A method to join the On-set and Off-set of an incompletely boolean function into a single BDD

Specific physical defects of high performance circuits manifest themselves as path delay faults (PDFs). A pair (v1, v2) of test patterns is required to detect a PDF. A PDF is robust testable if there is a test pair on which the fault manifestation does not depend on delays of other circuit paths. A PDF is non robust testable if manifestation of the fault on any test pair is possible only when all other paths of the circuit are fault free. For high quality delay testing it is desirable to detect delay of any path regardless of delays of other paths. In this paper, synthesis of a circuit by covering each nonterminal node of the reduced ordered binary decision diagram (ROBDD) or free binary decision diagram (Free BDD) system with the original subcircuit from gates is suggested. Delay testability of such circuits is investigated. It is specified that the PDF of each path of the circuit is detected without injecting additional inputs into the circuit. It is well known that the BDD is a directed acyclic graph based on the Shannon decomposition in each nonterminal node v: 0 1 i i x x v i v i v f x f x f = = = ∨ . Here fv is the function corresponding to the node v, and the decomposition variable xi marks the node v. The dashed edge drops in the node corresponding to the function 0 i x v f = , and the bold edge drops in the node corresponding to the function 1 i x v f = . The BDD is called ordered (OBDD) if variables are encountered in the same order in all paths connecting the BDD root with the terminal node. The BDD is called reduced (RBDD) if it does not contain either isomorphic subgraphs or nonterminal nodes such that 0 1 i i x x v v f f = = = . An example of reduced and ordered BDD (ROBDD) is shown in Fig. 1. When deriving the Free BDD, a suitable decomposition variable is chosen for each nonterminal node v independently of other nonterminal nodes. That may cut the number of Free BDD of nonterminal nodes in comparison with the ROBDD (for the same function). Let 1 { ,..., } m F f f = be a system of Boolean functions describing combinational circuit behavior. Derive the ROBDD using the same order of variables for each Boolean function from F. Join isomorphic subgraphs in different ROBDDs. Combine different ROBDDs of 1-terminal nodes into one 1-terminal node and their 0-terminal nodes into one 0-terminal node. As a result, we obtain the graph with m roots and two terminal nodes. This graph represents a system of m Boolean functions. It is called the shared ROBDD. Without loss of generality, we will further consider systems with one function as an example. In Fig. 1, the ROBDD for one Boolean function is shown. The ROBDD represents the disjoint sum of products (DSoP) of the function f as follows:

Bent Boolean functions exist only for an even number of variables, moreover, they are unbalanced. Therefore, they are used in coding theory and in many areas of computer science. General form of bent functions is still unknown. One way of representing Boolean functions is with a reduced ordered binary decision diagram (ROBDD). The strength of ROBDDs is that they can represent Boolean functions data with a high level of redundancy in a compact form, as long as the data is encoded in such a way that the redundancy is exposed. This paper investigates characteristics of bent functions with focus on their ROBDD parameters. Decision diagram experimental framework has been used for implementation of a program for calculation of the ROBDD parameters. The results presented in this paper are intended to be used to create methods for the construction of bent functions using a ROBDD as a data structure from which the bent functions can be discovered.

Reduced Ordered Binary Decision diagrams (ROBDDs) are nowadays one of the most common dynamic data structures for Boolean functions. Among the many areas of application are verification, model checking, and computer aided design. In the last few years, SAT checkers, based on the CNF representation of Boolean functions are getting more and more attention as an alternative to the ROBDD based methods. We show the difference between the CNF representation and the ROBDD representation in one of the most degenerate cases – random monotone 2CNF formulas. We examine this model and give almost matching lower and upper bounds for the ROBDD size in different cases, and show that as soon as the formulas are non-trivial the ROBDD size becomes exponential, thus showing perhaps one of the most fundamental advantages of SAT solvers over ROBDDs.

An ordered binary decision diagram (OBDD) is a directed acyclic graph that represents a Boolean function. Since OBDDs have many nice properties as data structures, they have been extensively studied for decades in both theoretical and practical fields, such as VLSI (Very Large Scale Integration) design, formal verification, machine learning, and combinatorial problems. Arguably, the most crucial problem in using OBDDs is that they may vary exponentially in size depending on their variable ordering (i.e., the order in which the variables are to be read) when they represent the same function. Indeed, it is NP hard to find an optimal variable ordering that minimizes an OBDD for a given function. Friedman and Supowit provided a clever deterministic algorithm with time/space complexity O^∗(3ⁿ), where n is the number of variables of the function, which is much better than the trivial brute-force bound O^∗(n!2ⁿ). This paper shows that a further speedup is possible with quantum computers by presenting a quantum algorithm that produces a minimum OBDD together with the corresponding variable ordering in O^∗(2.77286ⁿ) time and space with an exponentially small error probability. Moreover, this algorithm can be adapted to constructing other minimum decision diagrams such as zero-suppressed BDDs.

