Decision Diagrams (DDs) are used in many applications in CAD. Various types of DDs, e.g. BDDs, FDDs, KFDDs, differ by their decomposition types. In this paper we investigate the different decomposition types and prove that there are only three that really help to reduce the size of DDs. 1 Introduction Decision Diagrams (DDs) are successfully applied in many fields of design automation, e.g. [17, 4, 1, 14, 7, 24, 11, 2, 9]. The most popular type of DD is the Ordered Binary Decision Diagram (OBDD) allowing efficient representation and manipulation of Boolean functions [5]. The more recent techniques have made it possible to handle (some) large functions without any basic variation of the OBDD concept itself. The dynamic variable ordering with sifting introduced by Rudell [21] allows to represent examples which could not be represented by any previous heuristic methods. Moreover, the variable ordering in [21] is handled by the package itself, alleviating the need for variable ordering ...

Exclusive-Sums-Of-Products (ESOPs) play an important role in logic synthesis and design-for-test. This paper presents an improved version of the heuristic ESOP minimization procedure proposed in [1,2]. The improvements concern three aspects of the procedure: (1) computation of the starting ESOP cover; (2) increase of the search space for solutions by applying a larger set of cube transformations; (3) development of specialized datastructures for robust manipulation of ESOP covers. Comparison of the new heuristic ESOP minimizer EXORCISM-4 with other minimizers (EXMIN2 [3], MINT [4], EXORCISM-2 [1] and EXORCISM3 [2]) show that, in most cases, EXORCISM-4 produces results of comparable or better quality on average ten times faster.

We present methods for the construction of small Ordered Kronecker Functional Decision Diagrams (OKFDDs). OKFDDs are a generalization of Ordered Binary Decision Diagrams (OBDDs) and Ordered Functional Decision Diagrams (OFDDs) as well. Our approach is based on dynamic variable ordering and decomposition type choice. For changing the decomposition type we use a new method. We briefly discuss the implementation of PUMA, our OKFDD package. The quality of our methods in comparison with sifting and interleaving for OBDDs is demonstrated based on experiments performed with PUMA. 1 Introduction Decision Diagrams (DDs) are often used in CAD systems for efficient representation and manipulation of Boolean functions. The most popular data structure in this context are Ordered Binary Decision Diagrams (OBDDs) [5] that are used in many applications [6]. Nevertheless, some relevant classes of Boolean functions cannot be represented efficiently by OBDDs [2, 17]. As one alternative Ordered Function...

In this paper we investigate the complexity of problems on Ordered Functional Decision Diagrams (OFDDs) related to satisfiability problems, i.e. SAT-ONE, SAT-ALL and SAT-COUNT. We prove that SAT-ALL has a running time linear in the product of the number of satisfying assignments, and the size of the corresponding OFDD. Counting the satisfying assignments in an OFDD is proved to be #P-complete. I. Introduction The increasing complexity of modern VLSI circuitry is only manageable together with advanced CAD systems which as one important component contain (logic) synthesis tools. The problems to be solved can often be formulated in terms of Boolean functions. The efficiency of the representation and the manipulation algorithms performing (synthesis) operations largely depends on the type of data structure chosen. The most popular data structure is the Ordered Binary Decision Diagram (OBDD), which is a restricted form of a Binary Decision Diagram (BDD) [15, 1], also called branching progr...

AbstractÐThis paper investigates reduced ordered binary decision diagrams (OBDD) of partially symmetric Boolean functions when using variable orders where symmetric variables are adjacent. We prove upper bounds for the size of such symmetry ordered OBDDs (SymOBDD). They generalize the upper bounds for the size of OBDDs of totally symmetric Boolean functions and nonsymmetric Boolean functions proven by Heap and Mercer [14], [15] and Wegener [37]. Experimental results based on these upper bounds show that the nontrivial symmetry sets of a Boolean function should be located either right up at the beginning or right up at the end of the variable order in order to obtain best upper bounds. Index TermsÐBinary decision diagrams, variable ordering, upper worst case bounds, partial symmetric Boolean functions. 1

New symmetries of degree two are introduced, along with spectral techniques for identifying these symmetries. Some applications of these symmetries are discussed, in particular their application to the construction of binary decision diagrams and the implementation of Boolean functions. 1.