Manipulation Of Boolean Functions Research Articles

A new reliability evaluation method for networks with imperfect vertices using BDD

AbstractAs an efficient data structure for representation and manipulation of Boolean functions, binary decision diagrams (BDDs) have been applied to network reliability analysis. However, most of the existing BDD methods on network reliability analysis have assumed perfectly reliable vertices, which is often not true for real‐world networks where the vertices can fail because of factors such as limited resources (eg, power and memory) or harsh operating environments. Extensions have been made to the existing BDD methods (particularly, edge expansion diagram and boundary set–based methods) to address imperfect vertices. But these extended methods have various constraints leading to problems in accuracy or space efficiency. To overcome these constraints, in this paper, we propose a new BDD‐based algorithm called ordered BDD dependency test for K‐terminal network reliability analysis considering both edge and vertex failures. Based on a newly defined concept “dependency set”, the proposed algorithm can accurately compute the reliability of networks with imperfect vertices. In addition, the proposed algorithm has no restrictions on the starting vertex for the BDD model construction. Comprehensive examples and experiments are provided to show effectiveness of the proposed approach.

A simpler counterexample to a long-standing conjecture on the complexity of Bryantʼs apply algorithm

In 1986 in his seminal paper Bryant has introduced Ordered Binary Decision Diagrams (OBDDs) as a dynamic data structure for the representation and manipulation of Boolean functions which allow efficient algorithms for important operations like synthesis, the combination of two Boolean functions by a Boolean operator, and equivalence checking. Based on his empirical evaluation he has conjectured that his apply algorithm for the synthesis works in linear time with respect to the input and output size. Recently in 2012, Yoshinaka et al. have presented a counterexample which contradicts this conjecture but their example has the drawback that the chosen variable ordering for the OBDD representation of the input and output is bad. Therefore, they have raised the question whether Bryantʼs conjecture may still stand for reasonable variable orderings. Here, a negative answer is given by presenting a simple counterexample which works for such kind of variable orderings.

Reversed signal propagation BDD based low‐power pass‐transistor logic synthesis

AbstractBinary decision diagrams (BDDs) are the most frequently used data structure for the representation and manipulation of Boolean functions in the area of Very Large Scale Integration (VLSI) CAD. In this paper, we propose the reversed‐signal‐propagation (RSP) BDD‐based low‐power pass‐transistor logic (PTL) synthesis. In RSP BDD, the signal flow direction is opposite to that of the forward BDD. So the power supply is taken as a root node and the terminal nodes 0 and 1 represent the function in its normal and complement form. We propose an efficient way to construct the multi‐output function RSP BDD and its PTL realization. Simulation results of the proposed method give lower power consumption and high performance as a result of reduced switching activity than the existing techniques. The power improvement is about 70–80% compared to the existing techniques. © 2013 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Spectrum access policy reasoning for policy-based cognitive radios

Cognitive radio (CR) is one of the key enabling technologies for opportunistic spectrum sharing. In such a spectrum sharing paradigm, radios access spectrum opportunistically by identifying the under-utilized spectrum and then transmitting waveforms in that spectrum that are compliant to relevant spectrum access policies. Implementing such a flexible scheme requires changes in the current static spectrum management approach. As a result, declarative spectrum management through policy-based dynamic spectrum access has garnered significant attention recently. Policy-based spectrum access decouples spectrum access policies and Policy Processing Components from the radio platform. The Policies define conditions under which the radios are allowed to transmit in terms of frequencies used, geographic locations, time etc. The Policy Processing Components include a reasoning engine called the Policy Reasoner, which is responsible for enforcement, analysis and processing of the policies, as well as resolving policy conflicts. This paper describes the design and implementation of a novel policy reasoner that processes spectrum access policies efficiently by reframing the policy reasoning problem as a graph-based Boolean function manipulation problem. The proposed policy reasoner has the capability to respond to either under-specified or invalid transmission requests (sent by the system strategy reasoner) by returning multiple sets of opportunity constraints that prescribe different ways of modifying transmission parameters in order to make them policy compliant.

