In order to apply formal design verification, it is necessary to describe formally and correctly the specification of the circuit under verification. Especially when we apply conventional OBDD-based logic comparison method for verifying combinational circuits, another correct" logic circuits or Boolean formulae must be given as the specification. It is desired to develop an efficient automatic design verification method which interprets specification that can be described easier. This paper provides a new verification method which is useful for combinational circuits such as arithmetic circuits. The proposed method efficiently verifies whether a designed circuit satisfies a specification given by recurrence equations. This enables us to describe easily an error-free specification for arithmetic circuits. To perform verification efficiently using an ordinary OBDD package, an efficient truth-value rotation algorithm is developed. The truthvalue rotation algorithm efficiently generates an OBDD representing f(x + 1 (mod 2n)) from a given OBDD representing f(x). By experiments on SPARC station 10 model 51, it takes 180 secs to generate an OBDD for designed circuit of 23-bit square function, and additional 60 secs is sufficient to finish verifying that it satisfies the specification given by recurrence equations.

Exploring enormous state graphs represented implicitly by ordered binary decision diagrams (OBDDs) is one of the most successful applications of OBDDs. However, our worst-case analysis of implicit graph representations by OBDDs shows that there are cases where OBDD representations are not optimal and require more space than adjacency lists. As an improvement, we propose a new type of BDDs, called Patricia BDDs, which are capable of implicit representation of graphs while their worst-case sizes are kept equal or less than adjacency lists and OBDDs.

An ordered binary decision diagram (OBDD) is a directed acyclic graph for representing a Boolean function. OBDDs are widely used in various areas which require Boolean function manipulation, since they can represent efficiently many practical Boolean functions and have other desirable properties. However, there is very little theoretical research on the complexity of constructing an OBDD. In this paper, we prove that the optimal variable ordering problem of a shared BDD is NP-complete, and briefly discuss the approximation hardness of this problem and related OBDD problems.

This paper presents implicit representation of binary decision diagrams (implicit BDDs) as a new effecient data structure for Boolean functions. A well-known method of representing graphs by binary decision diagrams (BDDs) is applied to BDDs themselves. Namely, it is a BDD representation of BDDs. Regularity in the structure of BDDs representing certain Boolean functions contributes to significant reduction in size of the resulting implicit BDD repersentation. Since the implicit BDDs also provide canonical forms for Boolean functions, the equivalence of the two implicit BDD forms is decided in time proportional to the representation size. We also show an algorithm to maniqulate Boolean functions on this implicit data structure.

This paper presents Ternary Decision Diagrams which represent sets of products. This paper also presents manipulating methods for sum-of-products forms and ringsum-of-products forms using Ternary Decision Diagrams, and gives comparison results between Ternary Decision Diagrams and Binary Decision Diagrams.

Though tree covering is an efficient algorithm for technology mapping, phase assignments on tree boundaries are not taken into consideration. Several inverter minimization algorithms have been proposed so far, but they do phase optimization before or after technology mapping, and their cost function is not to minimize the total area but to minimize the number of inverters. This paper describes a new formulation of phase optimization problem aiming to minimize the total area during the technology mapping. Cost function representing area according to each phase assignment is introduced, and tree covering algorithm is modified to handle that cost function. Edge-Valued Binary Decision Diagram is used to represent the function implicitly. Experimental results show that proposed method reduces about 10% area on average compared with a state-of-the-art logic synthesis system sis.

Ordered binary decision diagrams (OBDDs) have been widely used in many CAD applications as efficient data structures for representing and manipulating Boolean functions. For the efficient use of the OBDD, it is essential to find a good variable order, because the size of the OBDD heavily depends on its variable order. Dynamic variable reordering is a promising solution to the variable ordering problem of the OBDD. Dynamic variable reordering with the sifting algorithm is especially effective in minimizing the size of the OBDD and reduces the need to find a good initial variable order. However, it is very time-consuming for practical use. In this paper, we propose two new implementation techniques for fast dynamic variable reordering. One of the proposed techniques reduces the number of variable swaps by using the lower bound of the OBDD size, and the other accelerates the variable swap itself by recording the node states before the swap and the pivot nodes of the swap. By using these new techniques, we have achieved the speed-up ranging from 2.5 to 9.8 for benchmark circuits. These techniques have reduced the disadvantage of dynamic variable reordering and have made it more attractive for users.

In this paper, we propose a method to design ternary cellular arrays by using Ternary Decision Diagrams (TDD's). Our cellular array has a rectangular structure composed of ternary switch cells. The ternary functions represented by TDD's are realized by mapping the TDD's to the arrays directly. That is, both the nodes and the edges in the TDD are realized by some sets of the cells. Since TDD's can represent easily multiple-output functions without large memory requirements, our arrays are wuitable for the realization of multiple-output functions. To evaluate our method, we apply our method to some benchmark circuits, and compare our arrays with the ternary PLA's. The experimental results show that our arrays have the advantage for their sizes, especially in the realization of symmetric functions. The results also clarify that the size of our arrays depends on the size of TDD's.

