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. On a negative side we have the following results. Elbassioni, Makino and Rauf ([1]) proved that this problem is coNP-complete when f is given by a depth-4 read-2 monotone Boolean formula. Gurvich ([2]) 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 [2] 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). On a positive side, from [3] we know that there is a polynomial time algorithm to recognize read-once functions when the input is a monotone depth-2 formula (that is, a DNF or a CNF). There are even very fast algorithms for this problem ( [4]). Our contribution consists of the following two results. 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). The second result states that this is a coNP-complete problem to decide whether the expression A∧Dn computes a read-once function, where Dn is as above and A is the ∧ − ∨ − ∧ depth-3 readonce monotone Boolean formula (so that the entire expression A∧Dn is depth-3 read-2). This result improves the result of [1] in the depth and the result of [2] in the readability of the input formula.