一句话：证明实数加法结构中有无整数谓词的可定义性可判定，并给出拓扑刻画与无中间结构结果。
本文研究实数加法结构 ⟨R,+,<,Z⟩ 与 ⟨R,+,<,1⟩ 的可定义性关系：判定前者中可定义的关系能否在后者中定义。作者在两类可定义关系族中给出 ⟨R,+,<,1⟩-可定义关系的拓扑刻画，并采用 Muchnik 方法证明该刻画可在 ⟨R,+,<,1,X⟩ 的逻辑中表达。由此得到可判定性，并证明两个结构之间不存在中间结构。此外，他们证明一个 ⟨R,+,<,Z⟩-可定义关系是 ⟨R,+,<,1⟩-可定义的当且仅当它与每条 ⟨R,+,<,1⟩-可定义直线的交集都是 ⟨R,+,<,1⟩-可定义的，从而给出非有效但简单的刻画。
We show that it is decidable whether or not a relation on the reals definable in the structure $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$ can be defined in the structure $\langle \mathbb{R}, +,<, 1 \rangle$. This result is achieved by obtaining a topological characterization of $\langle \mathbb{R}, +,<, 1 \rangle$-definable relations in the family of $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$-definable relations and then by following Muchnik's approach of showing that the characterization of the relation $X$ can be expressed in the logic of $\langle \mathbb{R}, +,<,1, X \rangle$. The above characterization allows us to prove that there is no intermediate structure between $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$ and $\langle \mathbb{R}, +,<, 1 \rangle$. We also show that a $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$-definable relation is $\langle \mathbb{R}, +,<, 1 \rangle$-definable if and only if its intersection with every $\langle \mathbb{R}, +,<, 1 \rangle$-definable line is $\langle \mathbb{R}, +,<, 1 \rangle$-definable. This gives a noneffective but simple characterization of $\langle \mathbb{R}, +,<, 1 \rangle$-definable relations.
@article{paperbot1310,
title = {Theories of real addition with and without a predicate for integers},
author = {Alexis Bès and Christian Choffrut},
journal = {Logical Methods in Computer Science},
volume = {Volume 17, Issue 2},
year = {2021},
doi = {10.23638/lmcs-17(2:18)2021}
}