The hardness of the Expected Decision Depth problem

Given a function f over n binary variables, and an ordering of the n variables, Added abstract we consider the Expected Decision Depth problem. Namely, what is the expected number of bits that need to be observed until the value of the function is determined, when bits of the input are observed according to the given order. Our main finding is that this problem is #P-hard. In particular, hardness holds when the function f is represented as a decision tree.