This paper describes an exact symbolic formulation of resource-constrained scheduling which allows speculative operation execution in arbitrary forward-branching control/data paths. The technique provides a closed-form solution set in which all satisfying schedules are encapsulated in a compressed OBDD-based representation. An iterative construction method is presented along with benchmark results. The experiments demonstrate the ability of the proposed technique to efficiently extract parallelism not explicitly specified in the input description.

The paper describes a new algorithm for Boolean matching, which is based on BDD structure manipulation. Pruning of the search space takes place after partial assignments if certain subgraphs of two BDD's become inequivalent. This pruning is different from existing techniques, so that the search space is further reduced. Another feature of this algorithm is topological filtering. Usually, many functions have no matchings and this is easily found by only counting the number of minterms. To check it quickly, upper and lower bounds of minterm count are calculated from topological information. Using these bounds, functions that have no matchings are discarded without building their BDD's.

A new approach is described for the datapath scheduling of behavioral descriptions containing nested conditional branches of arbitrary structures. This paper first investigates such a complex scheduling mechanism, and formulates an optimal scheduling problem as a 0-1 integer programming problem such that given a prescribed number of control steps, the total cost of functional units can be minimized. In this formulation, each constraint is expressed in the form of a Boolean function, which is set equal to 1 or 0 according as the constraint is satisfied or not, respectively, and a satisfiability problem is defined by the product of the Boolean functions. A procedure is then described, which intends to seek an optimal solution by means of a branch-and-bound method on a binary decision diagram representing the satisfiability problem. Experimental results are also shown, which demonstrate that our approach is of more practical use than the existing methods.

Binary decision diagrams (BDD's) are graph representations of Boolean functions, and at the same time they can be regarded as a computational model. In this paper, we discuss relations between BDD's and other computational models and clarify the computational power of BDD's. BDD's have the property that each variable is examined only once according to a total order of the variables. We characterize families of BDD's by on-line deterministic Turing machines and families of permutations. To clarify the computational power of BDD's, we discuss the difference of the computational power with respect to the way of reading inputs. We also show that the language TADGAP (Topologically Arranged Deterministic Graph Accessibility Problem) is simultaneously complete for both of the class U-PolyBDD of languages accepted by uniform families of polynomial-size BDD's and the clas DL of languages accepted by log-space bounded deterministic Turing machines. From the results, we can see that the problem whether U-PolyBDD U-NC1 is equivalent to a famous open problem whether DL U-NC1, where U-NC1 is the class of languages accepted by uniform families of log-depth constant fan-in logic circuits.

An Ordered Binary Decision Diagram (BDD) is a graph representation of a Boolean function. According to its good properties, BDD's are widely used in various applications. In this paper, we investigate the computational complexity of basic operations on BDD's. We consider two important operations: reduction of a BDD and binary Boolean operations based on BDD's. This paper shows that both the reduction of a BDD and the binary Boolean operations based on BDD's are NC1-reducible to REACHABILITY. That is, both of the problems belong to NC2. In order to extend the results to the BDD's with output inverters, we also considered the transformations between BDD's and BDD's with output inverters. We show that both of the transformations are also NC1-reducible to REACHBILITY.

This paper presents techniques for generating the input patterns for locating logic design errors (PLE's) by Boolean function manipulation based on binary decision diagrams (BDD's). One PLE has one Boolean variable X or and constant values. A primary output of a correct circuit takes value X, while the designed circuit takes either 0 or 1. By using PLE's, the X-algorithms locate single or multiple logic design errors in a combinational circuit. Although PLE's play the most important role in the X-algorithms, the condition under which PLE's exist has not been formalized. This paper gives a formal analysis on the existence condition of PLE's. It is shown that the condition is always satisfied by incorporating another type of PLE. From the condition, an implicit representation of PLE's is derived. In addition, two kinds of approaches are presented for generating PLE's by Boolean function manipulation based on BDD's. One is an approach for generating all the existing PLE's. The other is a heuristic approach to obtain a limited number of PLE's in a short time. Both approaches generate PLE's including don't cares. Incorporating them, a compact representation of PLE is achieved. Experimental results have shown the compactness of the proposed representations and the availability of the pattern generation techniques.

