be a problem. Then we obtain

Figures (3)

  • Figure 1: Problems $f:\subseteq X\rightrightarrows Y$ with respect to $\leq_{\rm W}^*$ and $\leq_{\rm sW}^*$.
  • Figure 2: Sets $A\subseteq{\Bbb{N}}$ with respect to many-one reducibility $\leq_{\rm m}$.
  • Figure 3: Determinacy properties in $\mathsf{ZF}+\mathsf{DC}$.

Theorems & Definitions (59)

  • Definition 1: Problems
  • Definition 2: Weihrauch reducibility
  • Theorem 4: Continuity and effective discontinuity
  • Theorem 5: Myhill 1955
  • Theorem 6
  • Theorem 7: Wadge games
  • Corollary 8
  • Conjecture 9: Solovay
  • Theorem 10
  • Theorem 11: smn-Theorem
  • ...and 49 more