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.

Source: arXiv.org:2002.04282

Volume: Volume 17, Issue 2

Published on: May 26, 2021

Accepted on: April 28, 2021

Submitted on: February 12, 2020

Keywords: Computer Science - Logic in Computer Science,Mathematics - Logic

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