The systematic design of visual languages applied to logical reasoning

Visual languages are distinguished by a number of graphical objects and their relations, usually arranged in the two-dimensional plane. While objects and relations are syntactical containers which are used to represent some information, the question arises how to systematically treat all possible syntactical containers given the richness and complexity of the underlying geometry. This paper adopts the intersection paradigm applied in the context of spatial reasoning, which ensures the systematic identification of all conceivable well-formed diagrams. This allows the exhaustive analysis of a visual language. As an example, it is shown how this method enables a thorough understanding of the relations of the graphical elements of linear diagrams which represent monadic first-order logic. The consideration of indeterminate sets even demonstrates the effectiveness of this approach for a representation that includes a total of 512 relations.     

eulerForce: Force-directed layout for Euler diagrams

Euler diagrams use closed curves to represent sets and their relationships. They facilitate set analysis, as humans tend to perceive distinct regions when closed curves are drawn on a plane. However, current automatic methods often produce diagrams with irregular, non-smooth curves that are not easily distinguishable. Other methods restrict the shape of the curve to for instance a circle, but such methods cannot draw an Euler diagram with exactly the required curve intersections for any set relations. In this paper, we present eulerForce, as the first method to adopt a force-directed approach to improve the layout and the curves of Euler diagrams generated by current methods. The layouts are improved in quick time. Our evaluation of eulerForce indicates the benefits of a force-directed approach to generate comprehensible Euler diagrams for any set relations in relatively fast time.     

Untangling euler diagrams

In many common data analysis scenarios the data elements are logically grouped into sets. Venn and Euler style diagrams are a common visual representation of such set membership where the data elements are represented by labels or glyphs and sets are indicated by boundaries surrounding their members. Generating such diagrams automatically such that set regions do not intersect unless the corresponding sets have a non-empty intersection is a difficult problem. Further, it may be impossible in some cases if regions are required to be continuous and convex. Several approaches exist to draw such set regions using more complex shapes, however, the resulting diagrams can be difficult to interpret. In this paper we present two novel approaches for simplifying a complex collection of intersecting sets into a strict hierarchy that can be more easily automatically arranged and drawn (Figure 1). In the first approach, we use compact rectangular shapes for drawing each set, attempting to improve the readability of the set intersections. In the second approach, we avoid drawing intersecting set regions by duplicating elements belonging to multiple sets. We compared both of our techniques to the traditional non-convex region technique using five readability tasks. Our results show that the compact rectangular shapes technique was often preferred by experimental subjects even though the use of duplications dramatically improves the accuracy and performance time for most of our tasks. In addition to general set representation our techniques are also applicable to visualization of networks with intersecting clusters of nodes.     

Minkowski-Type Theorems and Least-Squares Clustering

Dissecting Euclidean d -space with the power diagram of n weighted point sites partitions a given m -point set into clusters, one cluster for each region of the diagram. In this manner, an assignment of points to sites is induced. We show the equivalence of such assignments to constrained Euclidean least-squares assignments. As a corollary, there always exists a power diagram whose regions partition a given d -dimensional m -point set into clusters of prescribed sizes, no matter where the sites are placed. Another consequence is that constrained least-squares assignments can be computed by finding suitable weights for the sites. In the plane, this takes roughly O(n ² m) time and optimal space O(m) , which improves on previous methods. We further show that a constrained least-squares assignment can be computed by solving a specially structured linear program in n+1 dimensions. This leads to an algorithm for iteratively improving the weights, based on the gradient-descent method. Besides having the obvious optimization property, least-squares assignments are shown to be useful in solving a certain transportation problem, and in finding a least-squares fitting of two point sets where translation and scaling are allowed. Finally, we extend the concept of a constrained least-squares assignment to continuous distributions of points, thereby obtaining existence results for power diagrams with prescribed region volumes. These results are related to Minkowski's theorem for convex polytopes. The aforementioned iterative method for approximating the desired power diagram applies to continuous distributions as well. May 30, 1995; revised June 25, 1996.     

Implicit GSPN reachability set generation using decision diagrams

Implicit techniques for representing and generating the reachability set of a high-level model have become quite efficient. However, such techniques are usually restricted to models whose events have equal priority. Models containing events with differing classes of priority or complex priority structure, in particular models with immediate events, have thus been required to use less-efficient explicit reachability set generation techniques. In this paper, we present an efficient implicit technique, based on multi-valued decision diagram (MDD) representations for sets of states and matrix diagram representations for next-state functions, that can handle models with complex priority structure. We adapt an efficient Kronecker-based reachability set generation algorithm to work with matrix diagrams. If the model contains immediate events, the vanishing states can be eliminated either during generation, by manipulating the matrix diagram, or after generation, by manipulating the MDD. We apply both techniques to several models and give detailed experimental results.     

