A diagrammatic approach to investigate interval relations

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Journal of Visual Languages and Computing 17 (2006) 466–502

Journal of Visual Languages & Computing www.elsevier.com/locate/jvlc

A diagrammatic approach to investigate interval relations$ Zenon Kulpa Institute of Fundamental Technological Research of the Polish Academy of Sciences, ul. S´wi˛etokrzyska 21, 00-049 Warsaw, Poland Received 9 November 2004; received in revised form 28 July 2005; accepted 24 October 2005

Abstract This paper describes several diagrammatic tools developed by the author for representing the space of intervals and especially interval relations. The basic tool is a two-dimensional, diagrammatic representation of space of intervals, called an MR-diagram. A diagrammatic notation based on it, called a W-diagram, is the main tool for representing arrangement (or Allen’s) interval relations. Other auxiliary diagrams, like conjunction and lattice diagrams, are also introduced. All these diagrammatic tools are evaluated by their application to various representational and reasoning tasks of interval relations research, producing also certain new results in the field. r 2005 Elsevier Ltd. All rights reserved. MSC: primary 65G10; 65S05; 00A35 Keywords: Interval relations; Interval algebra; Time intervals; Interval diagrams; Diagrammatic representation; Diagrammatic reasoning; Diagrammatic notation

$ 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). Tel.: +48 22 8261281. E-mail address: [email protected]. URL: http://www.ippt.gov.pl/
zkulpa.

1045-926X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jvlc.2005.10.004

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1. 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

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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 [1,11–14]. 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].

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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 [20–23], 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. 2. Real intervals For the sake of rigour and clarity, let us formally define the basic notions and notation. Let ðE; pÞ 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 ðE; pÞ is an ordered pair u ¼ ½e1 ; e2 , where e1 ; e2 2 E, called endpoints of the interval, fulfill the condition e1 pe2 . The interval is called thick if e1 oe2 ; thin (or point) interval if e1 ¼ e2 . For most purposes, point intervals can be identified with the corresponding element of the base set, i.e., ½e; e ¼ e. The beginning and end of the interval u are denoted by u and u, respectively. Thus, u ¼ ½u; u. Usually, an interval can be identified with the set of elements lying between its endpoints (including the endpoints), namely u ¼ fej u pepug. When the base set E is taken to be the set of real numbers R, the corresponding intervals are usually defined alternatively as follows: Definition 2 (Real interval). A real interval is a closed, connected and bounded subset of R. The set of all real intervals is denoted by IR and called the (real) interval space. For real intervals, the midpoint and radius of the interval are also defined, respectively, as ^ follows, see Fig. 1 (with wid u ¼ 2u): u
¼ mid u ¼ ðu þuÞ=2, u^ ¼ rad u ¼ ðu  uÞ=2.

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Fig. 1. A real interval and its basic parameters.

(a)

(b)

Fig. 2. The E-diagram for a space of intervals: the interval space marked in grey (a), and constructions for reading off midpoint and radius parameters (b).

With these parameters, another notation for real intervals, called sometimes a centred notation (introduced already by Warmus [5]) becomes possible: ^ u
þ u. ^ u
 u^ ¼ ½u
 u;

(1)

3. 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 that space. As the endpoint representation of an interval has been the most common one, it has been natural to choose the endpoints as coordinates. This has led to an interval space representation called here an E-diagram (from Endpoints diagram), see Fig. 2(a). However, this representation has several drawbacks making it inconvenient to use for more complicated interval diagrams, especially in interval arithmetic [3,13,26–28]. The set of intervals occupies an inconveniently skewed upper-left half-plane above the diagonal, with the coordinate axes spanning it diagonally. This suggests some false symmetries in the interval

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°

471

°

Fig. 3. The MR-diagram for the interval space.

