Constructing o-minimal structures with decidable theories using generic families of functions from quasianalytic classes

TL;DR

This paper demonstrates that expanding the real ordered field with a generic family of quasianalytic functions results in a decidable theory if the functions are computably smooth, and it shows the abundance of such families.

Contribution

It establishes a link between generic quasianalytic function families and decidability of the resulting o-minimal structures, introducing new conditions for decidability.

Findings

Decidability of the theory is equivalent to the functions being computably smooth.

Many generic, computably smooth families of functions exist.

The framework applies to expansions of the real field with quasianalytic functions.

Abstract

Let $\RR_S$ denote the expansion of the real ordered field by a family of real-valued functions $S$, where each function in $S$ is defined on a compact box and is a member of some quasianalytic class which is closed under the operations of function composition, division by variables, and extraction of implicitly defined functions. It is shown that if the family $S$ is generic (which is a certain technically defined transcendence condition), then the theory of $\RR_S$ is decidable if and only if $S$ is computably $C^\infty$ (which means that all the partial derivatives of the functions in $S$ may be effectively approximated). It is also shown that, in a certain topological sense, many generic, computably $C^\infty$ families $S$ exist.