计算模型的真值表最小化
Truth Table Minimization of Computational Models
摘要：
复杂性理论为计算布尔函数提供了多种简洁的计算模型-分支程序、电路、决策树和有序二元决策图等等。对于任何这样的模型，在这种背景下出现的一个自然问题是：给定函数f：{0，1}^n \to {0，1}，我们能计算出在所讨论的计算模型中计算f的最优复杂度吗？（根据某种可取的措施）。关于这个问题的一个关键问题是如何精确地给出f，因为更详细的f描述允许算法使用更多的计算资源。在可能的表示中有黑盒访问f（例如在计算学习理论中），在期望的计算模型中表示f或在其他模型中表示f。人们可能会猜想，如果f作为其完整的真值表给出（即，f在其2^n个可能输入中的每一个上的值的列表），这是可以想到的最精细的描述，那么任何计算模型都可以被有效地计算，因为计算它的算法可以运行poly（2^n）时间。最近的几项研究表明，这与事实相去甚远-一些模型具有有效和简单的算法，可以产生所需的结果，其他模型被认为是困难的，并且对于某些模型，这个问题仍然存在。在这篇论文中，我们将针对几种常见的计算模型来讨论这个问题的计算复杂性。我们将提出几个新的硬度结果和有效的算法，以及新的证明和已知定理的扩展，为变种的决策树，公式和分支程序。
英文摘要：
Complexity theory offers a variety of concise computational models for computing boolean functions - branching programs, circuits, decision trees and ordered binary decision diagrams to name a few. A natural question that arises in this context with respect to any such model is this: Given a function f:{0,1}^n \to {0,1}, can we compute the optimal complexity of computing f in the computational model in question? (according to some desirable measure). A critical issue regarding this question is how exactly is f given, since a more elaborate description of f allows the algorithm to use more computational resources. Among the possible representations are black-box access to f (such as in computational learning theory), a representation of f in the desired computational model or a representation of f in some other model. One might conjecture that if f is given as its complete truth table (i.e., a list of f's values on each of its 2^n possible inputs), the most elaborate description conceivable, then any computational model can be efficiently computed, since the algorithm computing it can run poly(2^n) time. Several recent studies show that this is far from the truth - some models have efficient and simple algorithms that yield the desired result, others are believed to be hard, and for some models this problem remains open. In this thesis we will discuss the computational complexity of this question regarding several common types of computational models. We shall present several new hardness results and efficient algorithms, as well as new proofs and extensions for known theorems, for variants of decision trees, formulas and branching programs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
