Solving First-Order Constraints over the Monadic Class

  • Dimitri Chubarov
  • Andrei Voronkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2605)


First-order constraints over arbitrary theories or structures can be formalised as the formula instantiation problem as defined in [11]. Several re- sults have been previously obtained for the formula instantiation problem in the case of quantifier-free formulas of first-order logic. In this paper we prove the first general result on formula instantiation for quantified formulas, namely that formula instantiation is decidable for the monadic class without equality.


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  1. 1.
    Degtyarev, A., Voronkov, A.: Equality reasoning in sequent-based calculi. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, ch. 10, vol. I, pp. 609–704. Elsevier Science, Amsterdam (2001)Google Scholar
  2. 2.
    Gallier, J.H., Raatz, S., Snyder, W.: Theorem proving using rigid E-unification: Equational matings. In: Proc. IEEE Conference on Logic in Computer Science (LICS), pp. 338–346. IEEE Computer Society Press, Los Alamitos (1987)Google Scholar
  3. 3.
    Gurevich, Y.: The decision problem for the logic of predicates and operations, vol. 8, pp. 284–308 (1969)Google Scholar
  4. 4.
    Hodges, W.: Model theory. Cambridge University Press, Cambridge (1993)MATHCrossRefGoogle Scholar
  5. 5.
    Lewis, H.: Complexity results for classes of quantificational formulas. Journal of Computer and System Sciences 21, 317–353 (1980)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Löb, M.: Decidability of the monadic predicate calculus with unary function symbols. Journal of Symbolic Logic 32, 563 (1967)Google Scholar
  7. 7.
    Meyer, A.: Weak monadic second order theory of successor is not elementaryrecursive. In: Parikh, R. (ed.) Logic Colloquium: Symposium on Logic Held at Boston. Lecture Notes in Mathematics, vol. 453, pp. 132–154. Springer, Heidelberg (1975)Google Scholar
  8. 8.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society 141(1), 1–35 (1969)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Veanes, M.: The undecidability of simultaneous rigid E-unification with two variables. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) KGC 1997. LNCS, vol. 1289, pp. 305–318. Springer, Heidelberg (1997)Google Scholar
  10. 10.
    Voronkov, A.: Strategies in rigid-variable methods. In: Pollack, M.E. (ed.) Proc. of the Fifteenth International Joint Conference on Artificial Intelligence (IJCAI 1997), Nagoya, Japan, August 23-29, vol. 1, pp. 114–119 (1997)Google Scholar
  11. 11.
    Voronkov, A.: Herbrand’s theorem, automated reasoning and semantic tableaux. In: Proc. IEEE Conference on Logic in Computer Science (LICS), pp. 252–263. IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  12. 12.
    Voronkov, A.: The ground-negative fragment of first-order logic is \( II^p_2\)-complete. Journal of Symbolic Logic 64(3), 984–990 (1999)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Voronkov, A.: Simultaneous rigid E-unification and other decision problems related to Herbrand’s theorem. Theoretical Computer Science 224, 319–352 (1999)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Dimitri Chubarov
    • 1
  • Andrei Voronkov
    • 1
  1. 1.University of Manchester 

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