space (generated by the endpoint axes), while hiding the important ones. Also, reading of other interval parameters, notably the midpoint and the radius, is awkward, see Fig. 2b. If read on the diagonal, the scale for the width and the radius is different than for the endpoint axes, while reading them on the axes requires additional projections (shown in Fig. 2b), substantially complicating the use of the diagram. Therefore, the author has tried other choices. After the first experiments with the midpoint-width coordinates [1], the midpoint-radius combination proved soon to be much better [11,12], resulting in the diagram known as MR-diagram (from Midpoint-Radius diagram).1 This choice goes in accord with several opinions to the effect that the midpointradius representation has many advantages in various applications and theoretical considerations. A graphical definition of the MR-diagram is given in Fig. 3. It uses the midpoint-radius ðm; rÞ coordinate system, with the upper half-plane above the Om axis used to represent all thick intervals. The representation of an interval u ¼ u
 u^ ¼ ½u; u is also shown. The figure defines also the diagonal lines (constant Ib-lines and ub-lines). All intervals lying on these lines have the same value of their beginning, or end, respectively. Thus, also the endpoints of an interval are conveniently represented with these diagonal lines. Actually,
u; ^ u, and u of an interval u uniquely determines the any pair of the four basic parameters u; point representing that interval as the intersection of two of the lines shown. The standard interpretation of a real interval—as a segment of the real line (here, the Om axis, cf. Fig. 1)—can also be represented, and the endpoint representation of intervals can be easily handled, contrary to the situation with the E-diagram which is awkward to use with the centred representation, see Fig. 2b. In case of the MR-diagram, all these representations and interval parameters are marked on the same axis and in the same scale. The MR-diagram of the interval space and other diagrams derived from it have found many uses in the area of the interval relations tackled in this paper [3,11,12,14], as well as in studying properties of interval arithmetic operations [3,13,28] and interval linear equations [3,26,27]. 1 In the first papers using the diagram [11,12], it was called an IS-diagram (a shorthand for Interval Space diagram); however, as all diagrams of that type are interval space diagrams, the more specific name ‘‘MRdiagram’’ has been adopted later [13].

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4. Interval relations First, let us recall the basic notions of the algebra of relations. Let X ; Y ; Z; V ; W be sets, not necessarily different. Then P  V  X , R  X  Y , and Q  Y  Z are (binary) relations; xRy means ðx; yÞ 2 R. As relations are sets, the union and intersection of any two relations are defined in a straightforward way. For relations P and Q, xðP [ QÞy()xPy or xQy, and xðP \ QÞy()xPy and xQy. The notation R  Q (shortly: RQ) is used for the composition of relations: xRQz()ð9yÞxRy and yQz.

(2)

Of course, any of the resulting relations above may be empty if no pair ðx; yÞ or ðx; zÞ fulfils the appropriate condition. Note that composition is not commutative, i.e., usually RQaQR (except in some special cases, like when one of the relations is an equality). However, composition is associative, i.e.: (3)

ðPRÞQ ¼ PðRQÞ, 1

1

The relation R such that yR x()xRy is called the inverse (or converse) of the relation R. Obviously, ðR1 Þ1 ¼ R. Also, ðRQÞ1 ¼ Q1  R1 .

(4)

Note the reversing of the order of composition above. The notation: W R ¼ fy 2 Y j w 2 W and wRyg and RW ¼ fx 2 X j w 2 W and xRwg

ð5Þ

denotes the image and coimage, respectively, of the set W under the relation R. The image of a set under R coincides with its coimage under the inverse of R, i.e., W R ¼ R1 W . When W is a singleton, fwgR and Rfwg will be simplified to wR and Rw, respectively. The following distributive laws are also useful: W ðR [ QÞ ¼ W R [ W Q; W ðR  QÞ ¼ ðW RÞQ;

ðR [ QÞW ¼ RW [ QW ,

ðR  QÞW ¼ RðQW Þ.

(6) (7)

Obviously, we have also PðR [ QÞ ¼ PR [ PQ;

ðR [ QÞP ¼ RP [ QP.

(8)

For real intervals, an interval relation is a subset of IR  IR. There is an infinite number of interval relations. In the AI research, only a finite subset of all interval relations is usually considered and used. Here they are called arrangement interval relations. 4.1. Arrangement interval relations In most of the literature on reasoning with interval relations, the term interval relations is used to denote a specific (finite) subset of such relations only. This practice may be detrimental, since there is an infinity of other interval relations that do not belong to this subset. Therefore, in a more general discussion, a more specific term is needed to avoid possible confusion. Thus, in [11,12] the term ‘‘arrangement interval relation’’ (AIR) was proposed to denote any interval relation capturing certain mutual positioning of two