Reduction of OBDDs in linear time

The binary decision diagram is a graph representation of Boolean functions, it also can efficiently represent the logic functionality of digital circuits. The number of nodes in binary decision diagram depends on the variable ordering. A new method is presented in this paper for the variable ordering and the minimization of binary decision diagram, the method is based on adaptive hierarchy genetic algorithm, which uses two populations with hierarchy to evolve in parallel, and the operation parameters in the evolution procedure are adjusted dynamically as the changes of environment. The individual representation scheme corresponding to binary decision diagram, and the implementation steps of adaptive hierarchy genetic algorithm, are given in detail. Besides, the application of the method for the test generation of digital circuits is given. A lot of experimental results show that the binary decision diagrams with smaller number of nodes can be obtained by the method proposed in this paper, and the test vectors of faults in digital circuits can also be produced.

Symmetry detection in completely specified Boolean functions is important for several applications in logic synthesis, technology mapping, binary decision diagram (BDD) minimization, and testing. This paper presents a new algorithm to detect four basic types of two-variable symmetries. The algorithm detects all pairs of symmetric variables in one pass over the shared BDD of the multioutput function. The worst case complexity of this method is cubic in the number of BDD nodes, but on typical logic synthesis benchmarks the complexity appears to be linear. The computation is particularly efficient when the functions have multiple symmetries or no symmetries. Experiments show that the algorithm is faster than other known methods, and in some cases achieves a speedup of several orders of magnitude.

Branching problems are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e.g., verification), exact representations are required. For learning boolean functions f on the basis of classified examples, it is sufficient to produce the representation of a function g approximating f. This motivates the investigation of the size of the smallest branching program approximating f. Although several non-approximability results are contained in the papers on randomized branching programs, these results often do not work for the uniform distribution (which is the most important one in applications). Here, the following non-approximability results are presented. (1) It is proven that a simple function from the branching program literature requires exponential size to be approximated with respect to the uniform distribution by OBDDs, which are the most important type of branching programs in applications. (2) The first truly exponential lower bound on the size of approximating syntactic read-k-times branching programs with respect to the uniform distribution and error probability 1/2-2/sup -/spl Omega/(n)/, n the input size, is shown. In order to improve upon the so far best results for error probabilities smaller than 1/3, a strong combinatorial lemma from a recent paper of Ajtai on linear-length branching programs is exploited.

On the Nonapproximability of Boolean Functions by OBDDs and Read-k-Times Branching Programs

