Recognizing Read-Once Functions from Depth-Three Formulas

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.