# The Bernays-Sch\"onfinkel-Ramsey Fragment with Bounded Difference Constraints over the Reals is Decidable

## TL;DR

This paper proves that the satisfiability problem for a fragment of first-order linear real arithmetic with bounded difference constraints and uninterpreted predicates is decidable, extending the understanding of decidable logical fragments over the reals.

## Contribution

It introduces a decidable fragment of first-order real arithmetic with bounded difference constraints and uninterpreted predicates, which was previously undecidable.

## Key findings

- Decidability is achieved by bounding the ranges of universally quantified variables.
- The fragment includes difference constraints over reals with bounded intervals.
- Trivial instantiation procedures are insufficient due to infinite value ranges.

## Abstract

First-order linear real arithmetic enriched with uninterpreted predicate symbols yields an interesting modeling language. However, satisfiability of such formulas is undecidable, even if we restrict the uninterpreted predicate symbols to arity one. In order to find decidable fragments of this language, it is necessary to restrict the expressiveness of the arithmetic part. One possible path is to confine arithmetic expressions to difference constraints of the form $x - y \mathrel{\#} c$, where $\#$ ranges over the standard relations $<, \leq, =, \neq, \geq, >$ and $x,y$ are universally quantified. However, it is known that combining difference constraints with uninterpreted predicate symbols yields an undecidable satisfiability problem again. In this paper, it is shown that satisfiability becomes decidable if we in addition bound the ranges of universally quantified variables. As bounded intervals over the reals still comprise infinitely many values, a trivial instantiation procedure is not sufficient to solve the problem.