Frames: Reduction of Size of Binary Decision Diagrams using an Alternative Representation

Abstract

Abstract Binary Decision Diagrams (BDDs) have long been a fundamental data structure in computer science and computational logic, due to their compact representation and efficient manipulation of Boolean functions. This paper introduces the concept of frame, an ordered array of partitions of a set that facilitates a structured approach to representing and analyzing Boolean functions. Frames enable a generalization of BDD constructions, providing a more flexible structure. The paper presents a description of frame-based BDDs, together with algorithms for frame change operations and for constructing frames that maximize the efficiency of Boolean functions representation. Additionally, computational complexities associated with these algorithms are discussed. Notably, we propose an algorithm that, for a Boolean function on \(n\) variables, constructs a frame-based BDD with \(n+2\) nodes or less in \(O(n^5)\) functional operations. MSC Classification: 68R05 , 06E30 , 05A18 , 68W30 , 68R10.