Zero-suppressed BDDs (ZBDDs) have been used in the nonenumerative path delay fault (PDF) grading of VLSI circuits. One basic and one cut-based grading algorithm are proposed to grade circuits with polynomial and exponential number of PDFs, respectively. In this article, we present a new ZBDD-based basic PDF grading algorithm to enable grading of some circuits with exponential number of PDFs without using the cut-based algorithm. The algorithm overcomes the memory overflow problems by dynamically pruning the ZBDD at run-time. This new algorithm may give exact or pessimistic coverage depending on the statuses of the pruned nodes. Furthermore, we re-assess the performance of the static variable ordering heuristics in ZBDDs for PDF coverage calculation. The proposed algorithm combined with the efficient static variable ordering heuristics can avoid ZBDD size explosion in many circuits. Experimental results for ISCAS85 benchmarks show that the proposed algorithm efficiently grades circuits.

A new approach for ultra-low-power LSIs based on quantum devices is presented and its present status and critical issues are discussed with a brief background review on the semiconductor nanotechnology. It is a hexagonal binary decision diagram (BDD) quantum logic circuit approach suitable for realization of ultra-low-power logic/memory circuits to be used in new applications such as intelligent quantum (IQ) chips embedded in the ubiquitous network environment. The basic concept of the approach, circuit examples showing its feasibility, growth of high density nanostructure networks by molecular beam epitaxy (MBE) for future LSI implementation, and the key processing issues including the device isolation issue are addressed.

In this paper, we propose a compact representation of logic functions using Multi-valued Decision Diagrams (MDDs) called heterogeneous MDDs. In a heterogeneous MDD, each variable may take a different domain. By partitioning binary input variables and representing each partition as a single multi-valued variable, we can produce a heterogeneous MDD with 16% smaller memory size than a Reduced Ordered Binary Decision Diagram (ROBDD), and with comparable memory size to Free Binary Decision Diagrams (FBDDs). And also, heterogeneous MDDs have shorter Average Path Length (APL) than ROBDDs and FBDDs. We minimized a large number of benchmark functions to show the compactness of heterogeneous MDDs.

This paper presents a group theoretic approach to the design of Decision diagrams (DDs) with increased functionality of nodes. Basic characteristics of DDs determine their applications, and thus, the optimization of DDs with respect to different characteristics is an important task. Increased functionality of nodes provides for optimization of DDs. In this paper, the methods for optimization of binary DDs by pairing of variables are interpreted as the optimization of DDs by changing the domain group for the represented functions. Then, it is pointed out that, for Abelian groups, the increased functionality of nodes by using larger subgroups may improve some of the characteristics of DDs at the price of other characteristics. With this motivation, we proposed the use of non-Abelian groups for the domain of represented functions by taking advantages from basic features of their group representations. At the same time, the present methods for optimization of DDs, do not offer any criterion or efficient algorithm to choose among a variety of possible different DDs for an assumed domain group. Therefore, we propose Fourier DDs on non-Abelian groups to exploit the reduced cardinality of the Fourier spectrum on these groups.

OPKFDDs (Ordered Pseudo-Kronecker Functional Decision Diagrams) are a data structure that provides compact representation of Boolean functions. The size of OPKFDDs depends on a variable ordering and on decomposition type choices. Finding an optimal representation is very hard and the size of the search space is n! 32n-1, where n is the number of input variables. To overcome the huge search space of the problem, a genetic algorithm is proposed for the generation of OPKFDDs with minimal number of nodes.

Functional decomposition is an essential technique of logic synthesis and is important especially for FPGA design. Bertacco and Damiani proposed an efficient algorithm finding simple disjoint decomposition using Binary Decision Diagrams (BDDs). However, their algorithm is not complete and does not find all the decompositions. This paper presents a complete theory of simple disjoint decomposition and describes an efficient algorithm using BDDs.

A shared binary decision diagram (SBDD) represents a multiple-output function, where nodes are shared among BDDs representing the various outputs. A partitioned SBDD consists of two or more SBDDs that share nodes. The separate SBDDs are optimized independently, often resulting in a reduction in the number of nodes over a single SBDD. We show a method for partitioning a single SBDD into two parts that reduces the node count. Among the benchmark functions tested, a node reduction of up to 23% is realized.