Efficient false positive free set synchronization using an extended bloom filter approach

Synchronization between two sets is an important requirement for many distributed applications. A basic prerequisite is to find out which elements of set A are not in set B and vice versa. A very space efficient data structure for such membership queries that has been used a lot in networking applications is the Bloom filter. Unfortunately, the Bloom filter owes its high efficiency to the fact that there is a chance of false positives when querying the filter. This precludes the adoption of Bloom filters in applications that cannot tolerate such errors. In this paper we present an approach that augments Bloom filters with a trie-based mechanism that deterministically and efficiently finds the false positives after using the Bloom filter to synchronize two sets. We show that the added communication overhead for our approach is negligible compared to the overhead of a plain Bloom filter.     

Minimum feedback vertex sets in shuffle-based interconnection networks

Given a graph G, the problem is to construct a smallest subset S of vertices whose deletion results in an acyclic subgraph. The set S is called a minimum feedback vertex set for G.Tight upper and lower bounds on the cardinality of minimum feedback vertex sets have been previously obtained for some hypercube-like networks, such as meshes, tori, butterflies, cube-connected cycles and hypercubes. In this paper we construct minimum feedback vertex sets and determine their cardinalities in certain shuffle-based interconnection networks, such as shuffle-exchange, de Bruijn and Kautz networks.     

Supporting the reconciliation of models of object behaviour

This paper presents Reconciliation+, a method which identifies overlaps between models of software systems behaviour expressed as UML object interaction diagrams (i.e., sequence and/or collaboration diagrams), checks whether the overlapping elements of these models satisfy specific consistency rules and, in cases where they violate these rules, guides software designers in handling the detected inconsistencies. The method detects overlaps between object interaction diagrams by using a probabilistic message matching algorithm that has been developed for this purpose. The guidance to software designers on when to check for inconsistencies and how to deal with them is delivered by enacting a built-in process model that specifies the consistency rules that can be checked against overlapping models and different ways of handling violations of these rules. Reconciliation+ is supported by a toolkit. It has also been evaluated in a case study. This case study has produced positive results which are discussed in the paper.     

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.     

Online region computations for Euler diagrams with relaxed drawing conventions

Euler diagrams are an accessible and effective visualisation of data involving simple set-theoretic relationships. Efficient algorithms to quickly compute the abstract regions of an Euler diagram upon curve addition and removal have previously been developed (the single marked point approach, SMPA), but a strict set of drawing conventions (called well-formedness conditions) were enforced, meaning that some abstract diagrams are not representable as concrete diagrams. We present a new methodology (the multiple marked point approach, MMPA) enabling online region computation for Euler diagrams under the relaxation of the drawing convention that zones must be connected regions. Furthermore, we indicate how to extend the methods to deal with the relaxation of any of the drawing conventions, with the use of concurrent line segments case being of particular importance. We provide complexity analysis and compare the MMPA with the SMPA. We show that these methods are theoretically no worse than other comparators, whilst our methods apply to any case, and are likely to be faster in practise due to their online nature. The machinery developed for the concurrency case could be of use in Euler diagram drawing techniques (in the context of the Euler Graph), and in computer graphics (e.g. the development of an advanced variation of a winged edge data structure that deals with concurrency). The algorithms are presented for generic curves; specialisations such as utilising fixed geometric shapes for curves may occur in applications which can enhance capabilities for fast computations of the algorithms' input structures. We provide an implementation of these algorithms, utilising ellipses, and provide time-based experimental data for benchmarking purposes.     

基于扩展RED图的概率时间自动机可达性分析

RED图可以表示一个完整的时间自动机上的状态集,包括其连续时间部分和离散部分.在它基础上实现的模型检测工具RED,在时间自动机模型检测中表现出了优良的性能.另一方面,现有的概率时间自动机模型检测工具仍然使用不同的方法来分别表示概率时间自动机状态的连续时间和离散部分.我们在复用原始RED图的数据结构的基础上,对其做出了扩展,以令其支持概率状态的表达,同时保持其性能方面的优势.我们又为此实现了一个概率时间自动机可达性分析工具原型,并将其与两个概率模型检测工具（PRISM和Modest）就概率时间自动机可达性分析作实验对比,来评估该工具原型的性能.实验结果显示,我们的集成表示概率状态空间的方式,确实提高了概率时间自动机模型检测的时间效率和延展性.     

