Decomposition of Decidable First-Order Logics over Integers and Reals

We tackle the issue of representing infinite sets of real- valued vectors. This paper introduces an operator for combining integer and real sets. Using this operator, we decompose three well-known logics extending Presburger with reals. Our decomposition splits a logic into two parts : one integer, and one decimal (i.e. on the interval [0,1]). We also give a basis for an implementation of our representation

Decidability of definability issues in the theory of real addition

Given a subset of $X\subseteq \mathbb{R}^{n}$ we can associate with every point $x\in \mathbb{R}^{n}$ a vector space $V$ of maximal dimension with the property that for some ball centered at $x$, the subset $X$ coincides inside the ball with a union of lines parallel with $V$. A point is singular if $V$ has dimension $0$. In an earlier paper we proved that a $(\mathbb{R}, +,< ,\mathbb{Z})$-definable relation $X$ is actually definable in $(\mathbb{R}, +,< ,1)$ if and only if the number of singular points is finite and every rational section of $X$ is $(\mathbb{R}, +,< ,1)$-definable, where a rational section is a set obtained from $X$ by fixing some component to a rational value. Here we show that we can dispense with the hypothesis of $X$ being $(\mathbb{R}, +,< ,\mathbb{Z})$-definable by assuming that the components of the singular points are rational numbers. This provides a topological characterization of first-order definability in the structure $(\mathbb{R}, +,< ,1)$. It also allows us to deliver a self-definable criterion (in Muchnik's terminology) of $(\mathbb{R}, +,< ,1)$- and $(\mathbb{R}, +,< ,\mathbb{Z})$-definability for a wide class of relations, which turns into an effective criterion provided that the corresponding theory is decidable. In particular these results apply to the class of $k-$recognizable relations on reals, and allow us to prove that it is decidable whether a $k-$recognizable relation (of any arity) is $l-$recognizable for every base $l \geq 2$.Comment: added sections 5 and 6, typos corrected. arXiv admin note: text overlap with arXiv:2002.0428

Theories of real addition with and without a predicate for integers

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