A note on the size of OBDDs for the graph 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 the most common dynamic data structure for Boolean functions. Among the many areas of application are verification, model checking, computer-aided design, relational algebra, and symbolic graph algorithms. Analyzing the limits of symbolic graph algorithms for the all-pairs-shortest paths problem which work on OBDD-represented graph instances the so-called graph of integer multiplication has been investigated by Sawitzki [D. Sawitzki, Lower bounds on the OBDD size of graphs of some popular functions, in: Proc. of SOFSEM, LNCS, vol. 3381, 2005, pp. 298-309]. Using simple arguments his lower bound of ²n/768−1 on the size of OBDDs representing the graph of integer multiplication is improved up to ²n/24.     

New Results on the Most Significant Bit 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. Analyzing the limits of symbolic graph algorithms for the reachability problem Sawitzki (Proc. of LATIN, LNCS, vol. 3887, pp. 781–792, Springer, Berlin, 2006) has presented the first exponential lower bound on the π-OBDD size for the most significant bit of integer multiplication according to one predefined variable order π. Since the choice of the variable order is a main issue to obtain OBDDs of small size the investigation is continued. As a result a new upper bound method and the first non-trivial upper bound on the size of OBDDs according to an arbitrary variable order is presented. Furthermore, Sawitzki’s lower bound is improved.     

Lower bounds on the OBDD size of two fundamental functions' graphs

Ordered Binary Decision Diagrams (OBDDs) are a data structure for Boolean functions which supports many useful operations. Among others it finds applications in CAD, model checking, and symbolic graph algorithms. Nevertheless, many simple functions are known to have exponential OBDD size with respect to their number of variables. In order to investigate the limits of symbolic graph algorithms which work on OBDD-represented graph instances, it is useful to have simply-structured graphs whose OBDD representation has exponential size. Therefore, we consider two fundamental functions with exponential lower bounds on their OBDD size and transfer these results to their corresponding graphs. Concretely, we consider the Indirect Storage Access function and the Hidden Weighted Bit function.     

Exact OBDD Bounds for Some Fundamental Functions

Ordered binary decision diagrams (OBDDs) are nowadays one of the most common dynamic data structures or representation types for Boolean functions. Among the many areas of application are verification, model checking, computer aided design, relational algebra, and symbolic graph algorithms. Although many exponential lower bounds on the OBDD size of Boolean functions are known, there are only few functions where the OBDD size is asymptotically known exactly. In this paper the exact OBDD sizes of the fundamental functions multiplexer and addition of n-bit numbers are determined.     

Exponential space complexity for OBDD-based reachability analysis

Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Nevertheless, many basic graph problems are known to be PSPACE-hard if their input graphs are represented by OBDDs. Computing the set of nodes that are reachable from some source s∈V in a digraph G=(V,E) is an important problem in computer-aided design, hardware verification, and model checking. Until now only exponential lower bounds on the space complexity of a restricted class of OBDD-based algorithms for the reachability problem have been known. Here, the result is extended by presenting an exponential lower bound for the general reachability problem. As a by-product a general exponential lower bound is obtained for the computation of OBDDs representing all connected node pairs in a graph, the transitive closure.     

On the OBDD complexity of the most significant bit of integer multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations. Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for boolean functions. Among the many areas of application are verification, model checking, computer-aided design, relational algebra, and symbolic graph algorithms. In this paper it is shown that the OBDD complexity of the most significant bit of integer multiplication is exponential answering an open question posed by Wegener (2000) [18].     

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.     

On the size of randomized OBDDs and read-once branching programs for k-stable functions

In this paper, a simple technique which unifies the known approaches for proving lower bound results on the size of deterministic, nondeterministic, and randomized OBDDs and kOBDDs is described.?As an application of this technique, a generic lower bound on the size of randomized OBDDs with bounded error is established for a class of functions which has been studied in the literature on branching programs for a long time. These functions have been called “k-stable” by Jukna. It follows that several standard functions are not contained in the analog of the class BPP for OBDDs. Furthermore, exponential lower bounds on the size of randomized kOBDDs are presented.?It is well known that k-stable functions with large k are hard for deterministic read-once branching programs. This is no longer true in the randomized case. It is shown here that a certain k-stable function due to Jukna, Razborov, Savicky, and Wegener has randomized branching programs of polynomial size, even with zero error. It follows that for the analogs of these classes defined in terms of the size of read-once branching programs. Received: September 3, 1998.     