Random feature weights for decision tree ensemble construction

This paper proposes a method for constructing ensembles of decision trees, random feature weights (RFW). The method is similar to Random Forest, they are methods that introduce randomness in the construction method of the decision trees. In Random Forest only a random subset of attributes are considered for each node, but RFW considers all of them. The source of randomness is a weight associated with each attribute. All the nodes in a tree use the same set of random weights but different from the set of weights in other trees. So, the importance given to the attributes will be different in each tree and that will differentiate their construction. The method is compared to Bagging, Random Forest, Random-Subspaces, AdaBoost and MultiBoost, obtaining favourable results for the proposed method, especially when using noisy data sets. RFW can be combined with these methods. Generally, the combination of RFW with other method produces better results than the combined methods. Kappa-error diagrams and Kappa-error movement diagrams are used to analyse the relationship between the accuracies of the base classifiers and their diversity.     

Algorithms useful for calculating multi-component equilibria,phase diagrams and other kinds of diagrams

Phase diagrams are important in materials science. They consist of lines separating regions with different sets of stable phases and different kind of axes can be used. Many available phase diagrams are drawn directly from experimental information. However, they are closely related to the thermodynamic properties of the phases and can be calculated from thermodynamic models of the phases using the Calphad technique. This paper presents a set of algorithms to calculate equilibria and several kinds of diagrams using a very flexible set of conditions and axes. The algorithms can be applied to multi-component systems using model parameters in thermodynamic databases.     

Modelling concurrent interactions

In UML 2.0 sequence diagrams have been considerably extended but their expressiveness and semantics remains problematic in several ways. In other work we have shown how sequence diagrams combined with an OCL liveness template gives us a much richer language for inter-object behaviour specification. In this paper, we give a semantics of these enriched diagrams using labelled event structures. Further, we show how sequence diagrams can be embedded into a true-concurrent two-level logic interpreted over labelled event structures. The top level logic, called communication logic, is used to describe inter-object specification, whereas the lower level logic, called home logic, describes intra-object behaviour. An interesting consequence of using this logic relates to how state-based behaviour can be synthesised from inter-object specifications. Plans of extending the Edinburgh Concurrency Workbench in this context are discussed.     

Automated Theorem Proving in Euler Diagram Systems

Diagrammatic reasoning has the potential to be important in numerous application areas. This paper focuses on the simple, but widely used, Euler diagrams that form the basis of many more expressive logics. We have implemented a diagrammatic theorem prover, called Edith, which has access to four sound and complete sets of reasoning rules for Euler diagrams. Furthermore, for each rule set we develop a sophisticated heuristic to guide the search for a proof. This paper is about understanding how the choice of reasoning rule set affects the time taken to find proofs. Such an understanding will influence reasoning rule design in other logics. Moreover, this work specific to Euler diagrams directly benefits the many logics based on Euler diagrams. We investigate how the time taken to find a proof depends not only on the proof task but also on the reasoning system used. Our evaluation allows us to predict the best choice of reasoning system, given a proof task, in terms of time taken, and we extract a guide for defining reasoning rules for other logics in order to minimize time requirements.     

关于一般图形Voronoi图的离散构造法的研究

生成元为任意图形的一般图形Vomnoi图,由于其生成元的任意性,使得构造一般图形Voronoi图的算法均比较复杂。本文给出了在生成元边界上选取母点,利用点为生成元的Voronoi图的离散画法进行构造,从而得到一般图形Voronoi图的离散构造法。与其它算法相比,该算法的实现与生成元的形状无关,无需复杂计算,无需考虑误差控制,因而更加实用,效率也更高。实验结果表明,该算法简单,具有较高的理论价值和应用价值。     

Density Conscious Subspace Clustering for High-Dimensional Data

Instead of finding clusters in the full feature space, subspace clustering is an emergent task which aims at detecting clusters embedded in subspaces. Most of previous works in the literature are density-based approaches, where a cluster is regarded as a high-density region in a subspace. However, the identification of dense regions in previous works lacks of considering a critical problem, called "the density divergence problem” in this paper, which refers to the phenomenon that the region densities vary in different subspace cardinalities. Without considering this problem, previous works utilize a density threshold to discover the dense regions in all subspaces, which incurs the serious loss of clustering accuracy (either recall or precision of the resulting clusters) in different subspace cardinalities. To tackle the density divergence problem, in this paper, we devise a novel subspace clustering model to discover the clusters based on the relative region densities in the subspaces, where the clusters are regarded as regions whose densities are relatively high as compared to the region densities in a subspace. Based on this idea, different density thresholds are adaptively determined to discover the clusters in different subspace cardinalities. Due to the infeasibility of applying previous techniques in this novel clustering model, we also devise an innovative algorithm, referred to as DENCOS (DENsity COnscious Subspace clustering), to adopt a divide-and-conquer scheme to efficiently discover clusters satisfying different density thresholds in different subspace cardinalities. As validated by our extensive experiments on various data sets, DENCOS can discover the clusters in all subspaces with high quality, and the efficiency of DENCOS outperformes previous works.