Multiple-valued Decision Diagrams Research Articles

Reversible Circuit Synthesis Method Using Sub-graphs of Shared Functional Decision Diagrams

Abstract Reversible circuit synthesis methods based on decision diagrams achieve low quantum costs but do not account for quantum bit (qubit) limits for the application of reversible logic in quantum computing. Here, a synthesis method using sub-graphs of shared functional decision diagrams (SFDDs) is proposed for reducing the number of lines when synthesizing reversible circuits. An SFDD is partitioned into sub-graphs by exploiting the longest dominant-active paths, and the sub-graphs are mapped to reversible gate cascades. To further reduce the number of lines, template root matching is presented for reusing circuit lines. Experimental results indicate that the proposed method achieves the known minimum number of lines in many cases and has good scalability. Although the proposed method increases the quantum cost over a prior method based on functional decision diagrams, it significantly reduces the number of lines in most cases. Compared with the one-pass method using quantum multiple-valued decision diagrams, the proposed method reduces the quantum cost without increasing the number of lines in many cases. When compared with the lookup table-based method using a direct mapping flow, the method reduces the number of lines in a few cases. Thus, the method aids in the physical realization of a quantum circuit.

Multiple-valued Decision Diagrams for Multi-state System Performability Analysis

Background: In a multi-state system, either the entire system or its components will have more than two states. Different kind of multi-state systems have been investigated, such as multistate series-parallel systems, multi-state k-out-of-n systems and multi-state linear consecutive-k-outof- n systems. Methods: Performability analysis is concerned with the calculation of probability distribution of different system performance levels. In this paper, Multi-valued Decision Diagram (MDD) based approaches are discussed to conduct the performability analysis of multi-state systems. Results: Examples are provided to illustrate the use of MDD approaches for three types of multistate systems, e.g. multi-state series-parallel systems, multi-state k-out-of-n systems and multi-state linear consecutive-k-out-of-n systems. Conclusion: The empirical results show that the MDD approaches have lower computational complexity than the traditional methods in model generation and evaluation.

Multivalued decision diagram-based common cause failure analysis in phased-mission systems

Due to a shock or shared root cause, multiple system components may fail at the same time contributing significantly to the entire system failure. Such dependent component failures are referred to as common cause failures (CCFs). A rich body of research has been conducted for addressing effects of CCFs in reliability analysis of diverse types of systems. This paper first adapts multiple-valued decision diagrams (MDDs) for analyzing reliability of a non-repairable phased-mission system (PMS) subject to CCFs caused by external shocks. An explicit method and two implicit methods based on MDDs are proposed and compared. While the three MDD-based methods differ in terms of space and computation complexities, examples show that all these methods offer low computational complexity while addressing dynamics and dependencies caused by CCFs and phased operations.

Improved MDD Algorithm for Mission Reliability Estimation of an Escort Formation

An escort formation is a phased mission system of systems (PMSoS), which is composed of multiple ships with different functions. The configuration of the formation and success criteria for a mission may vary in different phases. Reliability estimation for PMSoS is complicated due to the strong phase dependence of multiple systems. This paper proposes an improved multiple-valued decision diagram (MDD) algorithm to perform reliability estimation of a nonrepairable escort formation. First, a phased fault tree is established to describe the failure mode of an escort formation throughout a mission, which is simplified according to the common failure basic mission (module) (CFBM). Bottom events are sorted based on the CFBM, and the case method is adopted to generate an MDD from the simplified fault tree model. On this basis, the MDD method is adopted to estimate mission reliability. The performance of the improved MDD method is compared with that of a binary decision diagram (BDD) method and a general MDD method. The results show that the improved MDD method can offer lower computational complexity as well as a simpler model construction over the BDD method and general MDD method. A case study of an escort formation PMSoS is analyzed to illustrate the proposed MDD method, and the sensitivity and composite importance measure (CIM) of each system are evaluated.