We have proposed a top-down design methodology for the RSFQ logic circuits based on the Binary Decision Diagram (BDD). In order to show the effectiveness of the methodology, we have designed a small RSFQ microprocessor based on simple architecture. We have compared the performance of the 8-bit RSFQ microprocessor with its CMOS version. It was found that the RSFQ system is superior in terms of the operating speed though it requires extremely large area. We have also implemented and tested a 1-bit ALU that is one of the important components of the microprocessor and confirmed its correct operation.

In this paper, we consider the complexity of recognizing ordered tree-shellable Boolean functions when Boolean functions are given as OBDDs. An ordered tree-shellable function is a positive Boolean function such that the number of prime implicants equals the number of paths from the root node to a 1-node in its ordered binary decision tree representation. We show that given an OBDD, it is possible to check within polynomial time if the function is ordered tree-shellable with respect to the variable ordering of the OBDD.

In this paper, we propose a method to minimize multiple-valued decision diagrams (MDDs) for multiple-output functions. We consider the following: (1) a heuristic for encoding the 2-valued inputs; and (2) a heuristic for ordering the multiple-valued input variables based on sampling, where each sample is a group of outputs. We first generate a 4-valued input 2-valued multiple-output function from the given 2-valued input 2-valued functions. Then, we construct an MDD for each sample and find a good variable ordering. Finally, we generate a variable ordering from the orderings of MDDs representing the samples, and minimize the entire MDDs. Experimental results show that the proposed method is much faster, and for many benchmark functions, it produces MDDs with fewer nodes than sifting. Especially, the proposed method generates much smaller MDDs in a short time for benchmark functions when several 2-valued input variables are grouped to form multiple-valued variables.

The invariant polynomials of discrete systems such as graphs, matroids, hyperplane arrangements, and simplicial complexes, have been theoretically investigated actively in recent years. These invariants include the Tutte polynomial of a graph and a matroid, the chromatic polynomial of a graph, the network reliability of a network, the Jones polynomial of a link, the percolation function of a grid, etc. The computational complexity issues of computing these invariants have been studied and most of them are shown to be #P-complete. But, these complexity results do not imply that we cannot compute the invariants of a given instance of moderate size in practice. To meet large demand of computing these invariants in practice, there have been proposed a framework of computing the invariants by using the binary decision diagrams (BDD for short). This provides mildly exponential algorithms which are useful to solve moderate-size practical problems. This paper surveys the BDD-based approach to computing the invariants, together with some computational results showing the usefulness of the framework.

This paper proposes a method to construct smaller binary decision diagrams for characteristic functions (BDDs for CFs). A BDD for CF represents an n-input m-output function, and evaluates all the outputs in O(n+m) time. We derive an upper bound on the number of nodes of the BDD for CF of n-bit adders (adrn). We also compare complexities of BDDs for CFs with those of shared binary decision diagrams (SBDDs) and multi-terminal binary decision diagrams (MTBDDs). Our experimental results show: 1) BDDs for CFs are usually much smaller than MTBDDs; 2) for adrn and for some benchmark circuits, BDDs for CFs are the smallest among the three types of BDDs; and 3) the proposed method often produces smaller BDDs for CFs than an existing method.

There are two approaches for formal verification of sequential designs or finite state machines: language containment checking and symbolic model checking. To verify designs of practical size, in these two approaches, designs are represented symbolically, in practice, by ordered binary decision diagrams. In the conventional algorithm for language containment checking, finite automata given as specifications are also represented symbolically. This paper proposes a new method, called partially explicit method for checking language containment. By representing states of finite automata given as specifications explicitly, this method can remove redundant computations, and as a result, provide better performance than the conventional method which uses the product machines of designs and specifications. The experimental results show that this approach is effective in checking language containment symbolically.

In several design methods for Pass-transistor Logic (PTL) circuits, Boolean functions are expressed as OBDDs in decomposed form and then the component OBDDs are directly mapped to PTL cells. The total size of OBDDs (number of nodes) corresponds to the circuit size. In this paper, we investigate a method for PTL synthesis based on exact minimization of Free BDDs (FBDDs). FBDDs are well-studied extension of OBDDs with free variable ordering on each path. We present statistics showing that more than 56% of 616126 NPN-equivalence classes of 5-variable Boolean functions have minimum FBDDs with less size than their OBDDs. This result can be used for PTL synthesis as libraries. We also applied the exact minimization algorithm of FBDDs to the minimization of subcircuits in the synthesis for MCNC benchmarks and found up to 5% size reduction.

