Functional decomposition is the process of breaking down a function $f$ into a composition $f=g(f_1,\dots,f_k)$ of simpler functions $f_1,\dots,f_k$ belonging to some class $\mathcal{F}$. This fundamental notion can be used to model applications arising in a wide variety of contexts, ranging from machine learning to formal language theory. In this work, we study functional decomposition by leveraging on the notion of functional reconfiguration. In this setting, constraints are imposed not only on the factor functions $f_1,\dots,f_k$ but also on the intermediate functions arising during the composition process. We introduce a symbolic framework to address functional reconfiguration and decomposition problems. In our framework, functions arising during the reconfiguration process are represented symbolically, using ordered binary decision diagrams (OBDDs). The function $g$ used to specify the reconfiguration process is represented by a Boolean circuit $C$. Finally, the function class $\mathcal{F}$ is represented by a second-order finite automaton $\mathcal{A}$. Our main result states that functional reconfiguration, and hence functional decomposition, can be solved in fixed-parameter linear time when parameterized by the width of the input OBDD, by structural parameters associated with the reconfiguration circuit $C$, and by the size of the second-order finite automaton $\mathcal{A}$.

翻译：函数分解是将函数 $f$ 拆解为属于某类 $\mathcal{F}$ 的较简单函数 $f_1,\dots,f_k$ 的复合 $f=g(f_1,\dots,f_k)$ 的过程。这一基本概念可用于建模从机器学习到形式语言理论等多种应用场景中产生的问题。在本工作中，我们基于函数重配置的概念研究函数分解。在此设定下，约束不仅施加于因子函数 $f_1,\dots,f_k$，也施加于复合过程中产生的中间函数。我们引入了一个符号化框架来处理函数重配置与分解问题。在该框架中，重配置过程中产生的函数采用有序二元决策图（OBDD）进行符号化表示；用于描述重配置过程的函数 $g$ 由布尔电路 $C$ 表示；而函数类 $\mathcal{F}$ 则由二阶有限自动机 $\mathcal{A}$ 表示。我们的主要结果表明，当以输入 OBDD 的宽度、重配置电路 $C$ 的结构参数以及二阶有限自动机 $\mathcal{A}$ 的规模为参数时，函数重配置问题（进而函数分解问题）可在固定参数线性时间内求解。