Countable Ordered Groups and Weihrauch Reducibility

TL;DR

This paper explores the connection between reverse mathematics and Weihrauch reducibility through Maltsev's theorem on countable ordered groups, revealing the importance of order-preserving functions in the computational strength of related problems.

Contribution

It demonstrates that the strength of Maltsev's theorem in Weihrauch degrees depends on the presence of order-preserving functions in the problem's output.

Findings

The theorem's strength is linked to having a dense linear order without endpoints.

Order-preserving functions are necessary for the problem to match the strength of al.

Without order-preserving functions, the problems are significantly weaker.

Abstract

This paper continues to study the connection between reverse mathematics and Weihrauch reducibility. In particular, we study the problems formed from Maltsev's theorem on the order types of countable ordered groups. Solomon showed that the theorem is equivalent to $\mathsf{\Pi_1^1}$-$\mathsf{CA_0}$, the strongest of the big five subsystems of second order arithmetic. We show that the strength of the theorem comes from having a dense linear order without endpoints in its order type. Then, we show that for the related Weihrauch problem to be strong enough to be equivalent to $\mathsf{\hat{WF}}$ (the analog problem of $\mathsf{\Pi_1^1}$-$\mathsf{CA_0}$), an order-preserving function is necessary in the output. Without the order-preserving function, the problems are very much to the side compared to analog problems of the big five.