Free BDDs (FBDDs) are an extension of ordered BDDs (OBDDs). FBDDs may have different orderings along each path. They allow a more efficient representation, while keeping (nearly) all of the properties of OBDDs. In some cases even an exponential reduction can be observed. In this paper we present for the first time an exact algorithm for finding a minimal FBDD representation for a given Boolean function. To reduce the huge search space, it makes use of a pruning technique. The algorithm also considers symmetries of the function. Since the algorithm is only applicable to small functions we also present a heuristic for FBDD minimization starting from an OBDD. Our experiments show that in many cases significant improvements can be obtained.

this paper we survey some state-of-the-art techniques used to perform automatic verification of combinational circuits. We classify the current approaches for combinational verification into two categories: functional and structural. The functional methods consist of representing a circuit as a canonical decision diagram. Two circuits are equivalent if and only if their decision diagrams are equal. The structural methods consist of identifying related nodes in the circuit and using them to simplify the problem of verification. We briefly describe some of the methods in both the categories and discuss their merits and drawbacks.

With the increase in the complexity of present day systems, proving the correctness of a design has become a major concern. Simulation based methodologies are generally inadequate to validate the correctness of a design with a reasonable confidence. More and more designers are moving towards formal methods to guarantee the correctness of their designs. In this paper we survey some state-of-the-art techniques used to perform automatic verification of combinational circuits. We classify the current approaches for combinational verification into two categories: functional and structural. The functional methods consist of representing a circuit as a canonical decision diagram. Two circuits are equivalent if and only if their decision diagrams are equal. The structural methods consist of identifying related nodes in the circuit and using them to simplify the problem of verification. We briefly describe some of the methods in both the categories and discuss their merits and drawbacks.

Introduction Various met2 dsexist t o represent mult#N3("E32vq functN3( [15]--[18]. Among tng( shared binary decision diagrams (SBDDs) [4], [11] are most commonly used, sincete(v sizes are usually smaller [18] t]( ot(qt ypes of BDDs, such as multNq("EwNEfi binary decision diagrams (MTBDDs) [16] and BDDs for charact"q#37fi functct (BDDs for CFs) [1], [19]. Some autEN3 [5] use te t erm "multv7N otv BDD"instvfi of SBDD. However, for some applicat"q7w SBDDs are stqE t o large and more compact representprese are required. Tofurtfi( reduce memory stv#73 we propose partr -(q#2 SBDDs, as amet2 d t represent multE#v( outE functv("q Eachpart represent a set ofout put and is optvNqN( independent ly. Such BDDs are considered as a special case ofpart27("v# BDDs [6], [12], [13] and free BDDs (FBDDs) [7], [8]. Not t hat BDD nomenclatNw is not unified. For example,t he te( "part77("v7 BDDs" has a di#erent meaning for certq# autq# [20]. Manuscript received March 13, 2002. Manuscript revised June 24,

Free Binary Decision Diagrams (FBDDs) are a data structure for the representation and manipulation of Boolean functions. Efficient algorithms for most of the important operations are known if only FBDDs respecting a fixed graph ordering are considered. However, the size of such an FBDD may strongly depend on the chosen graph ordering and efficient algorithms for computing good or optimal graph orderings are not known. In this paper it is shown that the existence of polynomial time approximation schemes for optimizing graph orderings or for minimizing FBDDs implies NP = ZPP or NP = P, respectively, and so such algorithms are quite unlikely to exist. 1. Introduction Many variants of Binary Decision Diagrams (BDDs) have been considered as a data structure for Boolean functions. Such data structures have several applications, in particular in computer aided hardware design. They are used in programs for, e.g., circuit verification, test pattern generation, model checking and logic synthes...

Ordered Binary Decision Diagrams (OBDDs) are the state-of-the-art data structure in VLSI CAD for representation and manipulation of Boolean functions. But due to the ordering restriction, many Boolean functions cannot be represented eciently. As one alternative read-k-times BDDs have been proposed. They are a generalization of OBDDs in the way that variables may occur up to k times on each path, while they may only occur once in OBDDs. More functions can be represented by read-k-times BDDs in polynomial space than by OBDDs, while many operations, like synthesis and satisability, still have polynomial worst case behavior. In this paper, we present a new technique for implementation of read-k-times BDD packages on top of standard OBDD implementations. Thus, highly optimized OBDD packages can be used and only few changes in the code are needed, while the new type of decision diagram allows much smaller representations. Experimental results are given to demonstrate the e- ciency of the a...

Decision diagrams are a natural representation of finite functions. The obvious complexity measures are length and size which correspond to time and space of computations. Decision diagrams are the right model for considering space lower bounds and time-space trade-o#s. Due to the lack of powerful lower bound techniques, various types of restricted decision diagrams are investigated. They lead to new lower bound techniques and some of them allow e#cient algorithms for a list of operations on boolean functions. Indeed, restricted decision diagrams like ordered binary decision diagrams (OBDDs) are the most common data structure for boolean functions with many applications in verification, model checking, CAD tools, and graph problems. From a complexity theoretical point of view also randomized and nondeterministic decision diagrams are of interest.

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