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

Let ${\text{RR}}_S$ 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 ${\text{RR}}_S$ is decidable if and only if $S$ is computably $C^{\mathrm{\infty }}$ (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 $C^{\mathrm{\infty }}$ families $S$ exist.