FroCoS 2017: Frontiers of Combining Systems pp 244-261

# The Bernays–Schönfinkel–Ramsey Fragment with Bounded Difference Constraints over the Reals Is Decidable

• Marco Voigt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10483)

## 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 $$x - y \mathrel {\triangleleft }c$$, where $$\mathrel {\triangleleft }$$ ranges over the standard relations $$<, \le , =, \ne , \ge ,>$$ and xy 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.

## Keywords

Bernays–Schönfinkel–Ramsey fragment Linear arithmetic constraints Difference constraints Combination of theories

## Notes

### Acknowledgement

The present author is indebted to the anonymous reviewers for their constructive criticism and valuable suggestions.

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