Advertisement

Testing Monotone Read-Once Functions

  • Dmitry V. Chistikov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)

Abstract

A checking test for a monotone read-once function f depending essentially on all its n variables is a set of vectors M distinguishing f from all other monotone read-once functions of the same variables. We describe an inductive procedure for obtaining individual lower and upper bounds on the minimal number of vectors T(f) in a checking test for any function f. The task of deriving the exact value of T(f) is reduced to a combinatorial optimization problem related to graph connectivity. We show that for almost all functions f expressible by read-once conjunctive or disjunctive normal forms, T(f) ~n / ln n. For several classes of functions our results give the exact value of T(f).

Keywords

Equivalence Relation Boolean Function Combinatorial Optimization Problem Inductive Assumption Conjunctive Normal Form 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bubnov, S.E., Voronenko, A.A., Chistikov, D.V.: Some test length bounds for nonrepeating functions in the \(\{\&, \lor\}\) basis. Computational Mathematics and Modeling 21(2), 196–205 (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chistikov, D.V.: Testing read-once functions over the elementary basis. Moscow University Computational Mathematics and Cybernetics (to appear)Google Scholar
  3. 3.
    Corneil, D.G., Lerchs, H., Stewart Burlingham, L.: Complement reducible graphs. Discrete Applied Mathematics 3(3), 163–174 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gurvich, V.A.: On repetition-free Boolean functions. Uspehi Matematicheskih nauk 32(1), 183–184 (1977) (in Russian)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Karchmer, M., Linial, N., Newman, I., Saks, M., Widgerson, A.: Combinatorial characterization of read-once formulae. Discrete Mathematics 114(1-3), 275–282 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ryabets, L.V.: Checking test complexity for read-once Boolean functions. Ser. Diskretnaya matematika i informatika, vol. 18. Izdatel’stvo Irkutskogo gosudarstvennogo pedagogicheskogo universiteta (2007) (in Russian)Google Scholar
  7. 7.
    Sachkov, V.N.: Probabilistic methods in combinatorial analysis. Encyclopedia of Mathematics and its Applications, vol. 56. Cambridge University Press (1997)Google Scholar
  8. 8.
    Voronenko, A.A.: Estimating the length of a diagnostic test for some nonrepeating functions. Computational Mathematics and Modeling 15(4), 377–386 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Voronenko, A.A.: On checking tests for read-once functions. In: Matematicheskie Voprosy Kibernetiki, Fizmatlit, Moscow, vol. 11, pp. 163–176 (2002) (in Russian)Google Scholar
  10. 10.
    Voronenko, A.A.: On the length of checking test for repetition-free functions in the basis \(\{0, 1, \&, \lor, \neg\}\). Discrete Mathematics and Applications 15(3), 313–318 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Voronenko, A.A.: Recognizing the nonrepeating property in an arbitrary basis. Computational Mathematics and Modeling 18(1), 55–65 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Voronenko, A.A., Chistikov, D.V.: Learning read-once functions individually. Uchenye zapiski Kazanskogo universiteta. Ser. Fiziko-matematicheskie nauki 151(2), 36–44 (2009) (in Russian)zbMATHGoogle Scholar
  13. 13.
    Voronenko, A.A., Chistikov, D.V.: On testing read-once Boolean functions in the basis B 5. In: Proceedings of the XVII International Workshop “Synthesis and complexity of control systems”, pp. 24–30. Izdatel stvo Instituta matematiki, Novosibirsk (2008) (in Russian)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dmitry V. Chistikov
    • 1
  1. 1.Faculty of Computational Mathematics and CyberneticsMoscow State UniversityRussia

Personalised recommendations