A. Srinivasan, T. Kam, S. Malik and R. Brayton, "Algorithms for Discrete Function Manipulation, " Proc. Int. Conf. CAD, pp. 92-95, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....calls in the past are used to determine a variation of sifting that is applied next. Experimental results are given to demonstrate that runtimes and memory consumption can be reduced by 30 on average when the proposed selection methods are used during MDD construction. 1 Introduction MDDs [19] are used to represent functions with multivalued input variables. These decision diagrams are derived from BDDs [4] and most of the efficient operations that make BDDs fundamental for many applications dealing with large binary functions, can be provided for MDDs, too. Hence, functions with ....

....As is well known, each Boolean function f : B 7 B can be represented by an ordered binary decision diagram (BDD) 4] i.e. a directed acyclic graph where a Shannon decomposition is carried out in each node. It is straightforward to extend BDDs to multi valued decision diagrams (MDDs) [19] representing functions f : f0; k 1g 7 f0; k 1g. For this, each internal node has k outgoing edges . In [19] it has been shown that the efficient operations known for BDDs can also be carried out In our application we restrict ourselves without loss of generality to the case that ....

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A. Srinivasan, T. Kam, S. Malik, and R. Brayton. Algorithms for discrete function manipulation. In Int'l Conf. on CAD, pages 92--95, 1990.

....is used to encode a single mv set membership function. The reader is referred to [CGD 02] for an empirical evaluation of the different choices. All four varieties of decision diagrams listed in Table 6. 1 have been proposed in the literature: MDDs were first described by Srinivasan et al. [SKMB90] They included MBTDDs as a special case, but these are discussed in more detail by Sasao and Butler [SB96] ADDs were proposed by Bahar et al. [BFG 93] these are also known under the name MTBDDs [FMY97] and BDDs were introduced by Akers [Ake78] and later by Bryant [Bry86] who suggested the ....

A. Srinivasan, T. Kam, S. Malik, and R.E. Brayton. "Algorithms for Discrete Function Manipulation". In IEEE/ACM International Conference on ComputerAided Design (ICCAD'90), pages 92--95, Santa Clara, CA, USA, November 1990. IEEE Computer Society.

.... represent all of the functions required [Brace et al. 1990; Karplus 1989; Minato et al. 1990; Reeves and Irwin 1987] adding labels to the arcs to denote Boolean negation [Brace et al. 1990; Karplus 1989; Minato et al. 1990; Madre and Billon 1988] and generalizing the concept to other finite domains [Srinivasan et al. 1990]. These refinements yield significant savings in the memory requirement generally the most critical resource in determining the complexity of the problems that can be solved. Applications that require generating over 1 million OBDD vertices are now routinely performed on workstation computers. ....

Srinivasan, A., Kam, T., Malik, S., and Brayton, R. K. 1990. Algorithms for discrete function manipulation. International Conference on Computer-Aided Design (Santa Clara, Nov.), IEEE, New York, pp. 92--95.

....by means of experiments. 1 Introduction Decision Diagrams (DDs) are used in many fields of design automation. This holds especially for (Ordered) Binary Decision Diagrams ( O)BDDs) 3] e.g. in the area of verification [8, 7] In the meantime also Ordered Multi Valued Decision Diagrams (OMDDs) [10] have been used for verification [4] The basic underlying algorithm of all the verification approaches is the symbolic simulation, i.e. a traversal of the circuit to be verified where the DDs are constructed for each internal signal. Obviously the sequence in which the DDs are combined at each ....

....Diagrams As well known each Boolean function f : B B can be represented by an Ordered Binary Decision Diagram (OBDD) 3] i.e. a directed acyclic graph where a Shannon decomposition is carried out in each node. It is straightforward to extend OBDDs to MultiValued Decision Diagrams (MDDs) [10] representing functions f : f0; k Gamma 1g f0; k Gamma 1g. Each internal node has k outgoing edges and there exist k terminal nodes, one for each logical value . In [10] it has been shown that the efficient operations known for BDDs can also be carried out on MDDs using a ....

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A. Srinivasan, T. Kam, S. Malik, and R.E. Brayton. Algorithms for discrete function manipulation. In Int'l Conf. on CAD, pages 92--95, 1990.

....In later work [15, 16] a more ecient encoding is introduced based on place invariants; however, the underlying logic is still based on binary variables. In our work, we instead use an extension of BDDs to non binary logic, as proposed for the Multi valued Decision Diagrams (MDDs) of Kam [11, 19], or the shared trees of Zampuni eris [20] Our title uses the term decision diagrams because the results we present, while particularly relevant to MDDs, apply also to BDDs. We are also building upon our own previous work on state space storage [8] By using a multilevel data structure based ....

....a binary encoding with dlog 2 (k 1)e variables. The former results in more boolean variables overall, but also in a simpler encoding of the function. 2. 3 Multi valued decision diagrams BDDs have been generalized to integer functions on integer (instead of binary) variables, resulting in MDDs [11, 19] (see also the shared trees in [20] MDDs can then represent functions of the form S 1 S 2 Sn f0; m 1g; a b c b c 0 0 0 0 1 1 1 1 2 2 2 2 Fig. 4. MDD representing min(a; b; c) 1. if F is a constant r then return G r = G = G then ....

