Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art data structure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branching programs, by ordering restrictions. In the context of EXOR-based logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD) becomes increasingly important. We study the relation between (ordered, free) BDDs and FDDs. Both, BDDs and FDDs, result from DDs by defining the represented function in different ways. If the underlying DD is complete, the relation between both types of interpretation can be described by a Boolean transformation . This allows us to relate the FDD-size of f and the BDD-size of (f) also in the case that the corresponding DDs are free or ordered, but not (necessarily) complete. We use this property to derive...

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...

In this paper methods for finding good variable orderings for ordered functional decision diagrams (OFDDs) are investigated. We present an algorithm for exact minimization of OFDDs that is applicable for functions up to n = 14 variables. We present an upper bound for the size of OFDDs representing tree-like circuits. Various methods for dynamic variable ordering based on the exchange of variables are presented. Experimental results are given to show the efficiency of our approaches.

This thesis investigates practical and theoretical concerns for the use of Binary Decision Diagrams (BDDs) for qualitative and quantitative risk assessments of complex systems. Boolean models describing failure relationships between components, and fault trees in particular, are boolean formulas whose variables are individual component failures; assessment of these models can be performed by analysis of the boolean function induced by the formula. Resource consumption for BDD computations, which is determined by the form of the boolean formula and the order imposed on its variables, is in many cases exponentially smaller than the truth table for the function. The use of Binary Decision Diagrams has made possible orders-of-magnitude increases in the complexity of systems that can be assessed efficiently. Nonetheless, the practical limits of straightforward use of BDDs for reliability analysis are often surpassed by real-world systems. Understanding why this happens is the first subject...

: We study two dissimilar techniques, one online and one offline, for improving boolean formula assessment in two formal verification domains, circuit verification and risk assessment. We establish, despite their differences, that the key advantages of these two techniques come from formula rewriting, regardless of whether it is online or offline. We formalize a unified framework for describing these two techniques, and shown how this provides a better understanding of the interaction between different improvements to boolean formulas. Experimental support for these claims is also shown. Topic Area: correctness preserving transformations; BDD- and FSMbased approaches Unifying Two Formula Rewriting Techniques for Circuit Verification and Risk Assessment Macha Nikolskaa 1 , Poul Frederick Williams 2 , and David James Sherman 1 1 Laboratoire Bordelais de Recherche en Informatique CNRS/Universite Bordeaux-1, Bordeaux, France macha|david@LaBRI.U-Bordeaux.FR 2 Departm...

We present a taxonomy of the heuristic analyses of boolean formulas that are used in BDD-based verifications of circuits and fault trees. This systematic characterization of known analyses gives greater insight into the nature of the problem and clearly indicates several opportunities for investigation. It also provides a formal basis for a novel software architecture for useful and extensible optimization tools for boolean formulas. 1 Introduction Building efficient BDD-based tools for circuit verification, reliability analysis, and model-checking critically depends on techniques for finding good variable orders: poor orders lead to intractably large BDD that are impossible to compute. Between a good and a poor variable order can lie an exponential factor in BDD size[Bry86, Bry91]. Efficiently building efficient tools requires intelligent heuristic analyses that work consistently for a given problem domain. Boolean formulas from different domains have different structural prop...

We present two new classes of 2-level AND/XOR expressions: the class RKRO of Reduced Kronecker Expressions and the class GKRO of Generalized Kronecker Expressions. GKRO contains RKRO and KRO, the well-known class of Kronecker Expressions [Sas93b] as a subclass. There is a close relation between RKROs, KROs, GKROs and Ordered Kronecker Functional Decision Diagrams (OKFDDs) [DST + 94], that together with efficient OKFDD algorithms can be utilized for (exact and heuristical) minimization of RKROs, KROs and GKROs. In this paper we concentrate on RKROs and KROs and propose several algorithms for their minimization. Experimental results are given to show the efficiency of our approach. For the first time efficient minimized 2-level AND/XOR expressions are determined for benchmark functions with more than 100 variables. Furthermore, we compare our solutions to results obtainable for other classes of AND/XOR forms. In particular, RKROs are much smaller than FPRMs and also turn ou...