Reliability Analysis of Cold Standby Systems based on Supplement Events using Multiple-Valued Decision Diagrams

Reliability Analysis of Cold Standby Systems based on Supplement Events using Multiple-Valued Decision Diagrams

QMDDs: Efficient Quantum Function Representation and Manipulation

Quantum mechanical phenomena such as phase shifts, superposition, and entanglement show promise in use for computation. Suitable technologies for the modeling and design of quantum computers and other information processing techniques that exploit quantum mechanical principles are in the range of vision. Quantum algorithms that significantly speed up the process of solving several important computation problems have been proposed in the past. The most common representation of quantum mechanical phenomena are transformation matrices. However, the transformation matrices grow exponentially with the size of a quantum system and, thus, pose significant challenges for efficient representation and manipulation of quantum functionality. In order to address this problem, first approaches for the representation of quantum systems in terms of decision diagrams have been proposed. One very promising approach is given by Quantum Multiple-Valued Decision Diagrams (QMDDs) which are able to efficiently represent transformation matrices and also inherently support multiple-valued basis states offered by many physical quantum systems. However, the initial proposal of QMDDs was lacking in a formal basis and did not allow, e.g., the change of the variable order—an established core functionality in decision diagrams which is crucial for determining more compact representations. Because of this, the full potential of QMDDs or decision diagrams for quantum functionality in general has not been fully exploited yet. In this paper, we present a refined definition of QMDDs for the general quantum case. Furthermore, we provide significantly improved computational methods for their use and manipulation and show that the resulting representation satisfies important criteria for a decision diagram, i.e., compactness and canonicity. An experimental evaluation confirms the efficiency of QMDDs.

A Multiple-Valued Decision Diagram Based Method for Efficient Reliability Analysis of Non-Repairable Phased-Mission Systems

Many practical systems are phased-mission systems (PMSs), where the mission consists of multiple, consecutive, and non-overlapping phases of operation. An accurate reliability analysis of a PMS must consider statistical dependence of component states across phases, as well as dynamics in system configurations, success criteria, and component behavior. This paper proposes a new method based on multiple-valued decision diagrams (MDDs) for the reliability analysis of a non-repairable binary-state PMS. Due to its multi-valued logic nature, the MDD model has recently been applied to the reliability analysis of multistate systems. In this work, we present a novel way to adapt MDDs for the reliability analysis of systems with multiple phases. Examples show how the MDD models are generated and evaluated to obtain the mission reliability measures. Performance of the MDD-based method is compared with an existing binary decision diagram (BDD)-based method for PMS analysis. Empirical results show that the MDD-based method can offer lower computational complexity as well as a simpler model construction and improved evaluation algorithms over those used in the BDD-based method.

A Multiple-Valued Decision-Diagram-Based Approach to Solve Dynamic Fault Trees

Dynamic fault trees (DFTs) have been used for many years because they can easily provide a concise representation of the dynamic failure behaviors of general non-repairable fault tolerant systems. However, when repeated failure events appear in real-life DFT models, the traditional modularization-based DFT analysis process can still generate large dynamic subtrees, the modeling of which can lead to a state explosion problem. Examples of these kinds of large dynamic subtrees abound in models of real-world dynamic software and embedded computing systems integrating with various multi-function components. This paper proposes an efficient, multiple-valued decision-diagram (MDD)-based DFT analysis approach for computing the reliability of large dynamic subtrees. Unlike the traditional modularization methods where the whole dynamic subtree must be solved using state-space methods, the proposed approach restricts the state-space method only to components associated with dynamic failure behaviors within the dynamic subtree. By using multiple-valued variables to encode the dynamic gates, a single compact MDD can be generated to model the failure behavior of the overall system. The combination of MDD and state-space methods applied at the component or gate level helps relieve the state explosion problem of the traditional modularization method, for the problems we explore. Applications and advantages of the proposed approach are illustrated through detailed analyses of an example DFT, and through two case studies.