A Pattern-Mining Method for High-Throughput Lab-on-a-Chip Data Analysis

Biochips are emerging as a useful tool for high-throughput acquisition of biological data and continue to grow in information quality and in discovering new applications. Recent advances include CMOS-based integrated biosensor arrays for deoxyribonucleic acid (DNA) expression analysis (Hassibi and Lee, 2005), (Schienle , 2004), and active research is ongoing for the miniaturization and integration of protein microarrays (Kiyonaka , 2004), (Rubina , 2003), (Scrivener , 2003), tissue microarrays (TMAs), (Chen , 2004), (Shergill , 2004), and fluorescence-based multiplexed cytokine immunoassays (Wang , 2002). The main advantages of microfluidic lab-on-a-chip include ease of use, speed of analysis, low sample and reagent consumption, and high reproducibility due to standardization and automation. Without effective data-analysis methods, however, the merit of acquiring massive data through biochips will be marginal. The high-dimensional nature of such data requires novel techniques that can cope with the curse of dimensionality better than conventional data-analysis approaches. In this paper, the authors proposed a pattern-mining method to analyze large-scale biological data obtained from high-throughput biochip experiments. In particular, when a data set is given as a matrix, the method can find patterns appearing in the form of (possibly overlapping) submatrices of the input matrix. The method exploits the techniques developed for the symbolic manipulation of Boolean functions. Leveraged by this approach, the method can find, given a data matrix, all patterns that satisfy specific input parameters. The authors tested the method with several large-scale biochip data and observed that the proposed method outperforms the alternatives in terms of efficiency and the number of patterns discovered.

Exact minimisation of path-related objective functions for binary decision diagrams

Reduced ordered binary decision diagrams (BDDs) are a data structure for efficient representation and manipulation of Boolean functions. They are frequently used in logic synthesis and formal verification. In recent practical applications, BDDs are optimised with respect to new objective functions. The exact optimisation of BDDs with respect to path-related objective functions is investigated. First, the path-related criteria are studied in terms of sensitivity to variable ordering. Second, a deeper understanding of the computational effort of exact methods targeting the new objective functions is aimed at. This is achieved by an approach based on dynamic programming that generalises the framework of Friedman and Supowit. A prime reason for the computational complexity can be identified using this framework. For the first time, experimental results give the minimal expected path length of BDDs for benchmark functions. They have been obtained by an exact branch and bound method that can be derived from the general framework. The exact solutions are used to evaluate a heuristic approach. Apart from a few exceptions, the results prove the high quality of the heuristic solutions.

Combining ordered best-first search with branch and bound for exact BDD minimization

Reduced-ordered binary decision diagrams (BDDs) are a data structure for efficient representation and manipulation of Boolean functions. They are frequently used in logic synthesis. The size of BDDs depends on a chosen variable ordering, i.e., the size may vary from linear to exponential, and the problem of improving the variable ordering is known to be NP-complete. In this paper, a new exact BDD minimization algorithm called A/sup stute/ is presented. Here, ordered best-first search, i.e., the A/sup */ algorithm, is combined with a classical branch-and-bound (B&B) algorithm. A/sup */ operates on a state space large parts of which are pruned by a best-first strategy expanding only the most promising states. Combining A/sup */ with B&B allows to avoid unnecessary computations and to save memory. Experimental results demonstrate the efficiency of our approach.

Towards the Exact Minimization of BDDs—An Elitism-Based Distributed Evolutionary Algorithm

Binary Decision Diagrams (BDDs) are the state-of-the-art data structure for representation and manipulation of Boolean functions. In general, exact BDD minimization is NP-complete. For BDD-based technology, a small improvement in the number of nodes often simplifies the follow-up problem tremendously. This paper proposes an elitism-based evolutionary algorithm (EBEA) for BDD minimization. It can efficiently find the optimal orderings of variables for all LGSynth91 benchmark circuits with a known minimum size. Moreover, we develop a distributed model of EBEA, DEBEA, which obtains the best-ever variable orders for almost all benchmarks in the LGSynth91. Experimental results show that DEBEA is able to achieve super-linear performance compared to EBEA for some hard benchmarks.

