Given a Boolean relational specification \(F(\textbf{X}, \textbf{Y})\), where \(\textbf{X}\) is a vector of inputs and \(\textbf{Y}\) is a vector of outputs, Boolean functional synthesis requires us to compute a vector of (Skolem) functions \(\varvec{\Psi }(\textbf{X})\), one for each output in \(\textbf{Y}\), such that \(F(\textbf{X}, \varvec{\Psi }(\textbf{X})) \leftrightarrow \exists \textbf{Y}\,F(\textbf{X},\textbf{Y})\) holds. This problem lies at the heart of many applications and has received significant attention in recent years. In this paper, we investigate the role of representation of \(F(\textbf{X}, \textbf{Y})\) and of \(\varvec{\Psi }(\textbf{X})\) in determining the computational hardness of Boolean functional synthesis. We start by showing that an efficient way of existentially quantifying variables from a Boolean formula in a given order yields an efficient solution to Boolean functional synthesis and vice versa. We then propose a semantic normal form, called SynNNF, that guarantees polynomial-time synthesis and characterizes polynomial-time existential quantification for a given order of quantification of variables. We show that several syntactic and other semantic normal forms for Boolean formulas studied in the knowledge compilation literature are subsumed by SynNNF, and that SynNNF is exponentially more succinct than most of them. We also investigate how the representation of the synthesized (Skolem) functions \(\varvec{\Psi }(\textbf{X})\) affects the complexity of Boolean functional synthesis, and present a map of complexity based on the representations of \(F(\textbf{X},\textbf{Y})\) and \(\varvec{\Psi }(\textbf{X})\). Finally, we propose an algorithm to compile a specification represented as a NNF (including CNF) circuit to SynNNF. We present results of an extensive set of experiments conducted using an implementation of the above algorithm, and two other tools available in the public domain.

给定布尔关系规范\(F(\textbf{X}, \textbf{Y})\)，其中\(\textbf{X}\)是输入向量，\(\textbf{Y}\)是输出向量，布尔函数综合要求我们计算 (Skolem) 函数向量\(\varvec{\Psi }(\textbf{X})\)，每个输出对应于\(\textbf{Y}\ )，使得\(F(\textbf{X}, \varvec{\Psi }(\textbf{X})) \leftrightarrow \exists \textbf{Y}\,F(\textbf{X},\textbf{ Y})\)成立。这个问题是许多应用的核心问题，近年来受到了极大的关注。在本文中，我们研究了\(F(\textbf{X}, \textbf{Y})\)和\(\varvec{\Psi }(\textbf{X})\)的表示在确定中的作用布尔函数综合的计算难度。我们首先证明，以给定顺序对布尔公式中的变量进行存在量化的有效方法可以产生布尔函数综合的有效解决方案，反之亦然。然后，我们提出了一种称为SynNNF 的语义范式，它保证多项式时间合成并表征给定变量量化顺序的多项式时间存在量化。我们表明，知识编译文献中研究的布尔公式的几种句法和其他语义范式都被SynNNF所包含，并且SynNNF比它们中的大多数要简洁得多。我们还研究了合成 (Skolem) 函数的表示\(\varvec{\Psi }(\textbf{X})\)如何影响布尔函数综合的复杂性，并根据\的表示呈现了复杂性图(F(\textbf{X},\textbf{Y})\)和\(\varvec{\Psi }(\textbf{X})\)。最后，我们提出了一种算法，将表示为NNF（包括CNF）电路的规范编译为SynNNF。我们展示了使用上述算法的实现以及公共领域中提供的其他两个工具进行的大量实验的结果。