Recognizing Read-Once Functions from Depth-Three Formulas

  • Alexander KozachinskiyEmail author
Part of the following topical collections:
  1. Computer Science Symposium in Russia (2018)


Consider the following decision problem: for a given monotone Boolean function f decide, whether f is read-once. For this problem, it is essential how the input function f is represented. Elbassioni et al. (J. Comb. Optim. 22(3), 293–304, 2011) proved that this problem is coNP-complete when f is given by a depth-4 read-2 monotone Boolean formula. Gurvich (2010) proved that this problem is coNP-complete even when the input is the following expression: CDn, where Dn = x1y1 ∨ … ∨ xnyn and C is a monotone CNF over the variables x1, y1, … , xn, yn (note that this expression is a monotone Boolean formula of depth 3; in Gurvich (2010) nothing is said about the readability of C, but the proof is valid even if C is read-2 and thus the entire formula is read-3). We show that we can test in polynomial-time whether a given expression CD computes a read-once function, provided that C is a read-once monotone CNF and D is a read-once monotone DNF and all the variables of C occur also in D (recall that due to Gurvich, the problem is coNP-complete when C is read-2). We also observe that from the so-called Sausage Lemma of Boros et al. (2009) it follows that the problem of recognizing read-once functions is coNP-complete when the input formula is depth-3 read-2.


Read-once functions NP-completeness Monotone Boolean functions Depth-three formulas 



The author would like to thank Nikolay Vereshchagin and Alexander Shen for help in writing this paper. The author would like to thank Vladimir Gurvich for pointing out to [1].


  1. 1.
    Boros, E., Elbassioni, K., Gurvich, V., Makino, K.: Generating vertices of polyhedra and related problems of monotone generation. In: Proceedings of the Centre de Recherches Mathématiques at the Université de Montréal, special issue on Polyhedral Computation (CRM Proceedings and Lecture Notes), vol. 49, pp 15–43 (2009)Google Scholar
  2. 2.
    Büning, H.K., Lettmann, T.: Propositional Logic: Deduction and Algorithms, vol. 48. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  3. 3.
    Elbassioni, K., Makino, K., Rauf, I.: On the readability of monotone boolean formulae. J. Comb. Optim. 22(3), 293–304 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gilmer, J., Kouckỳ, M., Saks, M.E.: A new approach to the sensitivity conjecture. In: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, pp. 247–254. ACM (2015)Google Scholar
  5. 5.
    Golumbic, M.C., Gurvich, V.: Read-Once Functions. Boolean Functions: Theory, Algorithms and Applications (2009)Google Scholar
  6. 6.
    Golumbic, M.C., Mintz, A., Rotics, U.: An improvement on the complexity of factoring read-once boolean functions. Discret. Appl. Math. 156(10), 1633–1636 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gurvich, V.: It is a conp-complete problem to decide whether a positive ∨-∧ formula of depth 3 defines a read-once or respectively quadratic boolean function. Rutcor Research Report (2010)Google Scholar
  8. 8.
    Gurvich, V.A.: Repetition-free boolean functions. Uspekhi Matematicheskikh Nauk 32(1), 183–184 (1977)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussian Federation

Personalised recommendations