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A. Srinivasan, T. Kam, S. Malik, and R. K. Brayton. Algorithms for discrete function manipulation. In International Conference on CAD, pages 92-95. IEEE Computer Society, 1990.

....Research Projects Agency (contract number N00014 87 K 0828) and by a gift from Mitsubishi Electronics. The first author was supported by an ONR Graduate Fellowship. Most of this work was done using equipment generously donated by Sun Microsystems. higher level BDD based verification, like MDDs [18, 15] or EVBDDs [16] which are extensions to the basic BDD data structure. To our initial surprise, the BDD based verification method has been disappointing a method which stores all of the reachable states explicitly in a hash table substantially outperforms the BDD methods on our real ....

....which state variables must be initialized (e.g. counters and status variables) and which can be allowed to start with an arbitrary value (e.g. data paths) Let s look at some concrete examples. 3.1 Type Invariants In all high level BDD based verifiers, integer types are encoded as bit vectors. [18, 15, 13] The most natural encoding scheme is simply to use the binary representation of the integer. In many instances, however, especially when performing high level verification, the number of possible values is not exactly a power of 2. For example, in a communication protocol, a variable might ....

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Arvind Srinivasan, TimothyKam, Sharad Malik, and Robert K. Brayton, "Algorithms for Discrete Function Manipulation," IEEE International Conference on Computer-Aided Design, 1990, pp. 92--95.

....package. 1. Introduction Reduced ordered binary decision diagrams (BDD) have been widely studied since their introduction by Bryant [2] in 1986. A good review can be found in [3] and the other articles included in that special issue. The extension to multiple valued logic has been considered [7,8,9,15]. In the MVL case, a function is represented by a directed acyclic graph called a multiple valued decision diagram (MDD) MDDs are ordered and reduced in a fashion analogous to the binary case and the resulting representation is termed a reduced ordered MDD. Since all diagrams considered in this ....

....the resulting representation is termed a reduced ordered MDD. Since all diagrams considered in this work are reduced and ordered, we shall for brevity use MDD for multiple valued decision diagrams and BDD for binary decision diagrams. The efficient implementation of BDDs has been widely studied [1,11,13,14,15] and several highly efficient packages are available, e.g . CUDD [13] Many binary techniques, or extensions thereof, are useful when implementing a package for the creation and manipulation of MDDs. But, there are issues new to the MDD case, particularly the choice of logic primitives to use in ....

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Srinivasan, A., T. Kam, S. Malik, and R.E. Brayton, "Algorithms for discrete function manipulation," Proc. ICCAD, pp. 92-95, 1990.

.... to represent functions f : B n f0; k 1g and the resulting graphs are denoted as Multi Terminal BDDs (MTBDDs) The operations on MTBDDs can be carried out as eciently as in the case of two terminals [5] It is straightforward to extend MTBDDs to MultiValued Decision Diagrams (MDDs) [21] representing 1 In the binary case MIN and MAX gates correspond to ANDand OR gates, respectively. 2 These LITERAL gates are also called window literals. 0 2 x 1 1 x x 2 2 Figure 1. Reduced OMDD functions f : f0; k 1g n f0; k 1g. For this each internal node has k outgoing ....

....1 In the binary case MIN and MAX gates correspond to ANDand OR gates, respectively. 2 These LITERAL gates are also called window literals. 0 2 x 1 1 x x 2 2 Figure 1. Reduced OMDD functions f : f0; k 1g n f0; k 1g. For this each internal node has k outgoing edges 3 . In [21] it has been shown that the ecient operations known for BDDs can also be carried out on MDDs using a case operator instead of the ite operator [1] A DD is called ordered if each variable is encountered at most once on each path from the root to a terminal and if the variables are encountered in ....

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A. Srinivasan, T. Kam, S. Malik, and R.E. Brayton. Algorithms for discrete function manipulation. In Int'l Conf. on CAD, pages 92-95, 1990.

.... diagrams against each other by using Kleenean ternary logic, much less the particular semantics we give to the U branch for our nonterminal vertices (below) More abstract efforts which seem to lay claim to general n ary decision diagrams for any cardinal n can be cited, such as the MDD of [SKMB90] and the NDD of [KP 92] Here we first see nonterminal vertices having more than two branches, although the definitions tend to be syntactical leaving us often to wonder whether any meaning [YM88] can be attached in general to such an n ary decision diagram. By our designation Ordered ....

Arvind Srinivasan, Timothy Kam, Sharad Malik, and Robert K. Brayton. Algorithms for Discrete Function Manipulation. In Proceedings ICCAD '90, pages 92--95, November 1990.