A single-photon fault-detection method for nanocircuits that use GaN material

As the complexity of nanocircuits continues to increase, developing tests for them becomes more difficult. Failure analysis and the localization of internal test points within nanocircuits are already more difficult than for conventional integrated circuits. In this paper, a new method of testing for faults in nanocircuits is presented that uses single-photon detection to locate failed components (or failed signal lines) by utilizing the infrared photon emission characteristics of circuits. The emitted photons, which can carry information about circuit structure, can aid the understanding of circuit properties and locating faults. In this paper, in order to enhance the strength of emitted photons from circuit components, test vectors are designed for circuits’ components or signal lines. These test vectors can cause components to produce signal transitions or switching behaviors according to their positions, thereby increasing the strength of the emitted photons. A multiple-valued decision diagram (MDD), in the form of a directed acrylic graph, is used to produce the test vectors. After an MDD corresponding to a circuit is constructed, the test vectors are generated by searching for specific paths in the MDD of that circuit. Experimental results show that many types of faults such as stuck-at faults, bridging faults, crosstalk faults, and others, can be detected with this method.

Multiple-Valued Logic mathematical approaches for multi-state system reliability analysis

A mathematical description of an examined system such as Multi-State System (MSS) permits the system reliability to be analyzed in more detail, because the MSS defines some performance levels (more than only working and failure). The structure function is one of the basic definitions and representations of MSS. But the dimension of the structure function increases critically depending on the number of system components. Therefore, the development of methods for examination and quantification of such a function is an actual problem in MSS reliability analysis. In this paper, a method for the analysis of the MSS structure function of high dimension is proposed. The principal point of this method is the interpretation of the MSS structure function as a Multiple-Valued Logic function. It allows effective and approved mathematical methods of Multiple-Valued Logic to be used for analysis and quantification of the MSS structure function. We propose to use two mathematical approaches of Multiple-Valued Logic for the MSS. One of them is a representation of the MSS structure function by a Multiple-Valued Decision Diagram. It is an effective approach for analysis and estimation of the function of high dimension in Multiple-Valued Logic. The other approach is Logic Differential Calculus. Logic Differential Calculus is a useful approach for analysis of the MSS state changes.

A New Verification Method of Digital Circuits Based on Cone-Oriented Partitioning and Decision Diagrams

The formal verification is able to check whether the implementation of a circuit design is functionally equivalent to an earlier version described at the same level of abstraction, it can show the correctness of a circuit design. A new circuit verification method based on cone-oriented circuit partitioning and decision diagrams is presented in this paper. First of all, the structure level of every signal line in a circuit is computed. Secondly, the circuit is partitioned into a lot of cone structures. The multiple-valued decision diagram corresponding to every cone structure is generated. The verification procedure is to compare the equivalence of the multiple-valued decision diagrams of two types of cone structures. Experimental results on a lot of benchmark circuits show the method presented in this paper can effectively perform the equivalence checking of circuits.

Test Method for Crosstalk Faults in VLSI Circuits Based on Multiple-Valued Decision Diagrams

The crosstalk fault in VLSI circuits is one of the interference effects being caused by parasitic capacitance and inductance coupling, it can lead to functional errors of circuits. It is necessary to detect the crosstalk faults in order to insure the functions of circuits. A new test method for crosstalk faults in VLSI circuits based on multiple-valued decision diagrams is presented in this paper, the test vectors of crosstalk faults are generated by building a multiple-valued decision diagram that is a difference operation of the two multiple-valued decision diagrams corresponding to the normal circuit and faulty circuit, respectively. One advantage of the test method is that it can get all test vectors of a given crosstalk fault, therefore for a digital circuit, the test set with minimal number of test vectors can be obtained. Experimental results on a lot of digital circuits demonstrate the feasibility of the method proposed in this paper.

A Systematic Design Method for Two-Variable Numeric Function Generators Using Multiple-Valued Decision Diagrams

This paper proposes a high-speed architecture to realize two-variable numeric functions. It represents the given function as an edge-valued multiple-valued decision diagram (EVMDD), and shows a systematic design method based on the EVMDD. To achieve a design, we characterize a numeric function ƒ by the values of l and p for which ƒ is an l-restricted Mp-monotone increasing function. Here, l is a measure of subfunctions of ƒ and p is a measure of the rate at which ƒ increases with an increase in the dependent variable. For the special case of an EVMDD, the EVBDD, we show an upper bound on the number of nodes needed to realize an l-restricted Mp-monotone increasing function. Experimental results show that all of the two-variable numeric functions considered in this paper can be converted into an l-restricted Mp-monotone increasing function with p = 1 or 3. Thus, they can be compactly realized by EVBDDs. Since EVMDDs have shorter paths and smaller memory size than EVBDDs, EVMDDs can produce fast and compact NFGs.

