Translation among CNFs, characteristic models and ordered binary decision diagrams

The Nonapproximability of OBDD Minimization

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:

Optimal ordered binary decision diagrams for read-once formulas

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.

The size of ordered binary decision diagrams (OBDDs) strongly depends on the chosen variable ordering. It is an obvious heuristic to use symmetric variable orderings, i.e., variable orderings where symmetric variables are arranged adjacently. In order to evaluate this heuristic, methods for estimating the OBDD size for random partially symmetric functions are presented. Characterizations of cases where, with high probability, only symmetric variable orderings and, with high probability, only nonsymmetric variable orderings lead to minimum OBDD size are obtained. For this analysis estimates for the number of different blocks of random Boolean matrices are used. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 49–70, 1998

The relationship between two important means of representing Boolean functions, combinational circuits and ordered binary decision diagrams (OBDDs), is studied. Circuit width is related to OBDD size, and it is shown how register allocation can be used to determine a good variable order of OBDD construction. It is shown that if C has n inputs, m outputs, and width w(C), then there is a variable ordering for which the directed-acyclic-graph-based representation for C has at most n*m*2/sup w(C)/ nodes. Since the width of a circuit is closely related to the number of registers required to evaluate the circuit, the result shows that register allocation techniques can be used to compute variable orderings with predicted behavior. The use of these ideas in decomposing a function either for representation as a set of OBDDs or for implementation in a cascode technology is outlined. A characterization is given of a class of multioutput functions which includes addition whose members have particularly small OBDDs. >

The complexity of minimizing and learning OBDDs and FBDDs

The ordered binary decision diagram (OBDD) has proven useful in many applications as an efficient data structure for representing and manipulating Boolean functions. A serious drawback of OBDD's is the need for application-specific heuristic algorithms to order the variables before processing. Further, for many problem instances in logic synthesis, the heuristic ordering algorithms which have been proposed are insufficient to allow OBDD operations to complete within a limited amount of memory. The paper proposes a solution to these problems based on having the OBDD package itself determine and maintain the variable order. This is done by periodically applying a minimization algorithm to reorder the variables of the OBDD to reduce its size. A new OBDD minimization algorithm, called the sifting algorithm, is proposed and appears especially effective in reducing the size of the OBDD. Experiments with dynamic variable ordering on the problem of forming the OBDD's for the primary outputs of a combinational circuit show that many computations complete using dynamic variable ordering when the same computation fails otherwise.

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.

Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far” (for a prespecified distance measure) from every object with that property. In this work we design and analyze an algorithm for testing functions for the property of being computable by a read-once width-2 Ordered Binary Decision Diagram (OBDD), also known as a branching program, where the order of the variables is not known to us. That is, we must accept a function f if there exists an order of the variables according to which a width-2 OBDD can compute f. The query complexity of our algorithm is $\tilde{O}({\rm log n}){\rm poly}(1/\epsilon)$. In previous work (in Proceedings of RANDOM, 2009) we designed an algorithm for testing computability by an OBDD with a fixed order, which is known to the algorithm. Thus, we extend our knowledge concerning testing of functions that are characterized by their computability using simple computation devices and in the process gain some insight concerning these devices.

The ordering of variables can have a significant effect on the size of the reduced binary decision diagram (BDD) that represents the set of solutions to a combinatorial optimization problem. It also influences the quality of the objective function bound provided by a limited-width relaxation of the BDD. We investigate these effects for the maximum independent set problem. By identifying variable orderings for the BDD, we show that the width of an exact BDD can be given a theoretical upper bound for certain classes of graphs. In addition, we draw an interesting connection between the Fibonacci numbers and the width of exact BDDs for general graphs. We propose variable ordering heuristics inspired by these results, as well as a k-layer look-ahead heuristic applicable to any problem domain. We find experimentally that orderings that result in smaller exact BDDs have a strong tendency to produce tighter bounds in relaxation BDDs.

Efficient Fault Tree Analysis of Complex Fault Tolerant Multiple-Phased Systems

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.

While most of the research efforts on network reliability analysis have focused on networks with imperfect links but perfectly reliable nodes, binary decision diagram–based methods have recently been developed to determine reliability of networks with imperfect links and imperfect nodes. The smaller the binary decision diagram size is, the better the performance of the binary decision diagram–based algorithm will be. The binary decision diagram size heavily depends on the chosen variable ordering. As finding the optimal variable ordering is a nondeterministic polynomial time-complete problem, a heuristic is typically used. Various heuristics based on depth-first search, breadth-first search, and network-driven search have been developed for network reliability analysis. However, they often assume the networks have perfectly reliable nodes. This article advances the state-of-the-art by proposing a new ordering heuristic for the binary decision diagram–based reliability analysis of networks with imperfect links/edges and nodes. Empirical studies show that the proposed new ordering heuristic can generate significantly smaller binary decision diagram size for a wide variety of network structures than the traditional depth-first search–based, breadth-first search–based, network-driven search–based heuristics for most cases, enabling efficient binary decision diagram–based reliability analysis of large-scale networks considering both link and node failures. Robustness of the proposed heuristic is also investigated through empirical studies.

Reduced ordered binary decision diagrams (BDDs) are a data structure for representation and manipulation of Boolean functions which are frequently used in VLSI design automation. The variable ordering largely influences the size of the BDD, varying from linear to exponential. In this paper we study BDD minimization problem based on scatter search optimization. The results we obtained are very encouraging in comparison with other heuristics (genetic and simulated annealing). This work is the first successful experience of using scatter search approach in design automation area. The approach can be applied to many design automation applications.