This special issue of the VLSI DESIGN: an International Journal of Custom-Chip Design, Simulation and Testing is devoted to spectral techniques, new types of decision trees and diagrams, their applications, efficient ways of their calculation and mutual relations between spectral techniques and classical decision diagrams. Manipulations and calculations with discrete functions are fundamental tasks in many areas of Computer Science and Engineering. Many problems in digital system design, simulation and testing can be expressed as a sequence of operations on discrete functions. The performance of Computer-Aided Design systems used in solving various problems in this area strongly depends on the efficiency of representations of discrete functions. Decision Diagrams (DDs) have proved very convenient data structure for discrete function representations, permitting manipulations and calculations with large functions efficiently in terms of time and space. In many applications, as for example those involving large matrices, conventional algorithms are significantly improved by using DDs. In logic design, such applications relate to basic problems of the design, verification, simulation and testing logical networks. Besides Boolean functions, some other types of discrete representations, such as multi-valued functions, cube sets, EXOR-based spectral and probabilistic formulas have to be dealt with. Hence the numbers of different decision trees and diagrams have been introduced for these various applications and representations. The spectral or polynomial representation of Boolean function is obtained by expressing it in terms of an orthogonal basis. The most popular basis is formed by discrete Walsh functions. Note that the spectral and the polynomial representation of a Boolean function are equivalent. The discrete orthogonal functions are referred to as the basis functions when the spectral or abstract harmonic terminology is used and as monomials when the polynomial terminology is used. Such polynomial representations have extensive applications in coding theory and cryptography as well as design of nonlinear filters. Spectral techniques based on Walsh, Haar, Arithmetic and Reed–Muller transforms have been used in digital logic design for more than 30 years. These techniques have been used for Boolean function classification, disjoint decomposition, identification of symmetries, parallel and serial linear decomposition, spectral translation synthesis (extraction of linear preand post-filters), multiplexer synthesis, threshold logic synthesis, state assignment and testing, and evaluation of logic complexity. The renewed interest in applications of spectral methods in design of VLSI digital circuits is caused by their excellent design for testability properties and in their efficiency in Boolean mapping problems. Furthermore, the practical applicability of these techniques was greatly improved with the development of efficient methods that allow calculating and operating on spectra of Boolean functions directly from reduced representations of such functions in the form of arrays of cubes or decision diagrams. Following from these recent developments, techniques have been presented for efficient spectral translation to decompose a circuit into a cascade of two sub circuits: linear block composed of EXOR gates fed by the primary input of the overall circuit. Such linear decomposition drastically simplifies the synthesis task. The choice of suitable linear transformation is based on a complexity measure assigned to each Boolean function that is heuristically related to the complexity of the final circuit implementation. To avoid the experimental costs of computing the Walsh transform, Binary Decision Diagrams (BDDs) based techniques are used. Recently the idea of linear transformations is applied to decision diagrams. By combining powerful spectral linear techniques with variable reordering techniques, it is possible to synthesize large Boolean functions with standard Computer-Aided Design tools that fail otherwise. The areas addressed by this special issue span spectral transforms, their applications and efficient calculation methods through decision diagrams. New spectral decision diagrams are also emerging that support larger data structures allowing more complex discrete representations to be implemented that was previously impossible.

We consider a model of computation for discrete functions—Ordered Binary Decision Diagrams (OBDD).We investigate comparative complexity of quantum, deterministic, probabilistic and nondeterministic (quantum and classical) OBDDs for total and partial functions. The measure of complexity is a width of OBDD. It is known that for total functions bounded error quantum OBDDs can be exponentially more effective than deterministic and bounded error probabilistic OBDDs. We show that such quantum OBDDs also can be exponentially more effective than nondeterministic OBDDs (both quantum and classical). For partial functions the gap can be more significant. For partial function depending on a parameter k exact quantum OBDD has the width two. Deterministic and bounded error probabilistic OBDD for this function must have a width exponentially depending on k.

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 OBDD-based 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 graph-based function representations. We focus on several recent advances of particular importance for formal verification and other CAD applications.

Optimal ordered binary decision diagrams for read-once formulas

The complexity of circuit and systems design increases rapidly. Therefore, a main focus of research in the area of electronic-design automation are efficient algorithms and data structures. Among these, binary decision diagrams (BDDs) have been used in a wide variety of applications and were intensively studied from a theoretical point of view. But mostly, when complexity issues were considered, only the number of nodes in a BDD has been analyzed. Here, we study minimizing the number of paths in BDDs from a theoretical and a practical point of view. Connections to different areas in computer-aided design are outlined, theoretical studies are carried out, and an algorithm to minimize the number of paths is presented. Experimental results show the efficiency of the algorithm.

Since their introduction in form of a canonical representation of logical functions, the Binary Decision Diagrams (BDDs) gained a wide acceptance in numerous industrial applications. This paper summarizes the properties of BDD representation of Minimal Cut Sets (MCS) of Fault Tree (FT) models most typically encountered in nuclear energetics. Cut sets from MCS are defined as paths from the top BDD node to terminal nodes in the BDD, on which a quantitative and qualitative FT analysis (FTA) is performed. The core of the FTA on the BDDs is performed with help of two fundamental algorithms, one for conditional probability evaluation and another for the selection of cut sets. The accuracy of conditional probability evaluation represents an essential feature for an unbiased quantitative analysis, such as the top event probability or the determination of event importance measures. The cut set selection algorithm is shown in a generic version introducing logical predicates for its selection criteria. As it is known, the efficiency of depicted algorithms depends only on the number of BDD nodes used for the FT representation. In order to appraise the compactness of the BDD representation of FT models, their characteristics have herein been evaluated on several real-life models from the Nuclear Power Plant Krško. The extraordinariness of the compactness of the BDD representation reflects in its ability to implement advanced dynamic analysis (i.e. what-if) of FT models. The efficiency of such an approach is recognized by commercial vendors upgrading their FT Tools to new versions by implementing BDD based algorithms.