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

Marco Voigt 1, 2
1 VERIDIS - Modeling and Verification of Distributed Algorithms and Systems
LORIA - FM - Department of Formal Methods , Inria Nancy - Grand Est, MPII - Max-Planck-Institut für Informatik
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 R c, where R ranges over the standard relations <, ≤, =, =, ≥, > 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.
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Marco Voigt. The Bernays–Schönfinkel–Ramsey Fragment with Bounded Difference Constraints over the Reals Is Decidable. FroCoS 2017 - 11th International Symposium on Frontiers of Combining Systems, Sep 2017, Brasilia, Brazil. pp.244-261, ⟨10.1007/978-3-319-66167-4_14⟩. ⟨hal-01592169⟩

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