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Verification of Arithmetic Functions with Binary Moment Diagrams
 IN DESIGN AUTOMATION CONF
, 1994
"... Binary Moment Diagrams (BMDs) provide a canonical representations for linear functions similar to the way Binary Decision Diagrams (BDDs) represent Boolean functions. Within the class of linear functions, we can embed arbitary functions from Boolean variables to real, rational, or integer values. BM ..."
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Cited by 110 (6 self)
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Binary Moment Diagrams (BMDs) provide a canonical representations for linear functions similar to the way Binary Decision Diagrams (BDDs) represent Boolean functions. Within the class of linear functions, we can embed arbitary functions from Boolean variables to real, rational, or integer values. BMDs can thus model the functionality of data path circuits operating over word level data. Many important functions, including integer multiplication, that cannot be represented efficiently at the bit level with BDDs have simple representations at the word level with BMDs. Furthermore, BMDs can represent Boolean functions with around the same complexity as BDDs. We propose
Binary decision diagrams in theory and practice
, 2001
"... Decision diagrams (DDs) are the stateoftheart data structure in VLSI CAD and have been successfully applied in many other fields.DDs are widely used and are also integrated in commercial tools.This special section comprises six contributed articles on various aspects of the theory and application ..."
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Cited by 31 (6 self)
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Decision diagrams (DDs) are the stateoftheart data structure in VLSI CAD and have been successfully applied in many other fields.DDs are widely used and are also integrated in commercial tools.This special section comprises six contributed articles on various aspects of the theory and application of DDs.As preparation for these contributions, the present article reviews the basic definitions of binary decision diagrams (BDDs). We provide a brief overview and study theoretical and practical aspects.Basic properties of BDDs are discussed and manipulation algorithms are described.Extensions of BDDs are investigated and by this we give a deeper insight into the basic data structure.Finally we outline several applications of BDDs and their extensions and suggest a number of articles and books for those who wish to pursue the topic in more depth.
How Many Decomposition Types Do We Need
 In European Design and Test Conference (EDTC
, 1995
"... Decision Diagrams (DDs) are used in many applications in CAD. Various types of DDs, e.g. BDDs, FDDs, KFDDs, di er by their decomposition types. In this paper we investigate the di erent decomposition types and prove that there are only three that really help to reduce the size of DDs. 1 ..."
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Cited by 24 (6 self)
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Decision Diagrams (DDs) are used in many applications in CAD. Various types of DDs, e.g. BDDs, FDDs, KFDDs, di er by their decomposition types. In this paper we investigate the di erent decomposition types and prove that there are only three that really help to reduce the size of DDs. 1
A Survey of Literature on Function Decomposition  Version IV
, 1995
"... This report surveys the literature on decomposition of binary, multiplevalued, and fuzzy functions. It gives also references to relevant basic logic synthesis papers that concern topics important for decomposition, such as for instance representation of Boolean functions, or symmetry of Boolean f ..."
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Cited by 12 (0 self)
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This report surveys the literature on decomposition of binary, multiplevalued, and fuzzy functions. It gives also references to relevant basic logic synthesis papers that concern topics important for decomposition, such as for instance representation of Boolean functions, or symmetry of Boolean functions. As a result of the analysis of the most successful decomposition programs for AshenhurstCurtis Decomposition, several conclusions are derived that should allow to create a new program that will be able to outperform all the existing approaches to decomposition. Creating such a superior program is necessary to make it practically useful for applications that are of interest to Pattern Theory group at Avionics Labs of Wright Laboratories. In addition, the program will be also able to solve problems that have been never formulated before. It will be a testbed to develop and compare several known and new partial ideas related to decomposition. Our emphasis is on the following topics: 1. representation of data and efficient algorithms for data manipulation, 2. variable ordering methods for variable partitioning to create bound and free sets of input variables; heuristic approaches and their comparison, 3. column compatibility problem, 4. subfunction encoding problem, 5. use of partial and total symmetries in data to decrease the decomposition search space, 6. methods of dealing with strongly unspecified functions which are typical for machine learning applications, 7. special types of decomposition, that can be efficiently handled (cascades, trees without variable repetition).
