Boolean Expression Diagrams

Abstract

This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) which can represent any Boolean circuit in linear space and still maintain many of the desirable properties of BDDs. Two algorithms are described for transforming a BED into a reduced ordered BDD. One closely mimics the BDD apply-operator while the other can exploit the structural information of the Boolean expression. The eficacy of the BED representation is demonstrated by verifying that the redundant and non-redundant versions of the ISCAS 85 benchmark circuits are identical. I n particular, it is verified that the two 16-bit multiplication circuits (c6288 and c6288nr) implement the same Boolean functions. Using BEDs, this verification problem is solved in less than a second, while using standard BDD techniques this problem is infeasible. BEDs are useful in applications where the end-result as a reduced ordered BDD is small, for example for tautology checking.

Equational Binary Decision Diagrams

“…By induction hypothesis, <P 1 and P2 are ordered, so no rule of the TRS ORDER is applicable to a strict subterm of <P. We now show that no rule (1)(2)(3)(4)(5)(6)(7)(8) We conclude with the complete algorithm to transform an arbitrary formula <P to EQ-OBDD, which is just a repeated application of TOPDOWN until a fixed point is reached:…”

“…This resembles the Binary Expression Diagrams (BEDs) of (2), for the pure boolean case. We have not thoroughly studied the relationship between our top-down algorithm and their up-one.…”

“…Consider the last equation in this sequence, which is either Xn-l = z E Th (T, a), in which case we are done, or it is z = Xn-l E Th(T, a). In this case, Xn-l doesn't occur in any other equation (it cannot occur positively above z = Xn-l in T by (2), nor can it occur below it by (1)). Hence n = 1 and z = x E Th (T,a).…”

Kodkod: A Relational Model Finder

“…Compact Boolean Circuits, described in Section 4.3, are a hybrid between Reduced Boolean Circuits (RBCs) [12] and Boolean Expression Diagrams (BEDs) [11]. Like RBCs, CBCs are a representational form for a quantifier-free logic, and they restrict variable vertices to the leaves of the graph.…”

“…· A new sparse-matrix representation of relations that is both simpler to implement and better performing than the 'atomization' used in Alloy [10]. · A new scheme for detecting opportunities for sharing in the constraint abstract syntax tree inspired by boolean expression diagrams [11] and reduced boolean circuits [12].…”