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Table 1 The table of the basic interval relations (BIRs), including the new symbols proposed for them and the definition of the conjunction diagram (after [11], modified)

intervals along the real line (or time line). The acronym ‘‘AIR’’ may be also read as Allen’s interval relation, after the author who originated their investigation [8]. Definition 3 (Arrangement interval relation). An arrangement interval relation is a relation between intervals (a subset of IR  IR in our case) that can be defined by an expression using only equality and ordering relations applied to endpoints of the interval arguments, and logical connectives. There are only 8191 arrangement interval relations (not counting the empty relation) [14,29]. 4.2. Basic interval relations The subset of AIRs containing interval relations minimal under inclusion within the set of all arrangement relations contains 13 basic interval relations (BIRs), see Table 1. The relations are grouped in pairs of mutually inverse relations, except for the equality relation which is symmetric, and therefore equal to its inverse. The table also contains new graphical symbols proposed for the relations.2 The symbols have been chosen to conform with the graphical arrangement of the intervals that belong to the given relation—compare them with the diagrams in the second column of the table. The definition of the conjunction diagram representing algebraic definitions of the relations is also provided. The diagrams 2

A set of LATEX commands for producing all the new symbols used in the paper is available from the author.

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(a)

(b) Fig. 4. The image (a) and coimage (b) W-diagrams of interval relations (after [11], modified).

are used to define the basic relations in the table—as can be seen, all BIRs are defined by a single conjunction that requires at most three elementary comparisons of the endpoints.3 Arrangement relations comprise all set-theoretic unions of the basic relations.4 They can be thus specified using a set-theoretic formula, like (see Fig. 7 further on), or, as is more customary in the literature on the subject [8], as a list of the constituent relations, like . This decomposition is unique; therefore, as we have 13 BIRs, the number of AIRs given in the previous subsection is now understandable: it equals 213  1, i.e., the number of all subsets of the 13-element set of BIRs, excluding the empty set. 4.3. The W-diagram Consider an arbitrary thick interval u represented as a point in the MR-diagram. Then, for any other point representing some interval v, we may determine some basic relation R 2 BIR that holds between u and v. Now select all such points, i.e. a region defined as ðuRÞ ¼ fv 2 IRjuRvg, and label it by the relation symbol R. Repeating the above procedure of selecting and marking the regions corresponding to all basic relations R 2 BIR, we obtain the image W-diagram, shown in Fig. 4(a) and named after its characteristic shape. Thus the lines and regions in the image W-diagram constitute the images of an arbitrary thick interval u under all 13 basic interval relations. The images are labelled by the graphical symbols of the corresponding relations. As uR ¼ R1 u, the coimage diagram (Fig. 4(b) is structurally the same as the image diagram, only with all the relation labels replaced by their inverses (the inverse relations are placed in the diagram symmetrically with respect to the point u, i.e., the ‘‘ ¼ ’’ relation). Two kinds of symmetry occur here, depending on the basic relation involved: one is a central symmetry with respect to the point u for the inclusion-type relations , and the other a mirror reflection (with respect to the vertical line going through the point u) for the precedencetype relations . The diagram has proved very useful in investigating the properties of AIRs. An important feature of the diagram, responsible for its usefulness, is that the structure and shape of its elements (images and coimages of some thick interval u) do not depend on the choice of the interval u, hence most of the 3 Contrary to the initial claim by Freksa [9, p. 202] that all basic relations can be defined by at most two relations between the interval endpoints, see [30]. 4 Note that Allen’s remark [8, p. 513] stating that ‘‘...these 13 relationships can be used to express any relationship that can hold between two intervals’’ is not true in general, as only arrangement interval relations can be expressed in this way.