An improved branch and bound algorithm for exact bdd minimization

Ordered binary decision diagrams (BDDs) are a data structure for efficient representation and manipulation of Boolean functions. They are frequently used in logic synthesis and formal verification. The size of the BDDs depends on a chosen variable ordering, i.e., the size may vary from linear to exponential, and the problem of Improving the variable ordering is known to be NP-complete. In this paper, we present a new exact branch and bound technique for determining an optimal variable order. In contrast to all previous approaches that only considered one lower bound, our method makes use of a combination of three bounds and, by this, avoids unnecessary computations. The lower bounds are derived by generalization of a lower bound known from very large scale integration design. They allow one to build the BDD either top down or bottom up. Experimental results are given to show the efficiency of our approach.

BDD minimization by scatter search

Reduced-ordered binary decision diagrams (BDDs) are a data structure for representation and manipulation of Boolean functions. The variable ordering largely influences the size of the BDD, varying from linear to exponential. In this paper, the authors study the BDD minimization problem based on scatter search optimization. Scatter search offers a reasonable compromise between quality (BDD reduction) and time. On smaller benchmarks it delivers almost optimal BDD size with less time than the exact algorithm. For larger benchmarks it delivers smaller BDD sizes than genetic algorithm or simulated annealing at the expense of longer runtime.

Lower Bounds for Linearly Transformed OBDDs and FBDDs

Linearly transformed ordered binary decision diagrams (LTOBDDs) have been suggested as a generalization of OBDDs for the representation and manipulation of Boolean functions. Instead of variables as in the case of OBDDs, parities of variables may be tested at the nodes of an LTOBDD. By this extension it is possible to represent functions in polynomial size that do not have polynomial size OBDDs, e.g., the characteristic functions of linear codes. In this paper lower bound methods for LTOBDDs and some generalizations of LTOBDDs are presented and applied to explicitly defined functions. By the lower bound results it is possible to compare the set of functions with polynomial size LTOBDDs and their generalizations with the set of functions with polynomial size representations for many other restrictions of BDDs.

Using lower bounds during dynamic BDD minimization

Ordered binary decision diagrams (BDDs) are a data structure for the representation and manipulation of Boolean functions, often applied in very large scale integration (VLSI) computer-aided design (CAD). The choice of variable ordering largely influences the size of the BDD; its size may vary from linear to exponential. The most successful methods to find good orderings are based on dynamic variable reordering, i.e., exchanging neighboring variables. This basic operation has been used in various variants, like sifting and window permutation. In this paper, we show that lower bounds computed during the minimization process can speed up the computation significantly. First, lower bounds are studied from a theoretical point of view. Then these techniques are incorporated in dynamic minimization algorithms. By the computation of good lower bounds, large parts of the search space can be pruned, resulting in very fast computations. Experimental results are given to demonstrate the efficiency of the approach.

A comment on "Graph-based algorithm for Boolean function manipulation"

In this paper, a slight error in the paper of Bryant (ibid., vol.35, no.8, p.677-691, Aug. 1986 is corrected: It was stated that, under a certain ordering restriction, composition of two Reduced Ordered Binary Decision Diagrams (ROBDDs) results in a reduced OBDD. We show a counterexample and explore under which conditions this statement is incorrect.

Improving the efficiency of BDD-based operators by means of partitioning

Binary decision diagrams (BDD's) are a state-of-the-art core technique for the symbolic representation and manipulation of Boolean functions, relations and finite sets. Many computer-aided design (CAD) applications resort to them, but size and time efficiency restrict their applicability to medium-small designs. We concentrate on complex operators used in symbolic manipulation. We analyze and optimize their performance by means of new dynamic partitioning strategies. We propose a novel quick algorithm for the estimation of cofactor size, and a technique to choose splitting variables according to their discrimination power, so that their cofactors may be optimized by different variable orderings (tending to the more flexible FBDDs). Furthermore, we analyze time efficiency and the impact of hashing/caching on BDD-based operators. We finally include an experimental observation of memory usage and running time for operators applied in symbolic manipulation.

