The desire to apply machine learning techniques in safety-critical environments has renewed interest in the learning of partial functions for distinguishing between positive, negative and unclear observations. We contribute to the understanding of the hardness of this problem. Specifically, we consider partial Boolean functions defined by a pair of Boolean functions $f, g \colon \{0,1\}^J \to \{0,1\}$ such that $f \cdot g = 0$ and such that $f$ and $g$ are defined by disjunctive normal forms or binary decision trees. We show: Minimizing the sum of the lengths or depths of these forms while separating disjoint sets $A \cup B = S \subseteq \{0,1\}^J$ such that $f(A) = \{1\}$ and $g(B) = \{1\}$ is inapproximable to within $(1 - \epsilon) \ln (|S|-1)$ for any $\epsilon > 0$, unless P=NP.

中文翻译：

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最小化定义局部布尔函数的DNF对或二元决策树对的不可近似性

在关键安全性环境中应用机器学习技术的愿望重新引起了人们对学习区分偏正，负和不清楚观察的部分功能的兴趣。我们有助于理解这个问题的难度。具体来说，我们考虑由一对布尔函数$ f，g \ colon \ {0,1 \} ^ J \ to \ {0,1 \} $定义的部分布尔函数，使得$ f \ cdot g = 0 $和这样$ f $和$ g $由析取范式或二元决策树定义。我们显示：在分离不交集的同时，使这些形式的长度或深度的总和最小化$ A \ cup B = S \ subseteq \ {0,1 \} ^ J $使得$ f（A）= \ {1 \}对于任何大于0的$ε，除非$和$ g（B）= \ {1 \} $近似在$（1-\ epsilon）\ ln（| S | -1）$之内，除非P = NP。