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(a) (b) Fig. 5. Constructing W-icons for the interval relation ‘‘overlapped by or same finish’’ (a), and finding possible interval arrangements from such icons (b).

properties of these images and coimages can be also safely generalized to the appropriate properties of the corresponding relations. As can be seen from the diagram, all BIRs are disjoint and cover the whole space of intervals, i.e., every pair of intervals belongs to exactly one of these relations. Strictly speaking, this holds only when one excludes the thin intervals coinciding with the endpoints u and u of an interval u. They belong to images (or coimages) of two relations ( and or and for images, and and or and for coimages, respectively), as can be seen in the diagrams. The exception, minor as it is, can be eliminated in two ways. The first way requires introduction of additional basic relations, namely separate relations between point intervals and thick intervals [10,31]. The other way is to restrict the analysis to thick intervals only. Here we adopt this approach, following Allen’s [8] argument that time points constitute a rather artificial theoretical construct, which in most practical cases can be replaced by (very) short intervals. This approach is also more uniform and simpler to analyse. The diagrams show that the basic relations fall into three classes according to the dimensionality of their images: the zero-dimensional relations (points: BIR0 ¼ f¼g), the one-dimensional ones (lines: BIR1 ¼ , and the two-dimensional ones (regions: BIR2 ¼ . The circular labels of the images in Fig. 4 are colour-coded according to the dimensionality of the appropriate relation, see also Fig. 6. The dimensionality corresponds (inversely) to the number of equality conditions in the conjunction of terms relating the interval endpoints in the definition of the relation, see Table 1. Note also that all images of BIRs are open, i.e. the regions do not contain their borders, and the lines—their endpoints.5 As AIRs are unions of BIRs, they can be represented in the W-diagram as regions consisting of the (sub)regions corresponding to their constituent basic relations, on the basis of the distributive law (6). Such a diagram may be used as a convenient icon (later called a W-icon) representing the given relation, see Fig. 5(a). White regions and lines in the icon correspond to the basic relations absent from the represented relation, while the constituent basic relations are marked in grey (for regions) or black (for lines and the central dot). The dotted diagonal lines do not denote any constituent relations—they only indicate the structure of the diagram for easier reference. For convenience, the W-icons are narrowed horizontally. A version with the left, right and top contours removed may be used to show that the diagram is actually unbounded in these directions. By convention, we will use the image W-diagram as the basis for the W-icons. It is easy to read off the possible arrangements of intervals defined by the W-icon, as shown with the onedimensional arrangements of intervals (akin to that in the second column of Table 1) below the icons in Fig. 5(b). The upper interval is the reference interval represented by the 5

After restricting the analysis to thick intervals, as explained before.

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(a)

(b)

Fig. 6. The lattice diagrams of basic interval relations LBIR , using relation symbols (a), and W-icons (b) (after [11], modified).

centre of the icon. The formulae involving the basic relation symbols and reflecting the structure of the representation are also provided. The W-icons are more informative than unstructured sequences of the basic relation symbols, as they show also the structure and properties of the arrangement relations they represent. As a result, applications of these icons to represent AIRs, their properties and operations on them are numerous, as the rest of the paper demonstrates. 4.4. The lattice diagram The regions in the W-diagram are related by the neighbourhood relationships (called ‘‘conceptual neighbourhoods’’ in [9,21]). In diagrammatic terms, the appropriate relationship is defined as follows. Definition 4 (Neighbour interval relations). Two basic interval relations are considered neighbours if their images on the W-diagram are diagonally adjacent, i.e., if there exists a pair of intervals placed along a diagonal line,6 one belonging to the image of the first relation, the other to the image of the second relation, such that the diagonal line joining these intervals is fully contained within the union of images of these two relations. By linking symbols of two basic relations with an edge when they are neighbours we obtain a graph7 (see Fig. 6(a)) which, when rotated into a vertical position (with the righthand node of the diagram at the top), can be considered as a Hasse diagram of the lattice of basic relations LBIR [17]. As in Fig. 4, the nodes of the lattice diagram in Fig. 6(a) are colour-coded according to the dimensionality of the corresponding relations. The neighbouring nodes differ in dimensionality by exactly one. The structure of the lattice diagram mirrors closely the structure of the W-diagram (compare Fig. 4(a) and Fig. 6(a)). An isomorphic dual lattice (with the lattice diagram put upside-down) corresponds to the coimage W-diagram. The nodes of the lattice diagram can be also labelled with the W-icons of the corresponding relations, as shown in Fig. 6(b). As the W-icons are more informative than sets of basic relation symbols, such combined diagrams (first introduced in [11]) have proved very useful in applications. They are especially useful to reason with other lattices of relations in which nodes are not basic relations, see e.g. lattices of ideals and filters in Fig. 21 in Section 6.1.2. 6 To be more orthodox, we should use general lozenges (see below, Section 6.1) instead of segments of diagonal lines (which are special cases of lozenges), but lines suffice here and allow for a simpler formulation. 7 Called the A-neighbourhood graph in [9], where it was defined with non-diagrammatic means.