Factored edgevalued binary decision diagrams
 Formal Methods in System Design
, 1997
"... : : : : : : : : : : 40 ..."
Dynamic Minimization of OKFDDs
, 1995
"... 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 decomposi ..."
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Cited by 11 (9 self)
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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...
OKFDDs versus OBDDs and OFDDs
, 1995
"... Ordered Decision Diagrams (ODDs) as a means for the representation of Boolean functions are used in many applications in CAD. Depending on the decomposition type, various classes of ODDs have been defined, the most important being the Ordered Binary Decision Diagrams (OBDDs), the Ordered Functiona ..."
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Cited by 9 (5 self)
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Ordered Decision Diagrams (ODDs) as a means for the representation of Boolean functions are used in many applications in CAD. Depending on the decomposition type, various classes of ODDs have been defined, the most important being the Ordered Binary Decision Diagrams (OBDDs), the Ordered Functional Decision Diagrams (OFDDs) and the Ordered Kronecker Functional Decision Diagrams (OKFDDs). In this paper we clarify the computational power of OKFDDs versus OBDDs and OFDDs from a (more) theoretical point of view. We prove several exponential gaps between specific types of ODDs. Combining these results it follows that a restriction of the OKFDD concept to subclasses, such as OBDDs and OFDDs as well, results in families of functions which lose their efficient representation.
Complexity Theoretical Aspects of OFDDs
, 1996
"... We extend the list of complexity results for OFDDs on problems arising in practical applications. We show that it is NPhard to decide whether a function represented by some OFDD can be represented by an OFDD of size s using another variable ordering. Given an OFDD representation for an incompletely ..."
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Cited by 9 (2 self)
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We extend the list of complexity results for OFDDs on problems arising in practical applications. We show that it is NPhard to decide whether a function represented by some OFDD can be represented by an OFDD of size s using another variable ordering. Given an OFDD representation for an incompletely defined function, it is NPhard to compute an optimal OFDD cover for this function respecting the same variable ordering. The replacement of variables by constants may cause an exponential blowup of the OFDD size. Finally, it is investigated how a local change of the variable ordering may change the OFDD size. This leads to simulated annealing algorithms to improve variable orderings. I. Introduction OBDDs (ordered binary decision diagrams) introduced by Bryant [5] are the most common representation of Boolean functions in many applications, in particular in hardware verification and model checking. Most of the complexity theoretic problems concerning OBDDs are already solved. Kebschull, ...
Verification of Arithmetic Circuits Using Binary Moment Diagrams
, 2001
"... Binary Moment Diagrams (BMDs) provide a canonical representations for linear functions similar to the way Binary Decision Diagrams (BDDs) represent Boolean functions. Within the class of linear functions, we can embed arbitrary functions from Boolean variables to real, rational, or integer values. ..."
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Cited by 9 (0 self)
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Binary Moment Diagrams (BMDs) provide a canonical representations for linear functions similar to the way Binary Decision Diagrams (BDDs) represent Boolean functions. Within the class of linear functions, we can embed arbitrary functions from Boolean variables to real, rational, or integer values. BMDs can thus model the functionality of data path circuits operating over word level data. Many important functions, including integer multiplication, that cannot be represented efficiently at the bit level with BDDs have simple representations at the word level with BMDs. Furthermore, BMDs can represent Boolean functions with around the same complexity as BDDs. We propose
On The Complexity Of The Hidden Weighted Bit Function For Various BDD Models
 THEORETICAL INFORMATICS AND APPLICATIONS
, 1998
"... Ordered binary decision diagrams (OBDDs) and several more general BDD models have turned out to be representations of Boolean functions which are useful in applications like verication, timing analysis, test pattern generation or combinatorial optimization. The hidden weighted bit function (HWB) is ..."
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Cited by 8 (2 self)
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Ordered binary decision diagrams (OBDDs) and several more general BDD models have turned out to be representations of Boolean functions which are useful in applications like verication, timing analysis, test pattern generation or combinatorial optimization. The hidden weighted bit function (HWB) is of particular interest, since it seems to be the simplest function with exponential OBDD size. The complexity of this function with respect to dierent circuit models, formulas, and various BDD models is discussed.