Minimization of Multiple-Valued Decision Diagrams Based on Matrix Computation

This paper proposes a new algorithm to simplify the multiple valued logic (MVL) decision diagrams. This algorithm is based on a new level coupling rule. By changing the designer’s perspective towards the design, this rule can make further simplification possible. In most of state of the art designs, the decision diagram plays a serious role in the implementation of the logical functions. The proposed algorithm uses the new level coupling rule and combines it with the existing ones, presenting a new method in simplifying and implementing the basic decision diagram.

Multiple Valued Logic: Concepts and Representations

Abstract Multiple Valued Logic: Concepts and Representations begins with a survey of the use ofmultiple-valued logic in several modern application areas including electronic design automation algorithms and circuit design. The mathematical basis and concepts of various algebras and systems of multiple valued logic are provided including comparisons among various systems and examples of their application. The book also provides an examination of alternative representations of multiple-valued logic suitable for implementation as data structures in automated computer applications. Decision diagram structures for multiple valued applications are described in detail with particular emphasis on the recently developed quantum multiple valued decision diagram.Table of Contents: Multiple Valued Logic Applications / MVL Concepts and Algebra / Functional Representations / Reversible andQuantum Circuits / Quantum Multiple-Valued Decision Diagrams / Summary / Bibliography

An Integrated Approach of Variable Ordering and Logic Mapping into LUT-Array-Based PLD

This paper presents an approach of logic mapping into LUT-Array-Based PLD where Boolean functions in the form of the sum of generalized complex terms (SGCTs) can be mapped directly. While previous mapping approach requires predetermined variable ordering, our approach performs mapping and variable reordering simultaneously. For the purpose, we propose a directed acyclic graph based on the multiple valued decision diagram (MDD) and an algorithm to construct the graph. Our algorithm generates candidates of SGCT expressions for each node in a bottom-up manner and selects the variables in the current level by evaluating the sizes of SGCT expressions directly. Experimental results show that our approach reduces the number of terms maximum to 71 percent for the MCNC benchmark circuits.

Conquering Uncertainty in Multiple-Valued Logic Design

In modern science, significant advances are typically made at cross-roads of disciplines. Thus, many optimization problems in Multiple-valued Logic Design have been successfully approached using ideas and techniques from Artificial Intelligence. In particular, improvements in multiple-valued logic design have been made by exploiting information/uncertainty measures. In this paper, we review well-known information measures in the multiple-valued domain and consider some methods of finding information measures for completely or incompletely specified functions with multiple-valued and continuous attributes. In this respect, the paper addresses the problem known as discretization and introduces a method of finding an optimal representation of continuous data in the multiple-valued domain. We also propose a technique for efficient calculation of different information measures using Multiple-valued Decision Diagrams. As one application of our technique, we outline an approach to synthesizing digital circuits derived from decision diagrams that can yield to reduction in power dissipation. The paper also shows the impact in several important areas of multiple-valued system design including (i) fuzzy logic, (ii) quantum computing systems, and (iii) data mining.

Heuristic minimization of multiple-valued relations

An approach to minimization that is based on a state-of-the-art paradigm for the two-level minimization of functions is presented. Some special properties of relations, in contrast to functions, which must be carefully considered in realizing a high-quality procedure for solving the minimization problem are clarified. An efficient heuristic method to find an optimal sum-of-products representation for a multiple-valued relation is proposed and implemented in the program GYOCRO. It uses multiple-valued decision diagrams (MDDs) to represent the characteristic functions for the relations. Experimental results are presented and compared with previous exact and heuristic Boolean relation minimizers to demonstrate the effectiveness of the proposed method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>