Symbolic model checking for self-stabilizing algorithms

A distributed system is said to be self-stabilizing if it converges to safe states regardless of its initial state. In this paper we present our results of using symbolic model checking to verify distributed algorithms against the self-stabilizing property. In general, the most difficult problem with model checking is state explosion; it is especially serious in verifying the self-stabilizing property, since it requires the examination of all possible initial states. So far applying model checking to self-stabilizing algorithms has not been successful due to the problem of state explosion. In order to overcome this difficulty, we propose to use symbolic model checking for this purpose. Symbolic model checking is a verification method which uses Ordered Binary Decision Diagrams (OBDDs) to compactly represent state spaces. Unlike other model checking techniques, this method has the advantage that most of its computations do not depend on the initial states. We show how to verify the correctness of algorithms by means of SMV, a well-known symbolic model checker. By applying the proposed approach to several algorithms in the literature, we demonstrate empirically that the state spaces of self-stabilizing algorithms can be represented by OBDDs very efficiently. Through these case studies, we also demonstrate the usefulness of the proposed approach in detecting errors     

Symbolic OBDD representations for mechanical assembly sequences

Assembly sequence planning is one typical combinatorial optimization problem, where the size of parts involved is a significant and often prohibitive difficulty. The compact storage and efficient evaluation of all the feasible assembly sequences is one crucial concern. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate the Boolean functions efficiently, and appears to give improved results for large-scale combinatorial optimization problems. In this paper, subassemblies, assembly states and assembly tasks are represented as Boolean characteristic functions, and the symbolic OBDD representation of assembly sequences is proposed. In this framework, the procedures to transform directed graph and AND/OR graph into OBDDs are presented. The great advantage of OBDD-based scheme is that the storage space of OBDD-based representation of all the feasible assembly sequences does not increase with the part count of assembly dramatically so quickly as that of both directed graph and AND/OR graph do. We undertake many experimental tests using Visual C++ and CUDD package. It was shown that the OBDD scheme represented all the feasible assembly sequences correctly and completely, and outperforms either directed graph or AND/OR graph in storage efficiency.     

多智能体系统时态认知规范高效符号模型检测的算法研究

Clarke和McMillan提出了利用mu演算和OBDDs符号模型检测时态逻辑的方法.这些方法是非常有效的,能用于验证许多具有极大状态空间的实际系统(状态个数可以超过1020).但是,这些方法不能检测知识逻辑.而时态认知逻辑能更精确地描述分布式领域中系统和协议的规范.文章首先讨论了Kripke结构和mu演算的扩展,然后提出了利用扩展mu演算和OBDDs符号模型检测时态认知逻辑的方法.     

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.     

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.     

A uniform framework for weighted decision diagrams and its implementation

This paper introduces a generic framework for OBDD variants with weighted edges. It covers many boolean and multi-valued OBDD-variants that have been studied in the literature and applied to the symbolic representation and manipulation of discrete functions. Our framework supports reasoning about reducedness and canonicity of new OBDD-variants and provides a platform for the implementation and comparison of OBDD-variants. Furthermore, we introduce a new multi-valued OBDD-variant, called normalized algebraic decision diagram, which supports min/max-operations and turns out to be very useful for, e.g., integer linear programming and model checking probabilistic systems. Finally, we explain the main features of an implementation of a newly developed BDD-package, called JINC, which relies on our generic notion of weighted decision diagrams, and realizes various synthesis algorithms, reordering techniques and transformation algorithms for boolean and multi-terminal OBDDs, with or without edge-attributes, and their zero-suppressed variants.     

Minimization Problems for Parity OBDDs

Parity Ordered Binary Decision Diagrams (⊕OBDDs) are a data structure for boolean functions that extends the well-known OBDDs and reduces the representation size for several functions. Both data structures share the problem that the representation size strongly depends on the chosen variable order. For ⊕OBDDs the number of edges and thus the representation size is also influenced by the choice of the basis of the represented vector space. In this paper the hardness of some minimization problems for ⊕OBDDs is proven, namely, that there is no polynomial time approximation scheme for minimizing the number of nodes by choosing the variable order and for minimizing the number of edges, where the variable order may be changed or is fixed, unless P=NP.     

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.     

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％。     

基于符号迁移图的互模拟验证算法

符号迁移图是传值进程的一种直观而简洁的语义表示模型，该模型由Hennessy和Lin首先提出，随后又被Lin推广至带赋值的符号迁移图，本文不但定义了符号迁移图各种版本（基／符号）的强操作语义和强互模拟，提出了相互的强互模拟算法，而且通过引入符号观察图和符号同余图，给出了其弱互模拟等价和观察同余的验证算法，给出并证明了了τ－循环和τ－边消去定理，在应用任何弱互模拟观察同余验证算法之前，均可利用这些定理对所给符号迁移图进行化简。     

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.     

一种基于符号关系图的快速符号数据聚类算法

由于在实际应用中有大量的符号数据生成,符号数据聚类成为了聚类分析的一个重要研究领域。目前,已有许多符号数据聚类算法被提出,但将它们应用于大数据环境时,仍然存在计算成本高、运行速度慢等问题。文中提出了一种基于符号关系图的快速符号数据聚类算法。该算法使用符号关系图替代原始数据,缩小数据集的规模,有效地解决了这一问题。大量的实验分析显示新算法相比其他算法是有效的。