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 DT(mon-C) for any circuit complexity class C, where the height of the tree is O(log n), and the query functions can be computed by monotone circuits in the class C. In the above context, we prove the following characterizations and bounds. For any Boolean function f, we show that the minimum monotone decision tree height can be exactly characterized (both in the adaptive and non-adaptive versions of the model) in terms of its alternation (alt(f) is defined as the maximum number of times that the function value changes, in any chain in the Boolean lattice). We also characterize the non-adaptive decision tree height with a natural generalization of certification complexity of a function. Similarly, we determine the complexity of non-deterministic and randomized variants of monotone decision trees in terms of alt(f). We show that DT(mon-C) = C when C contains monotone circuits for the threshold functions (for e.g., if C = TC0). For C = AC0, we are able to show that any function in AC0 can be computed by a sub-linear height monotone decision tree with queries having monotone AC0 circuits. To understand the logarithmic height case in case of AC0 i.e., DT(mon-AC0), we show that functions in DT(mon-AC0) have AC0 circuits with few negation gates.
翻译:在本文中, 我们开始研究适应性和非适应性单调决定树的计算能力。 这自然导致任何电路复杂 C 级的DT( mon- C) 形式的复杂等级, 其中每个查询的高度为 O( log n), 查询功能可以由 C 类的单调决定树高度( 或大小) 被视为对给定的 Boolean 函数的非调色度的测量。 我们还通过限制( 电路复杂性) 可在每个节点查询的单调函数。 这自然导致任何电路复杂 C 级的 DT( mon- C) 形式的复杂等级, 其中树的高度为 O( log n), 而查询功能可以由 C 级的单调电路电路电路来计算。 对于任何 Boole 函数, 我们通过 Oral C 的适应性高度可以精确地描述( 适应性和非适应性版本) A( 等电路的 Oral- deal- deal- deal c), 当我们无法在 C 级的决定函数中显示 C 的 Ral- deal- deal- deal- deal- deal- deal- deal didededeal ex y y ex ex ex yal y) a c) a c) a c. we. we.