一种基于熵的OBDD变量排序算法

有序二叉决策图（OBDD）是一种有效表示布尔函数的数据结构，其大小依赖于所采用的变量序。熵是定量描述布尔函数中变量重要性的一种方法。基于变量的熵值分析了高质量变量序的特征，给出了一种基于熵的OBDD变量排序算法。实验结果表明：该算法与模拟退火算法和遗传算法结果相当。时间仅为相应算法的80．84％和29．79％。     

基于SBDD图的布尔匹配

在逻辑验证和综合中,布尔匹配利用有序二叉判定图OBDD来检验两个给定的逻辑函数是否相等。为了提高匹配算法的效率,文中用最小项数作为标签标定变量（变量组）。对比两函数中变量（变量组）的“标签”,可以删除不可能的排序,从而加快匹配过程。在提取变量标签时,提出简约二分决策图－SBDD,并利用其节点少的特性进一步提高“标签”提取算法的效率。实验结果表明本算法执行速度快,变量区分能力强。     

基于OBDD的有向图的存储与操作研究

本文讨论了一种基于OBDD的有向图的存储结构,给出了基于OBDD的有向图的操作方法及搜索算法.实验结果表明,该存储结构与传统的邻接表的存储结构相比,在处理大规模的有向图时,具有较高的存储效率.     

基于重量分析的OBDD变量排序算法

有序的二叉判决图（ＯＢＤＤ）是布尔表达式的一种有效表示方法，但它的体积对变量排序具有较强的依赖性。本文提出一种电路结构图，并在此基础上定义了原始输入重量和节点重量等参数，并建立了用重量分析来指导的ＯＢＤＤ变量排序算法。由于从考虑变量对输出函数的影响出发与从考虑ＯＢＤＤ节点共享性出发对变量排序的要求不同，本文分别设计了两类算法。实验结果表明，本文对大多数标准电路变量排序的效果都优于国际上的同类算法，     

异步时序电路分析一种OBDD方法

对异步时序电路的分析和使用是一个比较困难的问题,所以,异步时序电路的实际应用范围远不如同步时序电路,通过改进JRBurch等提出的分析方法,使之适用于异步时序电路,该方法使用基于OBDD的布尔特征函数来表示电路的转移关系,并通过基于OBDD的布尔函数的运算涞确定异步时序电路的稳定状态,及当输入改变时电路的下一个稳定状态,由此可实现对电路特性的精确描述。     

约束满足问题求解的符号OBDD桶消元算法

桶消元算法是求解约束满足问题的一种典型推理方法.针对桶消元算法面临的状态空间爆炸问题,将有序二叉决策图( OBDD)技术与该算法结合起来,给出了约束满足问题的一种求解算法.通过对约束满足问题中变量和域值的编码,将CSP问题转化为命题可满足性问题,给出了约束满足问题的OBDD表示方法;基于桶消元的算法思想,在约束满足问题...     

基于OBDD的SMC中PRE操作的改进算法

提出一种基于排序二值判定图(OBDD)的符号模型检测中PRE操作的改进算法。该算法处理PRE步骤3(嵌套布尔存在量化)的方法是一次遍历“删除”所有被量化变量的节点,产生表示布尔函数与嵌套存在量化结果等价的不确定排序二值判定图,把不确定排序二值判定图转换成OBDD。实验表明,该算法能有效缩短计算时间,减少中间节点所需空间。     

基于OBDD的SMC中PRE 操作的改进算法

提出一种基于排序二值判定图（OBDD）的符号模型检测中PRE 操作的改进算法。该算法处理PRE 步骤3（嵌套布尔存在量化）的方法是一次遍历“删除”所有被量化变量的节点,产生表示布尔函数与嵌套存在量化结果等价的不确定排序二值判定图,把不确定排序二值判定图转换成OBDD。实验表明,该算法能有效缩短计算时间,减少中间节点所需空间。     

基于OBDD的SMC中PRE(E)操作的改进算法

提出一种基于排序二值判定图(OBDD)的符号模型检测中PRE(操作的改进算法.该算法处理PRE(步骤3(嵌套布尔存在量化)的方法是一次遍历"删除"所有被量化变量的节点,产生表示布尔函数与嵌套存在量化结果等价的不确定排序二值判定图,把不确定排序二值判定图转换成OBDD.实验表明,该算法能有效缩短计算时间,减少中间节点所需空间.     