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Fig. 7. Four example AIRs P; Q; Y, and V, represented in several ways.

In Fig. 7, four example AIRs are specified using several different representations for comparison: descriptive names (based on the idea proposed in [9]), unions of basic relations, W-icons, conjunction diagrams, and icons based on the lattice diagram (similar to those used in [9]). Having such a variety of representations is very useful in applications, as different representations reveal different aspects and properties of the represented objects. In this respect, it seems that the W-icons constitute the richest and most useful representation of arrangement relations, as their use in the sequel will demonstrate. 4.5. Operations on interval relations Set-theoretic operations on interval relations (see the beginning of Section 4) can be carried out in a straightforward way with the W-icons, as shown in Fig. 8. Arrangement relations can be treated as discrete sets of basic relations, represented in the W-diagram as appropriate sets of regions and lines (marked grey and black in the W-icons). Thus, settheoretic operations on them reduce to standard combinatorics on sets of these regions. E.g., intersection produces a set of regions that occur in both arguments, union groups all regions occurring in any of the arguments, difference leaves out those regions of the first argument that do not occur in the second, etc., as exemplified in the figure. Note the two different symmetries with respect to the centre of the W-icon used for finding the inverse of a relation in Fig. 8(c), as explained in Section 4.3. The operations can be also conducted using conjunction diagrams, but because the general procedure is rather involved for them, and hence of little practical importance, it is omitted here. However, in some cases it may be easy and instructive, see some examples later on, especially in Fig. 25. However, as the important operation of composition of relations (used in reasoning with relation networks, see Section 5) is not so straightforward to execute diagrammatically, the specific construction for performing it is explained in detail in a separate subsection.

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×Ι

(b)

(a)

−1 (c)

(d)

(e)

(f)

(g) Fig. 8. Examples of operations on relations using the W-diagram icons: the zero-argument ‘‘operations’’ of the empty relation (a) and the total relation (b); one-argument operations of inverse (c) and complement (d); and twoargument operations of intersection (e), union (f), and difference (g).

(a)

(b)

(c)

(d) Fig. 9. Diagrammatic composition of relations: the general procedure in four steps (a), composing of another two basic relations (b), basic and non-basic ones (c), and two non-basic ones (d).

4.5.1. Diagrammatic composition of interval relations The W-diagram representation can be also applied to derive an intuitive diagrammatic algorithm for producing a composition of arbitrary arrangement interval relations, not only the basic ones. The algorithm is based on the distributive law (7). The procedure is illustrated in detail with the example in Fig. 9(a), in four steps as indicated in the figure. (i) In diagrammatic terms, the expression means that in the image W-diagram centred at the interval u, the interval o must lie somewhere within the region

