Undecidabiliity for the additive AIA fragment of the theory of normed spaces

TL;DR

This paper proves that the validity problem remains undecidable for a specific logical fragment related to normed spaces, even when the language is restricted to additive operations only.

Contribution

It extends previous undecidability results to the additive sublanguage of the AIA fragment in the theory of normed spaces.

Findings

Undecidability of validity persists in the additive sublanguage.

The result applies to the AIA fragment with purely universal formulas.

Previous results for the full language are extended to the additive case.

Abstract

An AIA formula is one of the form 'A implies B' where A and B are purely universal. Up to a simple reduction AIA formula are both EA and AE. In an earlier paper Solovay, Harrison and I proved the undecidability of validity for the AIA fragment of a two-sorted first-order language for normed vector spaces. In this note we find that validity remains undecidable for AIA sentences in the additive sublanguage, i.e., when multiplication is disallowed.