基于OBDD的描述逻辑εL循环术语集推理

循环术语集推理是描述逻辑研究中面临的难点问题,尚未得到很好的解决.有序二叉决策图（ordered binary decision diagram,简称OBDD）是一种对布尔函数进行紧凑表示和高效操作的数据结构,适用于表示和处理大规模问题.将OBDD应用于描述逻辑循环术语集的推理.首先,针对描述逻辑εL中的循环术语集,给出了描述图上关于最大模拟关系的重要性质,并借助集合表示和集合运算对该性质进行了表述和证明.在此基础上,应用布尔函数对描述图进行编码,给出了基于OBDD求解最大模拟关系的方法,进而给出了最大不动点语义下基于OBDD对概念包含关系进行判定的算法;接下来,基于OBDD给出了求解描述图中可以到达循环路径的所有结点的方法,进而给出了最小不动点语义下基于OBDD对概念包含关系进行判定的算法;最后,对算法的正确性、复杂度等进行了分析和证明,并对算法进行了编程实现,给出了关于计算性能的实验结果.该工作为循环术语集的推理提供了一条有效途径,也为OBDD在逻辑推理中的应用提供了新的案例.     

On the Minimization of (Complete) Ordered Binary Decision Diagrams

Ordered binary decision diagrams (OBDDs) are a popular data structure for Boolean functions. Some applications work with a restricted variant called complete OBDDs which is strongly related to nonuniform deterministic finite automata. One of its complexity measures is the width which has been investigated in several areas in computer science like machine learning, property testing, and the design and analysis of implicit graph algorithms. For a given function the size and the width of a (complete) OBDD is very sensitive to the choice of the variable ordering but the computation of an optimal variable ordering for the OBDD size is known to be NP-hard. Since optimal variable orderings with respect to the OBDD size are not necessarily optimal for the complete model or the OBDD width and hardly anything about the relation between optimal variable orderings with respect to the size and the width is known, this relationship is investigated. Here, using a new reduction idea it is shown that the size minimization problem for complete OBDDs and the width minimization problem are NP-hard.     

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 Bryant?s 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.     

Efficient algorithms for the transformation between different types of binary decision diagrams

The problem of transforming an FBDD (free binary decision diagram) on variables or a OBDD (ordered binary decision diagram with respect to the variable ordering ) for the Boolean function into the reduced OBDD for is considered. The algorithms run in time (where, e.g., is the size of ) and need space , if may be an FBDD, or , if is known to be an OBDD. The problem is important for the improvement of given variable orderings, e.g., by simulated annealing or genetic algorithms, and in the situation where incompatible representations of functions have to be made compatible. Received: 13 October 1994 / 7 November 1995     

On the influence of the variable ordering for algorithmic learning using OBDDs

OBDDs with a fixed variable ordering are used successfully as data structure in experiments with learning heuristics based on examples. In this paper, it is shown that, for some functions, it is necessary to develop an algorithm to learn also a good OBDD variable ordering. There are functions with the following properties. They have OBDDs of linear size for optimal variable orderings. But for all but a small fraction of all variable orderings one needs large size to represent a list of randomly chosen examples. These properties are shown for simple functions like the multiplexer and the inner product.     

Size of ordered binary decision diagrams representing threshold functions

An ordered binary decision diagram (OBDD) is a graph representation of a Boolean function. In this paper, the size of ordered binary decision diagrams representing threshold functions is discussed. We consider two cases: the case when a variable ordering is given and the case when it is adaptively chosen. We show 1) O(2^n/2) upper bound for both cases, 2) Ω(2^n/2) lower bound for the former case and 3) Ω(n2^√n/2) lower bound for the latter case. We also show some relations between the variable ordering and the size of OBDDs representing threshold functions.     

The Nonapproximability of OBDD Minimization

The size of ordered binary decision diagrams (OBDDs) is determined by the chosen variable ordering. A poor choice may cause an OBDD to be too large to fit into the available memory. The decision variant of the variable ordering problem is known to be NP-complete. We strengthen this result by showing that, unless P=NP, for each constant c>1 there is no polynomial time approximation algorithm with the performance ratio c for the variable ordering problem, i.e., no polynomial time algorithm that guarantees the computation of a variable ordering so that the resulting OBDD size is larger than the minimum size by a factor of at most c. This result justifies, also from a theoretical point of view, the use of heuristics for the variable ordering problem.     

Translation among CNFs, characteristic models and ordered binary decision diagrams

We consider translation among conjunctive normal forms (CNFs), characteristic models, and ordered binary decision diagrams (OBDDs) of Boolean functions. It is shown in this paper that Horn OBDDs can be translated into their Horn CNFs in polynomial time. As for the opposite direction, the problem can be solved in polynomial time if the ordering of variables in the resulting OBDD is specified as an input. In case that such ordering is not specified and the resulting OBDD must be of minimum size, its decision version becomes NP-complete. Similar results are also obtained for the translation in both directions between characteristic models and OBDDs. We emphasize here that the above results hold on any class of functions having a basis of polynomial size.     

Larger lower bounds on the OBDD complexity of integer multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Recently, the question whether the OBDD complexity of the most significant bit of integer multiplication is exponential has been answered affirmatively. In this paper a larger general lower bound is presented using a simpler proof. Furthermore, we prove a larger lower bound for the variable order assumed to be one of the best ones for the most significant bit. Moreover, the best known lower bound on the OBDD complexity for the so-called graph of integer multiplication is improved.