K. Ravi, K. McMillan, T. Shiple, and F. Somenzi, "Approximation and decomposition of binary decision diagrams," in Proc. 35th ACM/IEEE Design Automation Conf., San Francisco, CA, June 1998, pp. 445--450.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....approximation is, the larger the dc is to minimize . The K maps in Fig. 12 represent a sequence of approximations starting with an initial exact approximation . The actual method presented in this paper uses BDDs to calculate function approximations; it is inspired on the approach presented in [24]. Fig. 13 presents an example for the same function depicted in Fig. 12. The approach consists of remapping some nodes of in such a way that the BDD size is reduced but the number of minterms of the new BDD is not increased too The approach for disjunctive decomposition is similar, but using ....

....by BDD minimization: This process is iteratively executed to generate a sequence of approximations as in Fig. 12. A cost function based on BDD sizes is used to select one of the approximations ( in the sequence. The actual BDD based approach used in this paper is similar to the one in [24], but considers many more nodes as candidates for replacement (same level, children, and grandchildren) It is important to notice that the approximation approach subsumes the conjunctive and disjunctive bi decompositions proposed by other authors [22] 25] in which the BDD transformations can ....

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K. Ravi, K. McMillan, T. Shiple, and F. Somenzi, "Approximation and decomposition of binary decision diagrams," in Proc. Design Automation Conf., 1998, pp. 445--450.

....is implemented by the function factor good in SIS [12] The second approach is computationally more expensive and based on BDD decompositions. Several approaches have been proposed in this direction. The one we have chosen has been inspired on the calculation of function approximations [11]. Fig. 9 illustrates a simple example on how a conjunctive decomposition for a function can be calculated by approximations. Given the BDD , we want to find such that ) pCS . For that, we need be over approximations of . The approach consists of remapping some nodes of in ....

....number of minterms is increased by two. Once p is known, can be calculated by BDD minimization: p . The approximations for disjunctive decomposition are similar (under approximations must be used instead) The actual BDD based approach used in this paper is similar to the one in [11], but considering many more nodes as candidates for replacement (same level, children and grand children) It is important to notice that the approximation approach subsumes the conjunctive and disjunctive bidecompositions proposed by other authors [9, 16] in which the BDD transformations can be ....

K. Ravi, K.L. McMillan, T.R. Shiple, and F. Somenzi. Approximation and decomposition of binary decision diagrams. In Design Automation Conference, pages 445--450, 1998.

....the approach of Yuan et al. 28] In view of this, we elected not to pursue BDD based FSM analysis. 5 In our work, we never report false negatives, i.e. report that an unreachable coverage goal is reachable. Several authors have developed procedures which never report false positives, e.g. [22, 14], but may report false negatives. A simple view of these papers is that they build an over estimate of the set of reachable states, e.g. by generic BDD approximation [22] or by aggressive existential quantification [14] Consequently, if the tool reports that the coverage goal is unreachable, it ....

....unreachable coverage goal is reachable. Several authors have developed procedures which never report false positives, e.g. 22, 14] but may report false negatives. A simple view of these papers is that they build an over estimate of the set of reachable states, e.g. by generic BDD approximation [22], or by aggressive existential quantification [14] Consequently, if the tool reports that the coverage goal is unreachable, it really is unreachable. However, such approaches may lead to false negatives; we are not aware of any good way of resolving these false negatives. 2.2 Test synthesis ....

Ravi, K., K. McMillan, T. Shiple, and F. Somenzi: 1998, `Approximation and Decomposition of Binary Decision Diagrams'. In: Proc. of the Design Automation Conf. 19

....the work was performed using RuleBase [4, 3] RuleBase was originally based on a version of SMV [14] After eight years of development, the original SMV code is a small part of the whole. Alternative model checking algorithms [17, 16, 1, 6] have been added, the original engine has been optimized [15, 10], the temporal logic has been enhanced [2] and features [5] supporting debugging of formulas (as opposed to debugging of the model) have been developed. The explicit model checker used in the work described in this paper was Spin [11] 2 The Veri ed Software The software veri ed is a ....

K. Ravi, K. McMillan, T. Shiple, and F. Somenzi. Approximation and decomposition of binary decision diagrams. In Proc. 35 Design Automation Conference (DAC). Association for Computing Machinery, Inc., June 1998.

....traversals may be a more convenient way of visiting the whole state set. The result is an interleaved breadth depth rst traversal, with better performance, especially in terms of memory. The real challenge, of course, is nding ecient problem decomposition and guided search strategies. In [1] [2], 3] the authors propose di erent techniques to control the size of BDD s during the whole traversal process. The key idea is to use subsetting based on high density . The density of a BDD is de ned as the ratio between the number of states and the number of BDD nodes. Whenever a BDD has a size ....

....coming from traversal themselves as a source for improved reachability analysis. We work on the correlation between reachable states and the transition relation within symbolic traversal operations (namely the relational product operator) whereas previous works focus on reachable state sets [1] [2], 3] 4] 7] or the transition relation [5] 6] 8] without any consideration about their combined impact on performance. This allows us to isolate sub behaviors of the FSM, and to follow the most (or the least) active ones, given a heuristic evaluation function. Concerning the ....

