Consider a wireless sensor network in which each sensor has a bit of information. Suppose all sensors with the bit 1 broadcast this fact to a basestation. If zero or one sensors broadcast, the basestation can detect this fact. If two or more sensors broadcast, the basestation can only detect that there is a "collision." Although collisions may seem to be a nuisance, they can in some cases help the basestation compute an aggregate function of the sensors' data.
Motivated by this scenario, we study a new model of computation for boolean functions: the 2+ decision tree. This model is an augmentation of the standard decision tree model: now each internal node queries an arbitrary set of literals and branches on whether 0, 1, or at least 2 of the literals are true. This model was suggested in a work of Ben-Asher and Newman but does not seem to have been studied previously.
Our main result shows that 2+ decision trees can "count" rather effectively. Specifically, we show that zero-error 2+ decision trees can compute the threshold-of-t symmetric function with O(t) expected queries (and that Ω(t) is a lower bound even for two-sided error 2+ decision trees). Interestingly, this feature is not shared by 1+ decision trees. Our result implies that the natural generalization to k+ decision trees does not give much more power than 2+ decision trees. We also prove a lower bound of Ω(t log(n/t)) for the deterministic 2+ complexity of the threshold-of-t function, demonstrating that the randomized 2+ complexity can in some cases be unboundedly better than deterministic 2+ complexity.
Finally, we generalize the above results to arbitrary symmetric functions, and we discuss the relationship between k+ decision trees and other complexity notions such as decision tree rank and communication complexity.