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Book ChapterDOI

Power of Decision Trees with Monotone Queries

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.
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Proceedings ArticleDOI
01 Jul 1992
TL;DR: It is shown that the minimum depth of a linear decision tree for this problem is Θ(n log(n/k), which can be established for k = O(n1/4−e) by a volume argument; for the whole range, the proof is more complicated and it involves the use of some topology as well as the theory of Mobius functions.
Abstract: We describe two methods for estimating the size and depth of decision trees where a linear test is performed at each node. Both methods are applied to the question of deciding, by a linear decision tree, whether given n real numbers, some k of them are equal. We show that the minimum depth of a linear decision tree for this problem is Θ(n log(n/k)). The upper bound is easy; the lower bound can be established for k = O(n1/4−e) by a volume argument; for the whole range, however, our proof is more complicated and it involves the use of some topology as well as the theory of Mobius functions.

113 citations

Journal ArticleDOI
TL;DR: The alphabet is called an alphabet of a formula in n variables words in the alphabet,I)~ which is defined by the following generating rules.
Abstract: 1. Let n be a natural number different from zero. We will call the alphabet {0, 1, xl , . . , xn, &, V, l, ( , )} an alphabet of a formula in n variables. We will denote this alphabet by ~ . We will call the letters Xl, • • • , x. of the alphabet • ~ variables and the letters 0, 1 constants. We will denote by formulas in n variables words in the alphabet ,I)~ which are defined by the following generating rules. F1. Single-lettered words

91 citations

Journal ArticleDOI
TL;DR: The results show that the property of a Boolean function having a concise Fourier representation is locally testable and an “implicit learning” algorithm is given that lets us test any subproperty of Fourier concision.
Abstract: We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We give the first efficient algorithms for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients) and for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of $\mathbb{F}_2^n$. In both cases we also prove lower bounds showing that any testing algorithm—even an adaptive one—must have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Building on these results, we give an “implicit learning” algorithm that lets us test any subproperty of Fourier concision. We also present some applications of these results to exact learning and decoding. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [V. Feldman, P. Gopalan, S. Khot, and A. Ponnuswami, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006, pp. 563-576].

85 citations

Journal ArticleDOI
TL;DR: The result presented here is the first lower bound of better than n log n given for an NP-complete problem for a model that is actually used in practice and derived by combining results on linear search tree complexity with results from threshold logic.

71 citations

Journal ArticleDOI
TL;DR: It is proved that the concept class of rank-r decision trees is contained within the class of r-decision lists (defined by Rivest) and one result of this note is that the simpler algorithm of Rivest can be used for both.

68 citations