On disguised double horn functions and extensions

  • Thomas Eiter
  • Toshihide Ibaraki
  • Kazuhisa Makino
Logic I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1373)


As a natural restriction of disguised Horn functions (i.e., Boolean functions which become Horn after a renaming (change of polarity) of some of the variables), we consider the class C DH R of disguised double Horn functions, i.e., the functions which and whose complement are both disguised Horn. We investigate the syntactical properties of this class and relationship to other classes of Boolean functions. Moreover, we address the extension problem of partially defined Boolean functions (pdBfs) in C DH R, where a pdBf is a function defined on a subset (rather than the full set) of Boolean vectors. We show that the class C DH R coincides with the class C 1–DL of 1-decision lists, and with the intersections of several well-known classes of Boolean functions. Furthermore, polynomial time algorithms for the recognition of a function in C DH R from Horn formulas and other classes of formulas are provided, while the problem is intractable in general. We also present an algorithm for the extension problem which, properly implemented, runs in linear time.


Boolean Function Polynomial Time Algorithm Extension Problem Disjunctive Normal Form Prime Implicant 
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.


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  1. 1.
    H. Aizenstein, T. Hegedüs, L. Hellerstein and L. Pitt, Complexity theoretic hardness results for query learning, to appear in Journal of Complexity.Google Scholar
  2. 2.
    D. Angluin, M. Frazier, and L. Pitt, Learning conjunctions of Horn clauses, Machine Learning, 9:147–164, 1992.zbMATHGoogle Scholar
  3. 3.
    M. Anthony and N. Biggs, Computational Learning Theory, Cambridge University Press, 1992.Google Scholar
  4. 4.
    M. Anthony, G. Brightwell and J. Shawe-Taylor, On specifying Boolean functions by labelled examples, Discr. Appl. Math., 61:1–25, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    B. Aspvall, Recognizing disguised NR(1) instance of the satisfiability problem, J. Algorithms, 1:97–103, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of partially defined Boolean functions, RUTCOR RRR 14-95, Rutgers University, 1995. Information and Computation, to appear.Google Scholar
  7. 7.
    S. Ceri, G. Gottlob, L. Tanca, Logic Programming and Databases, Springer, 1990.Google Scholar
  8. 8.
    Y. Crama, P. L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined Boolean functions, Annals of Operations Research, 16:299–326, 1988.CrossRefMathSciNetGoogle Scholar
  9. 9.
    D. W. Dowling and J.H. Gallier, Linear-time algorithms for testing the satisfyability of propositional Horn formulae, J. Logic Programming, 3:267–284, 1984.CrossRefMathSciNetGoogle Scholar
  10. 10.
    T. Eiter, Generating Boolean μ-expressions, Acta Informatica, 32:171–187, 1995.zbMATHMathSciNetGoogle Scholar
  11. 11.
    T. Eiter, T. Ibaraki, and K. Makino, Multi-Face Horn Functions, CD-TR 96/95, CD Lab for Expert Systems, TU Vienna, Austria, iii + 97 pages, 1996.Google Scholar
  12. 12.
    T. Eiter, T. Ibaraki, and K. Makino, Double Horn functions, RUTCOR Research Report RRR 18–97, Rutgers University 1997; to appear in Information and Computation.Google Scholar
  13. 13.
    T. Eiter, T. Ibaraki, and K. Makino, Two-face Horn extensions, to appear in Proceedings of ISAAC'97, Springer LNCS.Google Scholar
  14. 14.
    O. Ekin, P.L. Hammer and U. N. Peled, Horn functions and submodular Boolean functions, Theoretical Computer Science, 175(2):257–270, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    S. A. Goldman, On the complexity of teaching, J. Computer and System Sciences, 50:20–31, 1995.zbMATHCrossRefGoogle Scholar
  16. 16.
    M. Golumbic, P.L. Hammer, P. Hansen, and T. Ibaraki (eds), Horn Logic, search and satisfiability, Annals of Mathematics and Artificial Intelligence 1, 1990.Google Scholar
  17. 17.
    H. Hunt III and R. Stearns, The complexity of very simple Boolean formulas with applications, SIAM J. Computing, 19:44–70, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    H. A. Kautz, M. J. Kearns, and B. Selman, Horn approximations of empirical data, Artificial Intelligence, 74:129–145, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    H. Lewis, Renaming a set of clauses as a Horn set, JACM, 25:134–135, 1978.zbMATHCrossRefGoogle Scholar
  20. 20.
    K. Makino, K. Hatanaka and T. Ibaraki, Horn extensions of a partially defined Boolean function, RUTCOR RRR 27-95, Rutgers University, 1995.Google Scholar
  21. 21.
    S. Muroga, Threshold Logic and Its Applications, Wiley-Interscience, 1971.Google Scholar
  22. 22.
    R. L. Rivest, Learning decision lists, Machine Learning, 2:229–246, 1996.Google Scholar
  23. 23.
    B. Selman and H. J. Levesque, Support set selection for abductive and default reasoning, Artificial Intelligence, 82:259–272, 1996.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Thomas Eiter
    • 1
  • Toshihide Ibaraki
    • 2
  • Kazuhisa Makino
    • 3
  1. 1.Institut für InformatikUniversität GießenGermany
  2. 2.Department of Applied Mathematics and Physics, Graduate School of EngineeringKyoto UniversityKyotoJapan
  3. 3.Department of Systems and Human Science, Graduate School of Engineering ScienceOsaka UniversityToyonaka, OsakaJapan

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