# 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.

Topics: Tree (descriptive set theory) (57%), Monotone polygon (55%), Circuit complexity (55%), Boolean function (51%)

##### References

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TL;DR: Several complexity measures for Boolean functions are discussed: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial, and how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers.

Abstract: We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers.

704 citations

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01 Dec 1983-

TL;DR: All the apparently known lower bounds for linear decision trees are extended to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20].

Abstract: A topological method is given for obtaining lower bounds for the height of algebraic computation trees, and algebraic decision trees. Using this method we are able to generalize, and present in a uniform and easy way, almost all the known nonlinear lower bounds for algebraic computations. Applying the method to decision trees we extend all the apparently known lower bounds for linear decision trees to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20]. We also show how this new method can be used to establish lower bounds on the complexity of constructions with ruler and compass in plane Euclidean geometry.

574 citations

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TL;DR: The authors demonstrate that any function f whose $L_1 $-norm is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions.

Abstract: This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (i.e., summation of a subset of the input variables over $GF(2)$).This paper shows how to learn in polynomial time any function that can be approximated (in norm $L_2 $) by a polynomially sparse function (i.e., a function with only polynomially many nonzero Fourier coefficients). The authors demonstrate that any function f whose $L_1 $-norm (i.e., the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions. Moreover, it is shown that the functions with polynomial $L_1 $-norm can be learned deterministically.The algorithm can also exactly identi...

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06 Jan 2012-

Abstract: Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive description of basic lower bound arguments, covering many of the gems of this complexity Waterloo that have been discovered over the past several decades, right up to results from the last year or two. Many open problems, marked as Research Problems, are mentioned along the way. The problems are mainly of combinatorial flavor but their solutions could have great consequences in circuit complexity and computer science. The book will be of interest to graduate students and researchers in the fields of computer science and discrete mathematics.

335 citations

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TL;DR: A topological method is given for obtaining lower bounds for the height of algebraic decision trees and an Ω(n2) bound is obtained for trees with bounded-degree polynomial tests, thus extending the Dobkin-Lipton result for linear trees.

Abstract: A topological method is given for obtaining lower bounds for the height of algebraic decision trees. The method is applied to the knapsack problem where an Ω(n 2 ) bound is obtained for trees with bounded-degree polynomial tests, thus extending the Dobkin-Lipton result for linear trees. Applications to the convex hull problem and the distinct element problem are also indicated. Some open problems are discussed.

161 citations