用拟解析类的泛函数族构造具有可判定理论的O-极小结构
Constructing o-minimal structures with decidable theories using generic families of functions from quasianalytic classes
摘要：
令$\RR_S$表示实序域由实值函数族$S$展开，其中$S$中的每个函数定义在紧致盒上，并且是某个准解析类的成员，该准解析类在函数合成、被变量除和隐式定义函数的提取的操作下是封闭的。它表明，如果家庭$S$是通用的（这是一个特定的技术定义的超越条件），那么$\RR_S$理论是可判定的，当且仅当$S$是可计算的$C^\infty$（这意味着所有的偏导数的功能在$S$可以有效地近似）。它还表明，在一定的拓扑意义上，许多通用的，可计算的$C^\infty$家庭$S$存在。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
