A diagrammatic approach to investigate interval relations☆
Introduction
Diagrams. Diagrammatic representations and associated diagrammatic reasoning methods have recently become a field of intensive research, as they often provide more effective means for storing, using, and presenting complex information and knowledge than other representations [37]. Diagrammatic methods can also be used to represent (and reason about) various issues of interval analysis. A system of diagrammatic tools for this field of study has been developed by the author since some time [1].
Generally, a diagram is a kind of an analogical representation of knowledge, as opposed to more familiar propositional representations (like language or mathematical formulae). An analogical representation has a structure whose syntax parallels (models), to a significant extent, the semantics of the problem domain, while the structure of a propositional representation bears no direct correspondence to the semantics of the problem domain. Diagrams are structured on a two-dimensional Euclidean plane (the spatiality of diagrams) using graphical objects whose mutual spatial and graphical relations are directly interpreted as relations in the target structure. These features make the use of appropriate diagrammatic representations more productive than of propositional ones. This to a large extent follows from the direct correspondence between the conceptual space and diagrammatic space which transforms abstract problems into spatial ones, and people have extensive experience in solving spatial problems [2]. They also have an excellent apparatus of visual processing of even complex graphical information. Other advantages of these representations are discussed in extensive literature on the subject, see e.g. the summary in [3].
It is important to bear in mind that the distinction between propositional and analogical is neither absolute nor sharp. First, there are various degrees of analogicity, and second, the representation may contain elements of both propositional and analogical character, thus becoming a hybrid representation (called also heterogeneous or multimodal). This situation is ubiquitous in practice—it is very hard, if not impossible, to find examples of indisputably pure cases. In particular, in mathematical diagrams propositional components are often essential, or even indispensable. They should co-exist with diagrammatic ones, so that both complement each other, with the diagram providing a general view on the problem structure and the formulae adding precision in important details or specifying the limits of argument generalization.
Due to these features, diagrams have played an important role in science, and will be even more commonly used with the development of new computer tools to handle them. Their profound importance for science has been recently acknowledged also by philosophers of science, like Giere [4, Chapter 7].
Intervals. The field of interval analysis and computation, starting from early works, like [5], through a series of monographs like [6], [7], is already a well-established field, providing mathematical and computational tools for modelling systems with uncertainties and for full control of rounding errors in computations. Other important sources of interval research are reasoning with time intervals [8], [9] and qualitative spatial reasoning [10] in artificial intelligence, where a certain class of interval relations (later named arrangement interval relations, [11], [12]) has been extensively studied.
Contents of the paper. This paper is concerned with the use of diagrammatic methods for the study of interval relations. First, an integrated system of diagrammatic tools for representing the space of intervals in general, and interval relations in particular, as developed by the author in [1], [11], [13], is introduced. The basic tool is a two-dimensional, diagrammatic representation of the interval space, called an MR-diagram. A diagrammatic tool based on it, called a W-diagram, is the main tool for representing arrangement interval relations. Other auxiliary diagrammatic tools, like new graphical symbols for basic interval relations, a conjunction diagram, and lattice diagrams, are also introduced.
The usefulness of all these diagrammatic tools is evaluated by their application to various representational and reasoning tasks of interval relations research, including analysis of properties of, and operations on interval relations, characterization of certain important classes of such relations, and diagrammatic reasoning involving these relations. The above applications of the diagrammatic system have served as a proof of its usefulness for known problems, producing some new results as well. The new results include a direct graphical method for composition of interval relations (Section 4.5.1), introduction of the “in-between” relation and a related notion of lozenges (Section 6.1), and their use for characterization of interval relations, including some new characterizations of convex (Section 6.1.1), pointisable (Section 6.1.2) and pre-convex relations (Section 6.1.3), as well as a new graphical approach to analysis of relation networks (Section 5).
The target users of the system are mostly researchers and students in the fields of interval algebra and interval relations, including developers of reasoning systems in these fields. For the convenience of the latter, the exposition purposefully refrains from the theorem–proof style of presentation, concentrating instead on practical demonstrations of the appropriate notions and reasoning methods. Some of the relevant theorems and proofs can be found elsewhere [3], [12], [14].
Various results of the diagrammatic approach to the study of interval relations are spread across several publications by the author [1], [3], [11], [12], [14]. This paper recapitulates the main findings of these works, combining them into an integrated system. Several new, yet unpublished results obtained by the author in this area are also included, like, as mentioned above, diagrammatic composition of relations (Section 4.5.1), diagrammatic analysis of relation networks (Section 5), and new characterizations of some relation classes (Section 6).
The comparison of the various diagrammatic tools developed, and discussion of their relative merits for different applications, are also included in Section 7, with the proposed directions for further research given in Section 8.
