Power of Decision Trees with Monotone Queries.

In this paper we initiate study of the computational power of adaptive and non-adaptive monotone decision trees - decision trees where each query is a monotone function on the input bits. In the most general setting, the monotone decision tree height (or size) can be viewed as a measure of non-monotonicity of a given Boolean function. We also study the restriction of the model by restricting (in terms of circuit complexity) the monotone functions that can be queried at each node. This naturally leads to complexity classes of the form \\({\\mathsf {DT}}(\\textit{mon-}\\mathcal {C})\\) for any circuit complexity class \\(\\mathcal {C}\\), where the height of the tree is \\(\\mathcal {O}(\\log n)\\), and the query functions can be computed by monotone circuits in class \\(\\mathcal {C}\\). In the above context, we prove the following characterizations and bounds.