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representing the image , while the expression means that in the image W-diagram centred at the interval o, the interval v must lie somewhere within the region representing the image , as shown in the step (i) in Fig. 9(a). (ii) Hence, to obtain the image u we must first combine both diagrams in such a way that the diagram showing the expression superimposed on the diagram for is centred around the interval o. (iii) Now we must adjust the size of the second diagram, without moving the point o, so that the Om axes of them coincide. If necessary, the regions representing the second relation that extend to infinity should be also extended accordingly (see the example in Fig. 9(d)). We may then omit the superfluous lines from the second diagram, leaving only the image . These regions of the first diagram that are intersected by the image of the relation given by the second diagram define now the possible basic relations that may hold between u and v. We mark them by grey-filled circles here. (iv) Now it suffices to colour the marked regions accordingly to obtain the image Wdiagram of the composition, in this case the relation , or in the list notation. In Fig. 9(b), another example of composing two basic relations is shown, this time without indicating explicitly the intervals u, o and v. Note in step (iii) that the region representing the image of the relation does not contain its borders, hence it does not intersect the regions corresponding to relations , ¼ , and . In Fig. 9(c), the second relation is not a basic one. This does not cause any problems, as shown in the figure. Note that the central point (representing the relation ¼ ) is missing in the diagonal line representing the second relation. Therefore, the latter line does not have any common point with the line representing the first relation , whence that line does not appear in the composition. The last example in Fig. 9(d) shows the situation when both relations are not basic. In this case it is convenient to use another distributive law, namely (6), to decompose the first relation into its two constituents, compose them separately with the second relation, and then take the union of the results. The first constituent is the same as in Fig. 9(c), hence the first composition can use the results of that example, as indicated by the diagonal arrow pointing to the previous row. Composing the second component with the second relation is straightforward. Note how in step (iii) the line representing the second relation has been extended ‘‘to infinity’’ (actually, to the top side of the W-diagram). In the last column, the union of the two resulting W-diagrams produces the composition result. Note also that when any of the constituent relations contains the ‘‘ ¼ ’’ relation, its composition with the other is simply the latter relation, whence the step-by-step process described above would not be needed in that case. The process described above is quite easy (at least for the visual-minded people) and with some practice can be done in memory for any arrangement interval relations. The most tricky point here seems to be to remember that the regions and lines representing the basic relations are open, i.e. regions do not contain their borders, and lines their endpoints. Failing to properly take that into account may produce superfluous component relations in the result. With the above procedure in place, it is easy to compile the graphical version of the composition table of the basic relations, shown in Table 2. For the basic relations, both Allen’s and new graphical symbols of the relations are shown. The table seems to look much more informative than analogous tables based on other representations (cf. [8,9]). It

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Table 2 The composition table for basic relations using W-icon representation

The equality relation ‘‘ ¼ ’’ has been omitted for brevity.

provides many useful intuitions about the structure of the process of constraint propagation through relation networks (see Section 5). The whole table (including the basic relations labelling its rows and columns) is inverse symmetric, i.e., the relations lying symmetrically with respect to the main diagonal (which goes here from the lower left to the upper right) are mutual inverses. Not surprisingly, the main diagonal groups symmetric (self-inverse) relations. This property of the table is a consequence of formula (4).

5. 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

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Fig. 10. Allen’s example of a network consistency problem: what are the possible relations between L and R? (a), is this system of relations consistent? (b).

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: S: the time of touching the switch, L: the time the light was on, R: the time John was in the room, we can capture the information from the story by the following relational expressions: (a) ‘‘S overlaps or meets L,’’ or with the formula: . (b) ‘‘S is before, meets, is met by, or after R,’’ or with the formula: . This information can be directly represented with the help of W-icons, as in Fig. 10(a). Note that touching the switch is, in accordance with the remarks in Section 4.3, considered as a (possibly short) interval, not a time point, with the light assumed to go on either within that interval or at its end at the latest (in line with the behaviour of real electrical switches). The diagrammatic rendering of the information from the story can be also easily derived directly from it (after assigning the symbols S, L, and R to the appropriate intervals), without listing the relations symbolically. With such a network of relations, one can state various problems, especially [29,32,33]: Consistent scenarios. Is the given network consistent, i.e., is it possible to assign to every edge a single basic relation selected from among the relations assigned to it so that the resulting constraints on the constituent intervals can be realized in the world? If so, find any (or all) such assignment(s). Minimal labelling. What relations may hold between the given intervals (nodes in the network), e.g., between L and R in Fig. 10(a), given the constraints imposed by other parts of the network? Find the set of minimal (in the sense of the number of constituent basic relations) assignments of relations to edges of the network that covers all possibilities allowed by the initial labelling (i.e., that generates the same set of consistent scenarios). Relations that can hold between the intervals connected by an edge of the network are called feasible relations for that edge. To answer such questions, we must be able to find the third relation in situations like in Fig. 10(a) on the basis of the two relations given in the network. As the algebra of relations points out, see (2), the relations that may hold between two given intervals on the basis of the relations holding between them and some third interval are provided by the operation

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(a)

(b)

(c)

(f)

(d)

(e) Fig. 11. Subsequent steps of answering questions stated in Fig. 10 (see text).

