This contribution discusses the visualization of discrete sets. With diagrams, we realize filtering set elements with particular properties, study set-theoretic operations, and exhibit set elements and their internal relationships. These techniques allow software developers to explore crucial set properties diagramatically. Modelling languages like UML (Unified Modeling Language) [5, 2], which includes the OCL (Object Constraint Language) [7, 1], allow developers to represent discrete sets of objects with so-called object diagrams. In UML tools, one can construct and manipulate such object diagrams and with this the underlying object sets. In the context of our tool USE (Uml-based Specification Environment)1, this paper studies set visualization and set property analysis by (a) OCL queries in order to represent interesting subsets of a given set, (b) graphical representation of set theoretic operations like union or intersection, and (c) associations for representing crucial relationships between set elements. The rest of this contribution is structured as follows. Section 2 studies OCL query selection. Section 3 focuses on set-theoretic operations. Section 4 treats internal relationships in sets determined by derived associations. All presented options are implemented in USE. Section 5 discusses very few related approaches. Section 6 ends the paper with concluding remarks and future work.
Figure 1 presents a set of integers in form of 16 Int objects where each object has an object identity (in the top of the object rectangle) and a unique attribute val showing the represented integer. The integers are randomly chosen from the interval 0..255 and are placed randomly in the object diagram. Such a situation, where an unknown set has to be traversed, occurs frequently in software development. Systematic placement of elements will be discussed later. The purpose of the following queries and selections is to explore this integer set and its properties with diagrams, however considering the basic layout as fixed. 1 https://sourceforge.net/projects/useocl/ by 3. The union of these two sets is graphically constructed by (a) clearing the object diagram (pushing button Hide All in the first OCL window), (b) showing the integers divisible by 2 (pushing button Evaluate with option Show activated), and (c) adding the integers divisible by 3 (pushing button Evaluate also with option Show activated in the second OCL window). The result of the diagrammatically achieved result can be checked against the third OCL expression that computes the union by means of a logical disjunction. Analogously, other set-theoretic operations like intersection or difference can be performed (by activating other options like Crop or Hide). 4
of a given integer, i.e., its predecessor and successor. Of course, other derived associations could be defined. This derived association divides the previously unconnected elements into components. In the example, there are six components. Each single component constitutes a connected subset. Unconnected components have disjoint predecessors and successors.
Up to now we have considered integer sets. Similar techniques as discussed above
can be applied however to sets of strings. Figure 5 is based on the titles of the
papers at the Diagrams 2016 conference. It shows a weighted word cloud of the
28 words appearing in at least two different titles. In our view such a word cloud
is also a diagrammatic representation of a set of strings. However, we will explore
an alternative representation revealing more information.
Analogously to establishing a structure in a set of integers through its
predecessors or successors, one can introduce an internal structure on a string set,
e.g., through the common use of the word in different titles. Figure 6 shows the
28 words (each word appearing in at least two different titles) where a connection
in form of a derived link is established, if the two words have a title in common.
These links now classify the word set basically into four different components:
(
To understand the shown derived links, the right bottom part of Fig. 6 shows how the Perception-Clutter-Diagrams triangle (the relation between “Perception”, “Diagrams” and “Clutter”) from the upper right is built. Here, all words occur in a single title.
The technical definition for the derived, reflexive association on class Word looks as follows. A link indicates that the two words have a title in common. The link goes from a lexicographically lower string (role fst) to a lexicographically higher string (role lst). association WordWord between
Word [0..*] role fst -- first Word [0..*] role lst -- last
derived = self.t.w->select(e | self.W<e.W)->asSet() end 2 https://www.cs.kent.ac.uk/people/staff/pjr/linear/, http://www.eulerdiagrams.org/inductivecircles.html
Due to space limitations we only mention very few approaches. In previous own work [3] we have discussed different options in our tool with emphasis on metamodels. Here we focus on set representations. The work in [4] discusses linear diagrams that could be derived from our objects diagrams with components. The work in [6] also applies UML diagrams focusing on ontology representation. 6
The problem discussed in this contribution has been to offer good diagrammatic representations for discrete, finite sets. We have shown how to use a software design tool to represent finite sets with UML object diagrams. Future work includes finding a general way to determine for a given set crucial internal relationships, i.e., associations, that can be applied to guide the process leading from the set to a diagram representing the set and displaying with the diagram layout meaningful relationships between set elements. Furthermore, finding a systematic way to go from the component representation to linear and Euler diagrams seems to be promising for string sets. Last but not least, larger case studies and examples should check the applicability and usefulness of the proposed techniques.