[Paper Review] A topological view on algebraic computation models

Eike Neumann, Arno Pauly|arXiv (Cornell University)|Feb 25, 2016

Computability, Logic, AI Algorithms31 references4 citations

TL;DR

This paper establishes a formal connection between the BSS model of computation and computable analysis using Weihrauch reducibility, showing that for functions with solvability complexity index (SCI) ≥ 2, the SCI is identical across both models. It characterizes the computational power of BSS-machines and analytic machines via Weihrauch degrees, demonstrating that topological limitations—particularly discontinuity—underlie non-computability in both frameworks.

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.

Motivation & Objective

• To investigate the topological constraints on computability in algebraic computation models, particularly the BSS model.
• To compare BSS-computability with computable analysis (TTE) by analyzing their shared non-computability structure.
• To characterize the computational power of BSS-machines and analytic machines using Weihrauch degrees.
• To show that the solvability complexity index (SCI) is invariant across models when SCI ≥ 2, establishing common ground between BSS and TTE frameworks.

Proposed method

• Uses Weihrauch reducibility as a framework to compare computational strength across models.
• Characterizes BSS-machines with equality and order tests as having computational power equivalent to closed choice over ℕ (Cₙ).
• Shows that BSS-machines without order tests are characterized by the weaker LPO* degree.
• Introduces and analyzes the Weihrauch degree Sort* for analytic machines, capturing their computational power.
• Applies the Stone-Weierstrass theorem to approximate BSS-computable functions by rational polynomials, enabling transfer to TTE-computability.
• Uses absorption properties of limit operators (lim⋆lim⋆Sort ≡ lim⋆lim) to reduce complexity in SCI comparisons.

Experimental results

Research questions

• RQ1 How do topological obstructions—particularly discontinuity—limit BSS-computability?
• RQ2 To what extent do algebraic and topological reasons for non-computability differ across BSS variants?
• RQ3 What is the relationship between the solvability complexity index (SCI) in the BSS model and in computable analysis (TTE)?
• RQ4 Can the computational power of BSS-machines and analytic machines be fully captured by Weihrauch degrees?
• RQ5 Is there a common framework for non-computability in BSS and TTE models when SCI ≥ 2?

Key findings

• For functions with SCI ≥ 2, the solvability complexity index is identical in the BSS model and the TTE model.
• BSS-machines with equality and order tests compute exactly the functions Weihrauch-reducible to Cₙ, the closed choice on ℕ.
• BSS-machines without order tests are characterized by the strictly weaker LPO* degree.
• Analytic machines are characterized by the Weihrauch degree Sort*, which captures their ability to compute via limits of approximations.
• The BSS halting problem cannot be solved by any machine using only continuous operations, due to topological reasons.
• The composition lim⋆lim⋆Sort* is Weihrauch-equivalent to lim⋆lim, enabling reduction of higher-order limits in SCI analysis.