....are represented within the model checker by multivalued decision diagrams (MDDs) a multi valued extension of the binary decision diagrams (BDDs) Bryant, 1992] We give a detailed treatment of MDDs in [Chechik et al. 2001a] Here we illustrate them by means of a brief example. Definition 3. [Srinivasan et al. 1990] Given a finite domain #, the generalized Shannon expansion of a function # : # # # #, with respect to the first variable in the ordering, is #(# # ## # # ##### ### ) # # # (# # ###### ### )###### ##### (# # ###### ### ) where # # = # [# # ## # ] the function obtained by substituting the ....

A. Srinivasan, T. Kam, S. Malik, and R.E. Brayton. "Algorithms for Discrete Function Manipulation". In IEEE International Conference on Computer-Aided Design, pages 92--95, 1990.

....whether given lattices have some desired properties, e.g. distributivity, and to support various lattice based calculations. Our library is based on Freese s Lisp lattice library [11] The MDD library is described below. 5. 1 Data Structures There is an extensive literature dealing with MDDs [21], mostly in the field of circuit design. To our knowledge, the logics used in that literature are given by total orders (such as the integers modulo #) and not by arbitrary quasi boolean lattices, but we concede that this is a minor difference. Also, as far as we know, they have not been used in ....

....# on ##1 variables with # # set to #, and the same function with # # set to #. These functions are referred to as # # and ## , respectively. We write this expansion as #(# # ###### ### ) # ## (# # ########## ### )## # (# # ###### ### ) This notion is generalized as follows: Definition 5. [21] Given a finite domain #, the generalized Shannon expansion of a function # : # # # #, with respect to the first variable in the ordering, is: #(# # ## # ###### ### ) # # # (# # ###### ### )###### ##### (# # ###### ### ) where # # = # [# # ## # ] the function obtained by substituting the ....

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A. Srinivasan, T. Kam, S. Malik, and R.E. Brayton. "Algorithms for Discrete Function Manipulation ".InIEEE Int. Conf. on Computer-Aided Design, pages 92--95, 1990.

....SPFD can be computed for each node and can be used to remove wires in its fanin. A network of PLAs can be modeled as a multi level network of multi valued nodes. The binary SPFD techniques for computing and distributing SPFDs using BDDs [3] can be generalized to MV SPFD techniques using MDDs [15]. The details of the computation are discussed below. Consider a node h j in a multi level, multi valued logic network. We know that the MV SPFD of h j represents the set of multi valued minterms (henceforth equivalently referred to as minterms) that should be distinguished by h j in order to ....

A. Srinivasan, T. Kam, S. Malik, and R. K. Brayton. Algorithms for Discrete Function Manipulation. In Proc. of the Intl. Conf. on Computer-Aided Design, pages 92--95, Nov. 1990.

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A. Srinivasan, T. Kam, S. Malik and R. Brayton, "Algorithms for Discrete Function Manipulation, " Proc. Int. Conf. CAD, pp. 92-95, 1990.

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A. Srinivasan, T. Kam, S. Malik and R. Brayton, "Algorithms for Discrete Function Manipulation," Proc. Int. Conf. CAD, pp. 92-95, 1990.

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A. Srinivasan, T. Kam, S. Malik, and R.E. Brayton. Algorithms for discrete function manipulation. In Int'l Conf. on CAD, pages 92--95, 1990.

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A. Srinivasan, T. Kam, S. Malik, and R.E. Brayton. Algorithms for discrete function manipulation. In Int'l Conf. on CAD, pages 92--95, 1990.

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A. Srinivasan, T. Kam, S. Malik, and R. K. Brayton. Algorithms for discrete function manipulation. In ACM/IEEE 27th Design Automation Conference, 1990.

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SRINIVASAN, A., KAM, T., MALIK, S., and BRAYTON, R., Algorithms for Discrete Function Manipulation, in: Proceedings of the Int'l Conf. on CAD, 1990, pp. 92--95.

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A. Srinivasan, T. Kam, S. Malik, and R. Brayton, "Algorithms for discrete function manipulation," in Proceedings of the Int'l Conf. on CAD, 1990, pp. 92--95.

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A. Srinivasan, T. Kam, S. Malik, and R.E. Brayton. Algorithms for discrete function manipulation. In Int'l Conf. on CAD, pages 92--95, 1990.

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A. Srinivasan, T. Kam, S. Malik, and R. K. Brayton. Algorithms for Discrete Functions Manipulation. In Proceedings of the IEEE International Conference on Computer-Aided Design (ICCAD'90), 1990.

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A. Srinivasan, T. Kam, S. Malik, and R.K. Brayton. Algorithms for discrete function manipulation. In Int. Conf. on CAD, pages 92--95. IEEE Computer Society, 1990.

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A. Srinivasan, T. Kam, S. Malik, and R.E. Brayton. "Algorithms for Discrete Function Manipulation". In IEEE International Conference on Computer-Aided Design, pages 92-- 95, 1990.

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A. Srinivasan, T. Kam, S. Malik, and R. E. Brayton. Algorithms for discrete function manipulation. In International Conference on CAD, pages 92-95, 1990.

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A. Srinivasan, T. Kam, S. Malik, and R. K. Brayton. Algorithms for discrete function manipulation. In International Conference on CAD, pages 92-95. IEEE Computer Society, 1990.

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