Bi-Kronecker Functional Decision Diagrams: A Novel Canonical Representation of Boolean Functions

Abstract

In this paper, we present a novel data structure for compact representation and effective manipulations of Boolean functions, called Bi-Kronecker Functional Decision Diagrams (BKFDDs). BKFDDs integrate the classical expansions (the Shannon and Davio expansions) and their bi-versions. Thus, BKFDDs are the generalizations of existing decision diagrams: BDDs, FDDs, KFDDs and BBDDs. Interestingly, under certain conditions, it is sufficient to consider the above expansions (the classical expansions and their bi-versions). By imposing reduction and ordering rules, BKFDDs are compact and canonical forms of Boolean functions. The experimental results demonstrate that BKFDDs outperform other existing decision diagrams in terms of sizes.

Dynamic minimization of bi-kronecker functional decision diagrams

“…We have implemented an efficient BKF-DD package with the proposed algorithms based on the code of the CUDD package. The implementation of the CUDD package is highly optimised, and hence this version of BKFDD package is more efficient than the initial version [12] based on the code of CacBDD [15].…”

“…Any Boolean function can be decomposed into smaller scale subfunctions by replacing the variable of with a different element. In the theory of BKFDDs [12], it has been proved that there are six different ways to decompose , including the Shannon decomposition [7,22], the positive and negative Davio decomposition [16,19], and their biconditional variants [1,12].…”

“…(1) for each complete path and two distinct internal nodes on the path, their decision variables are different; (2) the decision variables of internal nodes on all complete paths in the DD appear following the same order. Since BKFDDs integrate classical and biconditional decompositions, it is required that the auxiliary function is either a Boolean constant 0 or a distinct variable [12]. To determine the auxiliary function of a node, we need the concept of variable orders with decomposition types (ODT).…”

“…By imposing reduction rules RI, RS and RD, weak reduced OBKF-DDs (weak ROBKFDDs) can be defined, and is a canonical form of Boolean functions [12]. A new reduction rule RC was identified in Huang et al [12], and applying this rule derives another canonical form of BKFDDs, namely strong reduced OBKFDDs (strong ROBKFDDs), more compact than weak ones. In this paper we just focus on weak ROBKFDDs since the implementation of strong ROBKFDDs is more complex.…”

“…Based on this decomposition, Biconditional Binary Decision Diagrams (BBDDs) are proposed in [1], and are proven to be more compact than BDDs for two common Boolean functions: Majority and Adder. Inspired by the Bi-Shannon decomposition, Huang et al [12] introduce the biconditional variant of Davio decomposition, and devise a novel canonical decision diagram, called Bi-Kronecker Functional Decision Diagrams (BKFDDs), fusing both classical and biconditional decompositions. The diversity of decompositions enables BKFDDs to be more general than the above decision diagrams.…”