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Checking Tests for Read-Once Functions over Arbitrary Bases

  • Dmitry V. Chistikov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)

Abstract

A Boolean function is called read-once over a basis B if it can be expressed by a formula over B where no variable appears more than once. A checking test for a read-once function f over B depending on all its variables is a set of input vectors distinguishing f from all other read-once functions of the same variables. We show that every read-once function f over B has a checking test containing O(n l ) vectors, where n is the number of relevant variables of f and l is the largest arity of functions in B. For some functions, this bound cannot be improved by more than a constant factor. The employed technique involves reconstructing f from its l-variable projections and provides a stronger form of Kuznetsov’s classic theorem on read-once representations.

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References

  1. 1.
    Aaronson, S.: Algorithms for Boolean function query properties. SIAM Journal on Computing 32(5), 1140–1157 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Angluin, D.: Queries and concept learning. Machine Learning 2(4), 319–342 (1988)Google Scholar
  3. 3.
    Angluin, D., Hellerstein, L., Karpinski, M.: Learning read-once formulas with queries. Journal of the ACM 40, 185–210 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bshouty, N.H., Hancock, T.R., Hellerstein, L.: Learning Boolean read-once formulas over generalized bases. Journal of Computer and System Sciences 50(3), 521–542 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    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)zbMATHCrossRefGoogle Scholar
  6. 6.
    Chegis, I.A., Yablonsky, S.V.: Logical methods for controlling electrical circuits. Trudy Matematicheskogo Instituta Steklova 51, 270–360 (1958) (in Russian)zbMATHGoogle Scholar
  7. 7.
    Chistikov, D.V.: On the relationship between diagnostic and checking tests of the read-once functions. Discrete Mathematics and Applications 21(2), 203–208 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chistikov, D.V.: Read-once functions with hard-to-test projections. Moscow University Computational Mathematics and Cybernetics 34(4), 188–190 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chistikov, D.V.: Testing Monotone Read-Once Functions. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2011. LNCS, vol. 7056, pp. 121–134. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Chistikov, D.V.: Testing read-once functions over the elementary basis. Moscow University Computational Mathematics and Cybernetics 35(4), 189–192 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Davies, R.O.: Two theorems on essential variables. Journal of the London Mathematical Society 41(2), 333–335 (1966)zbMATHCrossRefGoogle Scholar
  12. 12.
    Goldman, S.A., Kearns, M.J.: On the complexity of teaching. Journal of Computer and System Sciences 50(1), 20–31 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kuznetsov, A.V.: On read-once switching circuits and read-once compositions of functions in the algebra of logic. Trudy Matematicheskogo Instituta Steklova 51, 186–225 (1958) (in Russian)zbMATHGoogle Scholar
  14. 14.
    Voronenko, A.A.: On checking tests for read-once functions. In: Matematicheskie Voprosy Kibernetiki, vol. 11, pp. 163–176. Fizmatlit, Moscow (2002) (in Russian)Google Scholar
  15. 15.
    Voronenko, A.A.: Recognizing the nonrepeating property in an arbitrary basis. Computational Mathematics and Modeling 18(1), 55–65 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Voronenko, A.A.: Testing disjunction as a read-once function in an arbitrary unrepeated basis. Moscow University Computational Mathematics and Cybernetics 32(4), 239–240 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    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
  19. 19.
    Zubkov, O.V., Chistikov, D.V., Voronenko, A.A.: An upper bound on checking test complexity for almost all cographs. In: Wang, D., et al. (eds.) 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2011), pp. 323–330. IEEE Computer Society, Los Alamitos (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

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