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K. Ravi, K. L. McMillan, T. R. Shiple, and F. Somenzi. Approximation and Decomposition of Binary Decision Diagram. In Proc. EDA/SIGDA/ACM/IEEE DAC'98, pages 445-450, San Francisco, California, June 1998.

....the approach of Yuan et al. 29] In view of this, we elected not to pursue BDD based FSM analysis. 4 In our work, we never report false negatives, i.e. report that an unreachable coverage goal is reachable. Several authors have developed procedures which never report false positives, e.g. [22, 14], but may report false negatives. A simple view of these papers is that they build an over estimate of the set of reachable states, e.g. by generic BDD approximation [22] or by aggressive existential quantification [14] Consequently, if the tool reports that the coverage goal is unreachable, it ....

....unreachable coverage goal is reachable. Several authors have developed procedures which never report false positives, e.g. 22, 14] but may report false negatives. A simple view of these papers is that they build an over estimate of the set of reachable states, e.g. by generic BDD approximation [22], or by aggressive existential quantification [14] Consequently, if the tool reports that the coverage goal is unreachable, it really is unreachable. However, such approaches may lead to false negatives; we are not aware of any good way of resolving these false negatives. 2.2 Test synthesis ....

K. Ravi, K. McMillan, T. Shiple, and F. Somenzi. Approximation and Decomposition of Binary Decision Diagrams. In Proc. of the Design Automation Conf., 1998.

....global model checking, we still need to show that it works in practice. We are currently working on proving the conjecture that our tableau construction can be implemented with linear complexity. We also want to investigate heuristics for applying the split rule. The approximation techniques of [25, 26] are a good starting point. ....

K. Ravi, K. L. McMillan, T. R. Shiple, and F. Somenzi. Approximation and decomposition of binary decision diagrams. In DAC'98. ACM, 1998.

....simulation to increase state coverage. These methods all have the problem that they do not scale to large designs easily, they require a lot of work and expertise, and the abstraction often hides many bugs. Another model checking method uses liberal abstraction to handle BDD overflow. One method [14] tries to find the closest subset to the original function. Liberal abstraction is useful in model checking since it simply reduces the state space searched. However, in symbolic simulation, liberal abstraction results in the wrong answer being produced and it is not clear how to compensate for ....

K. Ravi, K. McMillan, T. Shiple, and F. Somenzi. Approximation and decomposition of binary decision diagrams. In Proc. of 35th Design Automation Conf., pages 445--450, 1998.

....in Section 4 is necessary. We are also working on a complete tableau construction for 10 A. Biere and E. M. Clarke and Y. Zhu eventualities with linear tableau size in the number of states. Finally, we want to investigate heuristics for applying the split rule. The approximation techniques of [24,25] are a good starting point. ....

K. Ravi, K. L. McMillan, T. R. Shiple, and F. Somenzi. Approximation and decomposition of binary decision diagrams. In DAC'98. ACM, 1998.

....global model checking, we still need to show that it works in practice. We are currently working on proving the conjecture that our tableau construction can be implemented with linear complexity. We also want to investigate heuristics for applying the split rule. The approximation techniques of [25, 26] are a good starting point. 12 ....

K. Ravi, K. L. McMillan, T. R. Shiple, and F. Somenzi. Approximation and decomposition of binary decision diagrams. In DAC'98. ACM, 1998.

....partitioning allows the construction of manageable implicit descriptions of the behavior (even 32 bit multipliers) and a speed up of test vectors identification. Related ideas have been successfully applied in the domain of formal verification, and in particular to perform approximate traversal [20]. The rest of the paper is organized as follows. Section 2 discusses the behavioral test generation algorithm and the analyzed VHDL subset. Section 3 overviews the proposed implicit TPG together with information on the adopted implicit internal representation. Section 4 shows the efficiency ....

K. Ravi, K.L. McMillan, T.R.Shiple, and F. Somenzi. Approximation and Decomposition of Binary Decision Diagrams. Proc. ACM/IEEE DAC, pages 445--450, 1998.

....to verify real time systems. That work used polyhedra for representing sets of real numbers along with BDDs, but approximation was used only for the polyhedra, not for the BDDs. Various approaches to approximate reachability and verification using BDDs have preceded this work. Ravi et al. [10, 11] use high density BDDs to compute an underapproximation of the forward reachabe set. Cho et al. [4] proposed symbolic forward reachability algorithms that induce an overapproximation. They partition the set of state bits into mutually disjoint subsets, and do a symbolic forward propagation on ....

Ravi, K., McMillan, K. L., Shiple, T. R., and Somenzi, F., "Approximation and Decomposition of Binary Decision Diagrams," DAC 1998, pp. 445-450.

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K. Ravi, K. McMillan, T. Shiple, and F. Somenzi, "Approximation and decomposition of binary decision diagrams," in Proc. 35th ACM/IEEE Design Automation Conf., San Francisco, CA, June 1998, pp. 445--450.

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