Decomposition of Decidable First-Order Logics over Integers and Reals

TL;DR

This paper presents a novel decomposition method for decidable first-order logics over integers and reals, enabling better representation and analysis of infinite real-valued vector sets.

Contribution

It introduces an operator to combine integer and real sets, decomposes extended Presburger logics, and provides a basis for implementation.

Findings

Decomposition effectively separates integer and decimal components.

The approach supports representation of infinite real-valued sets.

Implementation basis facilitates practical application.

Abstract

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.