Symbolic Functional Decomposition: A Reconfiguration Approach

TL;DR

This paper introduces a symbolic framework for functional decomposition using reconfiguration, leveraging OBDDs, Boolean circuits, and finite automata, and proves fixed-parameter linear time solvability based on key structural parameters.

Contribution

It presents a novel symbolic approach to functional reconfiguration and decomposition, integrating OBDDs, Boolean circuits, and automata, with complexity results.

Findings

Functional reconfiguration can be solved efficiently with fixed-parameter linear time.

The framework unifies various models of function decomposition.

Complexity depends on the width of OBDDs, circuit parameters, and automaton size.

Abstract

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 $\mathcal{A}$.