Power of Decision Trees with Monotone Queries
29 Aug 2020-pp 287-298
TL;DR: This paper begins study of the computational power of adaptive and non-adaptive monotone decision trees - decision trees where each query is amonotone function on the input bits and proves the following characterizations and bounds.
Abstract: 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.
References
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TL;DR: In this paper, the authors study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions.
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