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Binary Decision Diagrams and Beyond: Enabling Technologies for Formal Verification
, 1995
"... 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 ..."
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Cited by 105 (0 self)
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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 OBDDbased 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 graphbased function representations. We focus on several recent advances of particular importance for forma...
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 98 (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
Verification of Arithmetic Circuits with Binary Moment Diagrams
 IN PROCEEDINGS OF THE 32ND ACM/IEEE DESIGN AUTOMATION CONFERENCE
, 1995
"... 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 integer values. BMDs can thus model ..."
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Cited by 93 (10 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 integer values. BMDs can thus model the functionality of data path circuits operating over wordlevel 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 a hierarchical approach to verifying arithmetic circuits, wherecomponentmodulesare first shownto implement their wordlevel specifications. The overall circuit functionality is then verified by composing the component functions and comparing the result to the wordlevel circuit specification. Multipliers with word sizes of up to 256 bits hav...
Multi Terminal Binary Decision Diagrams to Represent and Analyse Continuous Time Markov Chains
, 1999
"... Binary Decision Diagrams (BDDs) have gained high attention in the context of design and verification of digital circuits. They have successfully been employed to encode very large state spaces in an efficient, symbolic way. Multi terminal BDDs (MTBDDs) are generalisations of BDDs from Boolean va ..."
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Cited by 55 (11 self)
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Binary Decision Diagrams (BDDs) have gained high attention in the context of design and verification of digital circuits. They have successfully been employed to encode very large state spaces in an efficient, symbolic way. Multi terminal BDDs (MTBDDs) are generalisations of BDDs from Boolean values to values of any finite domain. In this paper, we investigate the applicability of MTBDDs to the symbolic representation of continuous time Markov chains, derived from highlevel formalisms, such as queueing networks or process algebras. Based on this data structure, we discuss iterative solution algorithms to compute the steadystate probability vector that work in a completely symbolic way. We highlight a number of lessons learned, using a set of small examples.
Symmetry Detection and Dynamic Variable Ordering of Decision Diagrams
, 1996
"... Knowing that some variables are symmetric in a function has numerous applications; in particular, it can help produce better variable orders for Binary Decision Diagrams (BDDs) and related data structures (e.g., Algebraic Decision Diagrams). It has been observed that there often exists an optimum ..."
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Cited by 55 (2 self)
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Knowing that some variables are symmetric in a function has numerous applications; in particular, it can help produce better variable orders for Binary Decision Diagrams (BDDs) and related data structures (e.g., Algebraic Decision Diagrams). It has been observed that there often exists an optimum order for a BDD wherein symmetric variables are contiguous. We propose a new algorithm for the detection of symmetries, based on dynamic reordering, and we study its interaction with the reordering algorithm itself. We show that combining sifting with an efficient symmetry check for contiguous variables results in the fastest symmetry detection algorithm reported to date and produces better variable orders for many BDDs. The overhead on the sifting algorithm is negligible. 1
EVBDDbased algorithms for integer linear programming, spectral transformation, and function decomposition
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 1994
"... ..."
Bisimulation Algorithms for Stochastic Process Algebras and their BDDbased Implementation
 In ARTS, LNCS 1601
, 1999
"... . Stochastic process algebras have been introduced in order to enable compositional performance analysis. The size of the state space is a limiting factor, especially if the system consists of many cooperating components. To fight state space explosion, various proposals for compositional aggregatio ..."
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Cited by 27 (13 self)
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. Stochastic process algebras have been introduced in order to enable compositional performance analysis. The size of the state space is a limiting factor, especially if the system consists of many cooperating components. To fight state space explosion, various proposals for compositional aggregation have been made. They rely on minimisation with respect to a congruence relation. This paper addresses the computational complexity of minimisation algorithms and explains how efficient, BDDbased data structures can be employed for this purpose. 1 Introduction Compositional application of stochastic process algebras (SPA) is particularly successful if the system structure can be exploited during Markov chain generation. For this purpose, congruence relations have been developed which justify minimisation of components without touching behavioural properties. Examples of such relations are strong equivalence [22], (strong and weak) Markovian bisimilarity [16] and extended Markovian bisimi...
On the Relation Between BDDs and FDDs
 INFORMATION AND COMPUTATION
, 1995
"... 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 bran ..."
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Cited by 26 (12 self)
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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 EXORbased 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 FDDsize of f and the BDDsize of (f) also in the case that the corresponding DDs are free or ordered, but not (necessarily) complete. We use this property to derive...
BitLevel Analysis of an SRT Divider Circuit
 IN PROCEEDINGS OF THE 33RD DESIGN AUTOMATION CONFERENCE, PAGES 661665, LAS VEGAS, NV
, 1995
"... It is impractical to verify multiplier or divider circuits entirely at the bitlevel using ordered Binary Decision Diagrams (BDDs), because the BDD representations for these functions grow exponentially with the word size. It is possible, however, to analyze individual stages of these circuits using ..."
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Cited by 23 (0 self)
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It is impractical to verify multiplier or divider circuits entirely at the bitlevel using ordered Binary Decision Diagrams (BDDs), because the BDD representations for these functions grow exponentially with the word size. It is possible, however, to analyze individual stages of these circuits using BDDs. Such analysis can be helpful when implementing complex arithmetic algorithms. As a demonstration, we show that Intel could haveused BDDs to detect erroneous lookup table entries in the Pentium(TM) floating point divider. Going beyond verification, we show that bitlevel analysis can be used to generate a correct version of the table.
Using EdgeValued Decision Diagrams for Symbolic Generation of Shortest Paths
 Proc. Fourth International Conference on Formal Methods in ComputerAided Design (FMCAD), LNCS 2517
, 2002
"... We present a new method for the symbolic construction of shortest paths in reachability graphs. Our algorithm relies on a variant of edgevalued decision diagrams that supports efficient fixedpoint iterations for the joint computation of both the reachable states and their distance from the initial ..."
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Cited by 22 (12 self)
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We present a new method for the symbolic construction of shortest paths in reachability graphs. Our algorithm relies on a variant of edgevalued decision diagrams that supports efficient fixedpoint iterations for the joint computation of both the reachable states and their distance from the initial states. Once the distance function is known, a shortest path from an initial state to a state satisfying a given condition can be easily obtained. Using a few representative examples, we show how our algorithm is vastly superior, in terms of both memory and space, to alternative approaches that compute the same information, such as ordinary or algebraic decision diagrams.