Symbolic Functional Decomposition: A Reconfiguration Approach

2025-04-11

AAAI Conference on Artificial Intelligence

AI Summary

This study addresses the problem of efficiently decomposing Boolean functions under simultaneous constraints on both intermediate and factor functions. To this end, the authors propose a functional reconfiguration framework that formulates decomposition as a constrained combinatorial optimization problem, uniquely integrating symbolic representations with a functional reconfiguration mechanism for the first time. The approach encodes the target function using ordered binary decision diagrams (OBDDs), represents reconfiguration logic via Boolean circuits, and characterizes admissible function classes through second-order finite automata, thereby enabling explicit constraints on intermediate functions. When key parameters—including OBDD width, reconfiguration circuit structure, and automaton size—are fixed, the algorithm achieves decomposition in linear time with respect to input length, yielding the first fixed-parameter tractable linear-time solution to this problem.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Functional decomposition is the process of breaking down a function f into a composition f=g(f_1,...,f_k) of simpler functions f_1,...,f_k belonging to some class 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,...,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 F is represented by a second-order finite automaton 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 A.

Problem

• Functional Decomposition
• Functional Reconfiguration
• Symbolic Representation
• Ordered Binary Decision Diagrams
• Second-order Finite Automata

Innovation

• symbolic functional decomposition
• functional reconfiguration
• ordered binary decision diagrams
• Boolean circuits
• second-order finite automata