Interval diagrams in the literature. Simple diagrams of the interval space appeared in the interval literature from the very beginning [7], [15], [16], also in the time-interval research [17], [18], albeit rather sporadically, only as informal illustrations for some concepts and properties. They have been neither systematically investigated, nor widely applied in interval research and applications. This is in marked contrast to the development of the complex number theory and analysis, where the diagrammatic notation based on the complex plane diagram (called also Argand diagram) has played an important role in the acceptance of complex numbers as legitimate mathematical objects and in the development of their theory. Even today, new capabilities of this notation are being discovered (like new diagrammatic representations for complex integration, see [19]). The systematic investigation of interval space diagrams and their prospective applications has started seemingly only with the works by the present author, see especially [1], [11], [12], [13], [14]. Hopefully, they may play a similar role in the development of the interval algebra field as the tools based on the Argand diagram played in the development of the complex number theory [19].
There are some small exceptions to the general rule of neglecting diagrams in interval research. The most important of them is probably a series of works by Ligozat, among others [20], [21], [22], [23], where a specific kind of a differently shaped lattice diagram (see Sections 4.4 and 7) was introduced and used to characterize some classes of interval relations and their properties (see Section 6). Also Kaucher in his (unpublished) dissertation [24] used a number of interval space diagrams in the midpoint-radius coordinates to illustrate the lattice properties of the extended algebra of directed intervals (see also Section 8). However, further works published by Kaucher on the subject contained no diagrams at all. The first (and only other) use of the midpoint-radius coordinates appears as a single diagram in [16].
A simple diagram of the interval space using the endpoint coordinates (E-diagrams, see Fig. 2 in Section 3) appeared already in [16] as well as in other early works, like [7]. It was also used in [18]. These works were not related to interval relations research, however. For that field, a simpler, E-diagram based version of the W-diagram idea of Section 4.3 was used in [17], [18], [21], [22] to illustrate basic relations. In [9], Freksa used icons structurally similar to those based on the lattice diagram (see Figs. 7 and 22) in his analysis and encoding of the composition table of basic relations.
Schlieder [25] proposed an enhancement of the one-dimensional notation for intervals as segments of the real axis to help in reasoning with interval relations.
Section snippets
Real intervals
For the sake of rigour and clarity, let us formally define the basic notions and notation. Let be a partially ordered set (called in the sequel a base set). Then, an interval can be generally defined as follows. Definition 1 Interval An interval over the base set is an ordered pair , where , called endpoints of the interval, fulfill the condition .
The interval is called thick if ; thin (or point) interval if . For most purposes, point intervals can be identified with the
Interval space diagram
The first step towards constructing a diagrammatic representation and reasoning system for the interval algebra is to construct an appropriate diagram capable of representing arbitrary intervals and their sets in a uniform graphical way. In other words, we need a diagram for the interval space. Because two parameters are needed to uniquely characterize an interval, the interval space is basically two-dimensional. There are many possible choices of the two parameters to be used as coordinates of
Interval relations
First, let us recall the basic notions of the algebra of relations. Let be sets, not necessarily different. Then , , and are (binary) relations; means . As relations are sets, the union and intersection of any two relations are defined in a straightforward way. For relations and , , and .
The notation (shortly: ) is used for the composition of relations:Of course, any of the resulting relations
Qualitative reasoning with interval relations
One of the most thoroughly investigated kind of problems involving interval relations is qualitative temporal reasoning using interval relation networks [8], [29], [32]. We will introduce the diagrammatic approach to the problem with a simple example first discussed by Allen [8, Section 4.2]. Let us start with the following simple story:
(A) John was not in the room when I touched the switch to turn on the light.
Denoting the time intervals involved as follows:
- :
the time of touching the switch,
- :
the
Important classes of interval relations
As was shown (see e.g. [29], [32], [33], [34]), when any AIR is allowed to label an edge in the network of relations, the problems of determining consistency and finding minimal labelling are NP-complete, hence most probably9 computationally intractable, being of exponential complexity in the number of nodes (intervals) in the network. One of the directions the research
Summary and evaluation of the diagrammatic tools
On the basis of the experience with application of the developed diagrammatic tools to various interval problems, as discussed in the paper, let us now summarize and evaluate their relative merits and ways of using them in various interval applications.
New graphical symbols. As noted in Section 4.2, the symbols listed in the fourth column of Table 1 represent the basis relations using a direct analogy between their graphical structure and the arrangement of intervals that belong to the given
Avenues for further research
Despite the development of numerous diagrammatic tools and their application to various problems involving interval relations, there are still many unexplored or only partially explored areas here. Some of them will be listed as possible avenues for further research.
First, let us point out the problem of diagrammatic characterization of other classes of arrangement relations than the three discussed in the paper and diagrammatic investigation of their properties. Some such classes can be found
Conclusions
The diagrammatic notation for interval analysis developed by the author has found already several applications in various branches of the field [3], [12], [13], [14], [26], [27], [28]. One of them is the study of interval relations. Here the diagrammatic approach has led to the development of several diagrammatic tools to represent and reason with interval relations. They include new graphical symbols for relations, the conjunction diagram, the W-diagram and W-icons (based on the MR-diagram of
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The research leading to this paper was supported by the Research Projects No. T11F 006 08 (for the years 1995–1997) and No. Nr 8 T11F 006 15 (for the years 1998–2001), while its final writing was supported in part by the project No. 5 T07F 002 25 (for the years 2003–2006), all granted by KBN (State Committee for Scientific Research).