The classic method of Nelson and Oppen for combining decision procedures requires the theories to be stably-infinite. Unfortunately, some important theories do not fall into this category (e.g. the theory of bit-vectors). To remedy this problem, previous work introduced the notion of polite theories. Polite theories can be combined with any other theory using an extension of the Nelson-Oppen approach. In this paper we revisit the notion of polite theories, fixing a subtle flaw in the original definition. We give a new combination theorem which specifies the degree to which politeness is preserved when combining polite theories. We also give conditions under which politeness is preserved when instantiating theories by identifying two sorts. These results lead to a more general variant of the theorem for combining multiple polite theories.


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  1. 1.
    Barrett, C., Shikanian, I., Tinelli, C.: An abstract decision procedure for a theory of inductive data types. Journal on Satisfiability, Boolean Modeling and Computation 3, 21–46 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Enderton, H.B.: A mathematical introduction to logic. Academic Press, New York (1972)zbMATHGoogle Scholar
  3. 3.
    Jovanović, D., Barrett, C.: Polite theories revisited. Technical Report TR2010-922, Department of Computer Science, New York University (January 2010)Google Scholar
  4. 4.
    Krstić, S., Goel, A., Grundy, J., Tinelli, C.: Combined Satisfiability Modulo Parametric Theories. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 602–617. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Transactions on Programming Languages and Systems 1(2), 245–257 (1979)CrossRefzbMATHGoogle Scholar
  6. 6.
    Oppen, D.C.: Complexity, convexity and combinations of theories. Theoretical Computer Science 12(3), 291–302 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ranise, S., Ringeissen, C., Zarba, C.: Combining Data Structures with Nonstably Infinite Theories using Many-Sorted Logic. Research Report RR-5678, INRIA (2005)Google Scholar
  8. 8.
    Ranise, S., Ringeissen, C., Zarba, C.G.: Combining Data Structures with Nonstably Infinite Theories Using Many-Sorted Logic. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, pp. 48–64. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Stump, A., Dill, D.L., Barrett, C.W., Levitt, J.: A decision procedure for an extensional theory of arrays. In: Proceedings of the 16th IEEE Symposium on Logic in Computer Science (LICS 2001), June 2001, pp. 29–37. IEEE Computer Society, Boston (June 2001)Google Scholar
  10. 10.
    Tinelli, C., Harandi, M.T.: A new correctness proof of the Nelson–Oppen combination procedure. In: Frontiers of Combining Systems. Applied Logic, pp. 103–120. Kluwer Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  11. 11.
    Tinelli, C., Zarba, C.: Combining decision procedures for sorted theories. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 641–653. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Tinelli, C., Zarba, C.: Combining decision procedures for theories in sorted logics. Technical Report 04-01, Department of Computer Science, The University of Iowa (February 2004)Google Scholar
  13. 13.
    Tinelli, C., Zarba, C.G.: Combining nonstably infinite theories. Journal of Automated Reasoning 34(3), 209–238 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Dejan Jovanović
    • 1
  • Clark Barrett
    • 1
  1. 1.New York UniversityUSA

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