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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.
References
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Journal ArticleDOI
TL;DR: A new representation class of Boolean functions, monotone term decision lists, is introduced which combines compact representation size with tractability of essential operations and is an attractive alternative to traditional universal representation classes such as DNF formulas or decision trees.

10 citations

Book ChapterDOI
Marc Snir1
13 Jul 1981

9 citations

Journal ArticleDOI
TL;DR: It is proved that the Fourier dimension of any Boolean function with Fourier sparsity s is at most O( s logs), and any XOR function has a protocol of complexity O( √ r logr) in the simultaneous communication model, where r is the rank of its communication matrix.
Abstract: We prove that the Fourier dimension of any Boolean function with Fourier sparsity s is at most O( √ s logs). This bound is tight up to a factor of O(logs) since the Fourier dimension and sparsity of the address function are quadratically related. We obtain our result by bounding the non-adaptive parity decision tree complexity, which is known to be equivalent to the Fourier dimension. A consequence of our result is that any XOR function has a protocol of complexity O( √ r logr) in the simultaneous communication model, where r is the rank of its communication matrix. ACM Classification: F.2.3 AMS Classification: 68Q17

8 citations

Proceedings ArticleDOI
19 Jun 1995
TL;DR: A tight lower bound of /spl theta/(k log(n/k))) is proved for the required depth of a decision tree for the threshold-k function and a tighter lower bound for the "direct sum" problem of computing simultaneously k copies of threshold-2 is proved.
Abstract: We investigate decision trees in which one is allowed to query threshold functions of subsets of variables. We are mainly interested in the case where only queries of AND and OR are allowed. This model is a generalization of the classical decision tree model. Its complexity (depth) is related to the parallel time that is required to compute Boolean functions in certain CRCW PRAM machines with only one cell of constant size. It is also related to the computation using the Ethernet channel. We prove a tight lower bound of /spl theta/(k log(n/k)) for the required depth of a decision tree for the threshold-k function. As a corollary of the method we also prove a tight lower bound for the "direct sum" problem of computing simultaneously k copies of threshold-2 in this model. Next, the size complexity is considered. A relation to depth-three circuits is established and a lower bound is proven. Finally the relation between randomization, nondeterminism and determinism is also investigated, we show separation results between these models.

7 citations