Application of symbolic FSM Markovian analysis to protocol verification

Computer-aided design tools for Markovian analysis and verification of digital circuits have gained much interest in the last few years. This is mainly due to the advent of effective data structures, e.g., binary and algebraic decision diagrams, for Boolean and pseudo-Boolean function representation and manipulation. The authors illustrate how those tools can be successfully exploited to analyse the behavior and verify the correctness of a communication protocol. They first consider the case of single protocol entities running in isolation, and present a simple application example (i.e. the sender entity of the alternating bit protocol). Then, informally illustrate how the analysis approach can be extended to the general case of multi-layer protocol stacks and complete communication systems.

Auxiliary variables for BDD-based representation and manipulation of Boolean functions

BDDs are the state-of-the-art technique for representing and manipulating Boolean functions. Their introduction caused a major leap forward in synthesis, verification, and testing. However, they are often unmanageable because of the large amount of nodes. To attack this problem, we insert auxiliary variables that decompose monolithic BDDs in smaller ones. This method works very well for Boolean function representation. As far as combinational circuits are concerned, representing their functions is the main issue. Going into the sequential domain, we focus on traversal techniques. We show that, once we have Boolean functions in decomposed form, symbolic manipulations are viable and efficient. We investigate the relation between auxiliary variables and static and dynamic ordering strategies. Experimental evidence shows that we achieve a certain degree of independence from variable ordering. Thus, this approach can be an alternative to dynamic re-ordering. Experimental results on Boolean function representation, and exact and approximate forward symbolic traversal of FSMs, demonstrate the benefits both in terms of memory requirements and of CPU time.

The complexity of the inclusion operation on OFDD's

Ordered functional decision diagrams (OFDD's) are a data structure for representation and manipulation of Boolean functions. A polynomial time algorithm for the test whether f/spl les/g for functions f and g given by OFDD's is presented. This is the last basic operation on OFDD's whose complexity was unknown. The result also solves an open question on the complexity of minimizing OFDD's for incompletely specified functions.

Ordered Kronecker functional decision diagrams-a data structure for representation and manipulation of Boolean functions

Ordered Kronecker functional decision diagrams (OKFDD's) are a data structure for efficient representation and manipulation of Boolean functions. OKFDD's are a generalization of ordered binary decision diagrams (OBDD)s) and ordered functional decision diagrams and thus combine the advantages of both. In this paper, basic properties of OKFDD's and their efficient representation and manipulation are given. Starting with elementary manipulation algorithms, we present methods for the construction of small OKFDD's. Our approach is based on dynamic variable ordering and decomposition-type choice. For changing the decomposition type, we use an efficient reordering-based method. We briefly discuss the implementation of PUMA, an OKFDD package, which was used in all our experiments. These experiments demonstrate the quality of our methods in comparison to sifting and interleaving for OBDD's.

The I/O-Complexity of Ordered Binary-Decision Diagram Manipulation

Ordered Binary-Decision Diagrams (OBDD) are the state-of-the art<br />data structure for boolean function manipulation and there exist<br />several software packages for OBDD manipulation. OBDDs have<br />been successfully used to solve problems in e.g. digital-systems design, verification and testing, in mathematical logic, concurrent system design and in artificial intelligence. The OBDDs used in many of these applications quickly get larger than the available main memory and it becomes essential to consider the problem of minimizing the Input/Output (I/O) communication. In this paper we analyze why existing OBDD manipulation algorithms perform poorly in an I/O environment and develop new I/O-efficient algorithms.

Genetic algorithm for variable ordering of OBDDs

A genetic algorithm (GA) is applied to find a variable ordering that minimises the size of ordered binary decision diagrams (OBDDs). OBDDs are a data structure for representation and manipulation of Boolean functions often applied in CAD. The choice of the variable ordering largely influences the size of the OBDD (i.e. its size may vary from polynomial to exponential in the number of variables). Dynamic variable ordering is the state-of-the-art method for finding good variable orderings. In the paper it is shown by experimental results that better sizes can be obtained using GAs. The authors' GA approach is a practical alternative to the exact algorithm for variable ordering. It produced optimal results for most considered benchmarks, but it is also applicable to functions with more than 20 variables due to its short runtimes.