We show that finding finite Herbrand models for a restricted class of first-order clauses is ExpTime-complete. A Herbrand model is called finite if it interprets all predicates by finite subsets of the Herbrand universe. The restricted class of clauses consists of anti-Horn clauses with monadic predicates and terms constructed over unary function symbols and constants. The decision procedure can be used as a new goal-oriented algorithm to solve linear language equations and unification problems in the description logic \(\mathcal{FL}_0\). The new algorithm has only worst-case exponential runtime, in contrast to the previous one which was even best-case exponential.


Decision Procedure Description Logic Replacement Sequence Tree Automaton Unary Predicate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baader, F., Narendran, P.: Unification of concept terms in description logics. J. Symb. Comput. 31(3), 277–305 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Baumgartner, P., Fuchs, A., de Nivelle, H., Tinelli, C.: Computing finite models by reduction to function-free clause logic. J. Appl. Log. 7(1), 58–74 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Birget, J.: State-complexity of finite-state devices, state compressibility and incompressibility. Math. Syst. Theory 26(3), 237–269 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Borgwardt, S., Morawska, B.: Finding finite Herbrand models. LTCS-Report 11-04, TU Dresden (2011),
  5. 5.
    Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. J. ACM 28(1), 114–133 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Comon, H., Dauchet, M., Gilleron, R., Löding, C., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications (2007),
  7. 7.
    Dreben, B., Goldfarb, W.D.: The Decision Problem: Solvable Classes of Quantificational Formulas. Addison-Wesley (1979)Google Scholar
  8. 8.
    Joyner Jr., W.H.: Resolution strategies as decision procedures. J. ACM 23(3), 398–417 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ladner, R.E., Lipton, R.J., Stockmeyer, L.J.: Alternating pushdown and stack automata. SIAM J. Comput. 13(1), 135–155 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Leitsch, A.: The Resolution Calculus. Springer, Heidelberg (1997)zbMATHCrossRefGoogle Scholar
  11. 11.
    Peltier, N.: Model building with ordered resolution: Extracting models from saturated clause sets. J. Symb. Comput. 36(1-2), 5–48 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Petri, C.A.: Kommunikation mit Automaten. Ph.D. thesis, Uni Bonn (1962)Google Scholar
  13. 13.
    Reisig, W.: Petri Nets: An Introduction. Springer, Heidelberg (1985)zbMATHGoogle Scholar
  14. 14.
    Slutzki, G.: Alternating tree automata. Theor. Comput. Sci. 41, 305–318 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Vardi, M.Y., Wolper, P.: Automata theoretic techniques for modal logics of programs (extended abstract). In: Proc. STOC 1984, pp. 446–456. ACM (1984)Google Scholar
  16. 16.
    Zhang, J.: Constructing finite algebras with FALCON. J. Autom. Reasoning 17, 1–22 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stefan Borgwardt
    • 1
  • Barbara Morawska
    • 1
  1. 1.Theoretical Computer ScienceTU DresdenGermany

Personalised recommendations