In this paper, an exact algorithm for finding minimum test set which detects all testable stuck-at faults of a given combinational circit is presented. So far several heuristic algorithms for this problem are proposed, but no efficient exact algorithms are known. To solve this exactly, minimum test set problem is formalized as a minimum set covering problem, and then implicit manipulation technique using binary decision diagrams(BDDs) is applied. The algorithm presented in the paper has two contributions. One is utilization of maximal compatible fault set, which can drastically reduce the number of candidates for minimum test set. A new BDD based algorithm for extracting all maximal compatible fault sets is shown. The other is a new implicit manipulation technique handling with huge covering matrix. Actually, the algorithm using this technique can handle minimum set covrering problem with over ten thousand columns in a few minutes. Experiments using ISCAS benchmark circuits show that the algorithm is quite efficient for small(100-300 gates) circuits. A computational complexith of minimum test set problen is much higher than that of ordinary test pattern generation problem, so that practical signifcance of this method is not high. But the algorithm is still useful for evaluation of other heuristic algorithms. furthermore, this implicit manipulation technique can also be applied to other minimumset covering problems.

Recently, there has been a lot of research on solving combinatorial problems using Binary Decision Diagrams (BDDs), which are very efficient representations of Boolean functions. We have already developed a Boolean Expression Manipulator, which calculates and reduces Boolean expressions quickly based on BDD techniques. This greatly aids our works on developing VLSI CAD systems and solving combinatorial problems. Any combinatorial problem can be described in Boolean expressions; however, arithmetic operations, such as addition, subraction, multiplication, equality and inequality, are also used for describing many practical problems. Arithmetic operations provide simple descriptions of problems in many cases. In this paper, we present an arithmetic Boolean expression manipulator (BEM-), based on BDD techniques. BEM- calculates Boolean expressions containing arithmetic operations and then displays the results in various formats. It can solve problems represented by a set of equalities and inequalities, which are dealt with using 0-1 linear programming. We show the efficient data structure based on BDD representation, algorihms for manipulating Boolean expressions with arithmetic operations, and good formats for displaying the results. Finally we present the specification of BEM- and an example of application to the 8-Queens problem. BEM- is customizable to various applicationa. It has good computation performance in terms of the total time for programming and execution. We expect BEM- to be a helpful tool in research and development on digital systems.

Since the time required for testing logic circuits is proportional to the number of test vectors, the size of test sets as well as test generation time is one of the most important factors to be considered in test generation. The size of test sets becomes an essential issue, especially for scan designed circuits, because of the need to shift a test vector serially into the scan path. In this paper, we propose new methods of generating compact test sets to detect al the irredundant single stuck-at faults in combinational circuits. The proposed algorithms calculate a test function for each fault which corresponds to the set of all test vectors for the fault and generate a compact test set by analyzing the test functions. The analysis is based on finding a test vector which detects the largest number of remaining faults. Since our methods select a test vector among all the test vectors, represented by a test function, for a target fault, smaller test sets can be generated, in general, than that by conventional test set compaction methods. The experimental results show that the size of test sets generated by our method is about one-third as large as that without compaction.

In this paper, a new method of synthesizing multi-level logic circuits directly from binary decision diagrams (BDDs) is proposed. In the simple multiplexer implementation, the depth of the synthesized circuit was always O (n), where n is the number of input variables. The new synthesis method attempts to reduce the depth of circuits. The depth of the synthesized circuits is O (log n log w) where w is the maximum width of given BDDs. The synthesized circuits are 2-rail-input 2-rail-output logic circuits. The circuits have good testability; it is proved that the circuits are robustly path-delay fault testable and also totally self-checking for single stuck-at faults.

This paper tends to analyze the trends of the research in logic synthesis. The first part is devoted to an expertise of the efficiency of factorization methods developed during the last decade and to the proposal of dedicated methods for complex logic blocks. The second part shows the importance of Binary Decision Diagrams as representation of Boolean functions. Their use in the technology mapping phase of multiplexor-based FPGAs in an industrial tool is taken as illustration.

Manipulation of Boolean functions is one of the most important techniques for implementing of VLSI logic design systems. This paper presents a fast method for generating prime-irredundant covers from Binary Decision Diagrams (BDDs), which are efficient representation of Boolean functions. Prime-irredundant covers are forms in which each cube is a prime implicant and no cube can be eliminated. This new method generates compact cube sets from BDDs directly, in contrast to the conventional cube set reduction algorithms, which commonly manipulate redundant cube sets or truth tables. Our method is based on the idea of a recursive operator, proposed by Morreale. Morreale's algorithm is also based on cube set manipulation. We found that the algorithm can be improved and rearranged to fit BDD operations efficiently. The experimental results demonstrate that our method is efficient in terms of time and space. In practical time, we can generate cube sets consisting of more than 1,000,000 literals from multi-level logic circuits which have never previously been flattened into two-level logics. Our method is more than 10 times faster than ESPRESSO in large-scale examples. It gives quasi-minimum numbers of cubes and literals. This method should find many useful applications in logic design systems.