of composition of relations, see Section 4.5.1. Using that operation, we may proceed to answer the questions posed in Fig. 10. The step-by-step process of analysis of our simple network of relations is shown in Fig. 11. In that figure, the relations (represented by W-icons) are labelled for easy reference. First, in order to obtain the relation (denoted as R in Fig. 11 (a)) binding the intervals L and R we should compose the relation between L and S with the relation S between S and R. However, we are not given the relation between L and S, only the relation L between S and L. To obtain the former, we must therefore find the inverse relation L1 . As has been explained in Section 4.5, Fig. 8, inverse relations lay in the W-diagram symmetrically with respect to its centre, so the inverse is easy to obtain diagrammatically (see Fig. 11(b)). Now we can find the required composition R ¼ L1  S, either directly or using the composition table, as explained in Section 4.5.1. Assume now that another piece of evidence has arrived, in the form of a continuation of the story that has been started by the sentence (A) at the beginning of this section: (B) But John was in the room later while the light went out. This piece of narrative can be more formally rendered as: (c) ‘‘L overlaps, starts, or is during R,’’ or with the formula: Hence it can be represented as the relation R0 in Fig. 11(c) (cf. Fig. 10(b)). As we see, this information is markedly different than the relation R obtained from the composition of relations L1 and S. First, it restricts considerably the set of possibilities allowed by R, but then allows the relation which is forbidden by R. To combine this new piece of evidence with that provided by the network, we thus have to intersect the relation R0 with R to obtain R00 ¼ R0 \ R, as shown in Fig. 11(c). Incorporating R00 into the network possibly restricts other relations in the network as well. To take that into account, we should now produce the composition of this new constraint R00 with L, obtaining a new relation between S and R, namely S0 ¼ L  R00 , see Fig. 11(d). Again, we must check that against the old data on the relationship between S and R. As shown in Fig. 11(e), intersecting S

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Fig. 12. Eight possible assignments of basic relations to the edges of the network in Fig. 11 (f): four inconsistent (a) and four consistent ones (b). Note that in each case the first row represents two assignments.

and S0 produces S00 , which replaces the old S in the network, as shown in Fig. 11(f). The reader may check as an exercise that the alternative route of taking the composition S  ðR00 Þ1 to restrict L does not lead to a refinement of the network, because S  ðR00 Þ1 \ L ¼ L. As all the relations in the network are non-empty, the network is consistent (but see the next example). The reasoning in this example produces finally the following reduced set of possible relations between the events following from the sentences (A) and (B): (d) , i.e. ‘‘Light started during or at the end of touching the switch.’’ (e) S R, i.e. ‘‘John appeared in the room at the end of touching the switch or after that.’’ (f) L R, i.e. ‘‘John appeared in the room at the beginning of, or during the period of the light being on, and remained till the light went out.’’ It is easy to verify (by finding the compositions of all three pairs of relations in the final diagram in Fig. 11(f) and comparing them with the third relation)8 that the final labelling of the edges cannot be further simplified in this way. This does not mean, however, that all possible eight assignments of basic relations to the edges of the network are realizable. Actually, four of them are not realizable, as is shown by composing the appropriate relations in Fig. 12(a). Despite that, the realizable configurations contain all the basic components of the labelling relations (see Fig. 12(b)), hence the labelling is minimal. The sentence ‘‘As all the relations in the network are nonempty, the network is consistent’’ concluding the procedure shown in Fig. 11 is true only for such simple three-node networks. For larger networks, the consistency of every three-node subnetwork does not guarantee the global consistency of the network, hence also the minimality of the labelling, as shown by another Allen’s example [8, Section 4.3]. In Fig. 13, the four-node network representing feasible relations between four intervals A; B; C, and D is shown. The results of composing pairs of relations selected from every three-node subnetwork show their (local, or three-node) consistency. However, there is no consistent assignment of single basic relations to the edges of the whole network. To show that, let us start with assigning the relation during ð Þ to the edge A ! B (Fig. 14(a)). Composing its inverse with the relation A ! C, we obtain that only the assignment of contains ð Þ to the edge B ! C is feasible (Fig. 14(b)). In a similar way, we are forced to assign the relation meets ð Þ to 8

Due to formula (4), it suffices to consider compositions in one direction only for each edge of the network.

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→ (a)

(d)

→ (a)

(b)

(c)

(c)

(b) (d)

→

−1

→ →

(d)

(d)

Fig. 13. Allen’s locally consistent four-node network example.

(c)

(b)

(a) (c) (d)

(b) (d)

(e)

(e) Fig. 14. First case (for A

(a)

B) of the (global) inconsistency proof for the network in Fig. 13.

