Ordered Binary Decision Diagrams (OBDDs) have found widespread use in CAD applications such as formal verification, logic synthesis, and test generation. OBDDs represent Boolean functions in a form that is both canonical and compact for many practical cases. They can be generated and manipulated by efficient graph algorithms. Researchers have found that many tasks can be expressed as series of operations on Boolean functions, making them candidates for OBDD-based methods. The success of OBDDs has inspired efforts to improve their efficiency and to expand their range of applicability. Techniques have been discovered to make the representation more compact and to represent other classes of functions. This has led to improved performance on existing OBDD applications, as well as enabled new classes of problems to be solved. This paper provides an overview of the state of the art in graph-based function representations. We focus on several recent advances of particular importance for forma...

The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the graphical model explicitly and may sometime reduce the search space exponentially. Indeed, most

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 is a generalized version of the BDD apply-operator while the other can exploit the structural information of the Boolean expression. This ability is demonstrated by verifying that two di erent circuit implementations of a 16-bit multiplier implement the same Boolean function. Using BEDs, this veri cation problem is solved in less than a second, while using standard BDD techniques this problem is infeasible. Generally, BEDs are useful in applications, for example tautology checking, where the end-result as a reduced ordered BDD is small.

The combinational logic-level equivalence problem is to determine whether two given combinational circuits implement the same Boolean function. This problem arises in a number of CAD applications, for example when checking the correctness of incremental design changes (performed either manually or by a design automation tool). This paper introduces a data structure called Boolean Expression Diagrams (BEDs) and two algorithms for transforming a BED into a Reduced Ordered Binary Decision Diagram (OBDD). BEDs are capable of representing any Boolean circuit in linear space and can exploit structural similarities between the two circuits that are compared. These properties make BEDs suitable for verifying the equivalence of combinational circuits. BEDs can be seen as an intermediate representation between circuits (which are compact) and OBDDs (which are canonical). Based on a large number of combinational circuits, we demonstrate that BEDs either outperform or achieve results comparable to...

Embedded systems make a heavy use of software to perform Real-Time embeddedcontrol tasks. Embedded software is characterized by a relatively long lifetime and by tight cost, performance and safety constraints. Several superoptimization techniques for embedded softwares based on Multi-valuedDecision Diagram (MDD) representations have been described in the literature, but they all share the same basic limitation. They arebased on standardOrderedMDD (OMDD) packages, and hencerequireafixedorder of evaluation for the MDD variables on every execution path. Free MDDs (FMDDs) lift this limitation, and henceopen up more optimization opportunities. Finding the optimal variable ordering for FMDDs is a very difficult problem. Henceinthis paper we describe a heuristic procedure that performs well in practice, and is based on FMDD cost estimation applied to recursive cofactoring. Experimental results show that our new variable ordering method obtains often smaller embedded softwarethanprevious (sifting-based) methods.

Computational complexity is concerned with the complexity of solving problems and computing functions and not with the complexity of verifying circuit designs. The importance of formal circuit verification is evident. Therefore, a framework of a complexity theory for formal circuit verification with binary decision diagrams is developed. This theory is based on read-once projections. For many problems it is determined whether and how they are related with respect to read-once projections. It is proved that multiplication can be reduced to squaring but squaring is not a read-once projection of multiplication. This perhaps surprising result is discussed. For most of the common binary decision diagram models of polynomial size complete problems with respect to read-once projections are described. But for the class of functions with polynomial-size free binary decision diagrams (read-once branching programs) no complete problem with respect to read-once projection exists. Supported by DF...

We propose a new class of decision diagrams called canonical cube transformation binary decision diagrams (canonical TBDD's), which is an extension of TBDD's proposed by Meinel et al [11, 3]. The core idea of TBDD's is to transform a function to another function in a new domain by an injective mapping and to represent the transformed function in a standard OBDD. If the new domain is larger than the original domain, canonicity is lost, which makes combinational verification difficult. In this paper we show that canonicity can be maintained by characterizing the care set of the new domain. Transformations of practical interest which guarantee polynomial size canonical TBDD's are introduced. We also give a new interpretation of TBDD's, which leads to an effective heuristic for extracting promising transformations automatically from highlevel specifications. Finally a combinational verification technique using canonical TBDD's is proposed, the effectiveness of which is justified by verifyi...

We discuss an open problem in construction of Reduced Ordered Binary Decision Diagrams (ROBDDs) using composition, and prove that the worst case complexity of the construction is truly cubic. With this insight we show that the process of composition naturally leads to the construction of (even exponentially) compact partitioned-OBDDs (POBDDs) [12]. Our algorithm which incorporates dynamic partitioning, leads to the most general (and compact) form of POBDDs - graphs with multiple root variables. To show that our algorithm is robust and practical, we have analyzed some well known problems in Boolean function representation, verification and finite state machine analysis where our approach generates graphs which are even orders of magnitude smaller.

The goal of Knowledge Compilation is to represent a Boolean expression in a format in which it can answer a range of online-queries in PTIME. The online-query of main interest to us is model counting, because of its application to query evaluation on probabilistic databases, but other online-queries can be supported as well such as testing for equivalence, testing for implication, etc. In this paper we study the following problem. Given a database query q, decide whether its lineage can be compiled efficiently into a given target language. We consider four target languages, of strictly increasing expressive power(when the size of compilation is constrained to be polynomial in the input size): Read-Once Boolean formulae, OBDD, FBDD and d-DNNF. For each target, we study the class of database queries that admit polynomial size representation: these queries can also be evaluated in PTIME over probabilistic databases. When queries are restricted to conjunctive queries without self-joins, it was known that these four classes collapse to the class of hierarchical queries, which is also the class of PTIME queries over probabilistic databases. Our main result in this paper is that, in the case of Unions of Conjunctive Queries (UCQ), these classes form a strict hierarchy. Thus, unlike conjunctive queries without self-joins, the expressive power of UCQ differs considerably w.r.t. these target compilation languages. Moreover, we give a complete characterization of the first two target languages, based on the query’s syntax.

BDDs (binary decision diagrams) and their variants are the most frequently used representation types or data structures for boolean functions. Research on BDD variants has turned out to be one of the areas where the symbiosis between theoretical investigations in algorithm design and analysis, complexity theory, and applications has led to progress in theory and in applications. Here the different roots of the interest in BDDs are described, the main BDD variants and their algorithmic properties are presented, the representation size of selected functions is investigated, lower bound techniques are discussed and applications to algorithmic graph problems and hardware verification problems are presented.