This paper reports computational results of a new approach of analyzing network reliability against probabilistic link failures. This problem is hard to solve exactly when it is large-scale, which is shown from complexity theory, but the approach enables us to analyze networks of moderate size, as demonstrated by our experimental results. Furthermore, this approach yields a polynomial-time algorithm for complete graphs, whose reliability provides a natural upper bound for simple networks, and also leads to an efficient algorithm for computing the dominant part of the reliability function when the failure probability is sufficiently small. Computational results for these cases are also reported. This approach thus establishes a fundamental technology of analyzing network reliability in practice.

This paper describes a method to represent m output functions using shared multi-terminal binary decision diagrams (SMTBDDs). The SMTBDD(k) consists of multi-terminal binary decision diagrams (MTBDDs), where each MTBDD represents k output functions. An SMTBDD(k) is the generalization of shared binary decision diagrams (SBDDs) and MTBDDs: for k=1, it is an SBDD, and for k=m, it is an MTBDD. The size of a BDD is the total number of nodes. The features of SMTBDD(k)s are: 1) they are often smaller than SBDDs or MTBDDs; and 2) they evaluate k outputs simultaneously. We also propose an algorithm for grouping output functions to reduce the size of SMTBDD(k)s. Experimental results show the compactness of SMTBDD(k)s. An SMTBDDmin denotes the smaller SMTBDD which is either an SMTBDD(2) or an SMTBDD(3) with fewer nodes. The average relative sizes for SBDDs, MTBDDs, and SMTBDDs are 1. 00, 152. 73, and 0. 80, respectively.

A formal verification approach that combines verification based on binary decision diagrams (BDDs) and theorem-prover-based verification has been developed. This approach is called the incremental formal verification approach. It uses an incremental verifier based on BDDs and a conventional theorem-prover-based verifier. Inputs to the incremental verifier are specifications in higher-level descriptions given in terms of arithmetic expressions, lower-level design descriptions given in terms of Boolean expressions, and constraints. The incremental verifier limits the behavior of the design by using the constraints, and compares the partial behavior limited by the constraints with the specifications by using BDD-based Boolean matching. It also replaces the matched part of the lower design description with equivalent constructs in the higher descriptions. Successive uses of the incremental verifier with different constraints can produce higher design descriptions from the lower design descriptions in a step-by-step manner. These higher descriptions are then input to the theorem-prover-based verification which enables faster treatment of larger circuits. Preliminary experimental results show that the incremental verifier can successfully check the partial equivalence and replace the matched parts by higher constructs.

This paper proposes a method of constructing single-electron logic subsystems on the basis of the binary decision diagram (BDD). Sample subsystems, an adder and a comparator, are designed by combining single-electron BDD devices. It is demonstrated by computer simulation that the designed subsystems successfully produce, through pipelined processing, an output data flow in response to the input data flow. The operation error caused by thermal agitation is estimated. An output interface for converting single-electron transport into binary-voltage signals is also designed.

Minato has proposed canonical representation for polynomial functions using zero-suppressed binary decision diagrams (ZBDDs). In this paper, we extend binary moment diagrams (BMDs) proposed by Bryant and Chen to handle variables with degrees higher than l. The experimental results show that this approach is much more efficient than the previous ZBDDs' approach. The proposed approach is expected to be useful for various problems, in particular, for computer algebra.

This paper presents an FPGA architecture for high-speed systems, such as next-generation B-ISDN telecommunications systems. Such a system requires an LSI in which an over-10K-gate circuit can be implemented and that has a clock cycle rate of 80MHz. So far, the FPGA architecture has only been discussed in terms of its circuit structure. In contrast we consider the circuit structure of the FPGA along with the performance of its dedicated CAD system. We evaluate several FPGA logic-element structures with a technology mapping method. From these experiments, a multiplexor-based logic-element is found to be suitable for implementing such a high-speed circuit using the BDD-based technology mapping method. In addition, we examine how to best utilize the characteristics of the selected logic-cell structure in designing the wiring structure. It is found that the multiplexor-based cell can be connected efficiently in a clustered wiring structure.