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

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


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.


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



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


  1. 1.
    Alur, R., Dill, D.L.: A theory of timed automata. Theoret. Comput. Sci. 126(2), 183–235 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bachmair, L., Ganzinger, H., Waldmann, U.: Refutational theorem proving for hierarchic first-order theories. Appl. Algebra Eng. Commun. Comput. 5, 193–212 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baumgartner, P., Waldmann, U.: Hierarchic superposition with weak abstraction. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 39–57. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38574-2_3 CrossRefGoogle Scholar
  4. 4.
    Bradley, A.R., Manna, Z., Sipma, H.B.: What’s decidable about arrays? In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 427–442. Springer, Heidelberg (2005). doi: 10.1007/11609773_28 CrossRefGoogle Scholar
  5. 5.
    Downey, P.J.: Undecidability of Presburger arithmetic with a single monadic predicate letter. Technical report, Center for Research in Computer Technology, Harvard University (1972)Google Scholar
  6. 6.
    Fietzke, A., Weidenbach, C.: Superposition as a decision procedure for timed automata. Math. Comput. Sci. 6(4), 409–425 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ge, Y., de Moura, L.: Complete instantiation for quantified formulas in satisfiabiliby modulo theories. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 306–320. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02658-4_25 CrossRefGoogle Scholar
  8. 8.
    Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory, 2nd edn. A Wiley-Interscience publication, New York (1990)zbMATHGoogle Scholar
  9. 9.
    Horbach, M., Voigt, M., Weidenbach, C.: On the combination of the Bernays-Schönfinkel-Ramsey fragment with simple linear integer arithmetic. In: Automated Deduction (CADE-26) (to appear)Google Scholar
  10. 10.
    Horbach, M., Voigt, M., Weidenbach, C.: The universal fragment of Presburger arithmetic with unary uninterpreted predicates is undecidable. ArXiv preprint, arXiv:1703.01212 [cs.LO] (2017)
  11. 11.
    Kroening, D., Strichman, O.: Decision Procedures. Texts in Theoretical Computer Science. An EATCS Series, 2nd edn. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-50497-0 CrossRefzbMATHGoogle Scholar
  12. 12.
    Kruglov, E., Weidenbach, C.: Superposition decides the first-order logic fragment over ground theories. Math. Comput. Sci. 6(4), 427–456 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Niebert, P., Mahfoudh, M., Asarin, E., Bozga, M., Maler, O., Jain, N.: Verification of timed automata via satisfiability checking. In: Damm, W., Olderog, E.-R. (eds.) FTRTFT 2002. LNCS, vol. 2469, pp. 225–243. Springer, Heidelberg (2002). doi: 10.1007/3-540-45739-9_15 CrossRefGoogle Scholar
  14. 14.
    Pratt, V.R.: Two Easy Theories Whose Combination is Hard. Technical report, Massachusetts Institute of Technology (1977)Google Scholar
  15. 15.
    Putnam, H.: Decidability and essential undecidability. J. Symbolic Logic 22(1), 39–54 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Voigt, M.: The Bernays-Schönfinkel-Ramsey Fragment with Bounded Difference Constraints over the Reals is Decidable. ArXiv preprint, arXiv:1706.08504 [cs.LO] (2017)
  17. 17.
    Voigt, M., Weidenbach, C.: Bernays-Schönfinkel-Ramsey with Simple Bounds is NEXPTIME-complete. ArXiv preprint, arXiv:1501.07209 [cs.LO] (2015)

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Max Planck Institute for Informatics and Saarbrücken Graduate School of Computer ScienceSaarbrückenGermany

Personalised recommendations