Abstract
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: C ∨ Dn, 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 C ∨ D 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.
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Notes
This is due to the fact that the SAT-problem is NP-complete even for read-3 (non-monotone) CNFs.
The proof of this lemma is almost identical to the proof of Lemma 4. Actually, it is possible to formulate a single lemma which implies both of them, but then the formulation of the lemma becomes immense.
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Acknowledgements
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].
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This article is part of the Topical Collection on Computer Science Symposium in Russia (2018)
The study has been funded by the Russian Academic Excellence Project ’5-100’. Supported in part by the Russian Foundation for Basic Research grant 16-01-00362. The preliminary version of this paper was accepted for presentation at CSR 2018.
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Kozachinskiy, A. Recognizing Read-Once Functions from Depth-Three Formulas. Theory Comput Syst 64, 3–16 (2020). https://doi.org/10.1007/s00224-019-09923-1
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DOI: https://doi.org/10.1007/s00224-019-09923-1