(b)

(c)

Fig. 15. Admissible transformations of the interval network: inverse of a label (a), composition of labels (b), and intersection of labels (c).

both D ! A and D ! C (Fig. 14(c) and (d)). Next, the composition of D ! A with A ! C produces the relation starts ð Þ as the only feasible label of D ! C, which, however, contradicts the previous assignment. Similar reasoning shows inconsistency of the second possible assignment to the edge A ! B (of the relation contains ð Þ). The actual reasoning (leading to the same contradiction at the edge D ! C) is left as an exercise for the reader. Summarizing the examples, analysis of such networks proceeds by iteratively applying the following basic transformations: Inverse of the relation labelling an edge, with the corresponding reverse of the direction of the edge, see Fig. 15(a).

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Composition of the relations along a path consisting of two consecutive edges, and assignment of the resulting relation to a new edge joining the nodes at the beginning and the end of the path, see Fig. 15(b). Intersection of the relations labelling edges of the same direction joining the same two nodes and assigning the resulting relation to a new edge replacing the used ones, see Fig. 15(c). If during the process of applying the above operations the resulting label becomes an empty relation, the network is inconsistent. However, non-emptiness of all the resulting labels does not guarantee consistency in networks of more than three nodes. The process outlined diagrammatically above is called the path consistency algorithm, and constitutes a basis of various algorithms for solving the problems specified in this section. It can be codified in the form of an algorithm guaranteed to find all simplifications of the labels that are possible to obtain with the operations of Fig. 15, see [8,29]. 6. Important classes of interval relations As was shown (see e.g. [29,32–34]), 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 then took was finding useful subclasses of AIRs for which these tasks are tractable. Many such classes have been found (probably the most complete enumeration can be found in [34]). Three most important of them are convex interval relations (CIRs, Section 6.1.1), pointisable interval relations (PIRs, Section 6.1.2), and preconvex interval relations (PCIRs, Section 6.1.3). The sharp inclusions BIR  CIR  PIR  PCIR  AIR hold for these classes. It is also interesting to compare the numbers of relations in the classes (note that we have excluded the empty relation from the enumeration) and the operations on relations under which the classes are closed: BIR: 13 ð Þ1 ; \ CIR: 82 ð Þ1 ; \;  PIR: 187 ð Þ1 ; \;  PCIR: 868 ð Þ1 ; \;  AIR: 8191 ð Þ1 ; not; \; [; n;  Before proceeding to the description of the above classes, we must introduce some new notions concerned with the metric of the interval space IR. 6.1. Ordering relations and lozenges Real numbers (or time points) are totally ordered (by the p relation). The real interval space IR can be ordered in several ways. One way is provided by the basic interval relations and . However, these relations hold only for disjoint intervals which is too restrictive in many situations. Other ordering relations can be derived from the notion of the 9

‘‘Most probably’’ because the hypothesis that ‘‘NP-complete’’ equals ‘‘of exponential complexity’’ has not been proven formally yet.

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Fig. 16. Examples of lozenges (a) with one-dimensional representation included for comparison (b) (after [11], modified).

in-between relation on intervals. Intuitively, for two intervals u and v, another interval, say x, lies in-between them when its endpoints lie between corresponding endpoints of u and v. More formally: Definition 5 (In-between interval relation). An interval x is said to lie in-between u and v when minfu; vgp x p maxfu; vg and minfu; vgpxp maxfu; vg, i.e., the endpoints of the interval x lie between the corresponding endpoints of u and v. The set of intervals that lie in-between some given intervals is called here a lozenge [11,12] (or metaregion) defined by these intervals. Definition 6 (Lozenge). A lozenge (metaregion) hhu; vii, defined by a pair od intervals ðu; vÞ, is the set of intervals which lie in-between the intervals u and v, i.e. hhu; vii ¼ fxj minfu; vgp x p maxfu; vg and minfu; vgpxp maxfu; vgg. A lozenge does not depend on the order of its arguments, that is, hhu; vii ¼ hhv; uii. Some examples of lozenges are shown in Fig. 16(a). Closer inspection of the diagram in Fig. 16(a) and its comparison with the W-diagram reveals that a lozenge is an intersection of images of two arrangement interval relations which impose certain ordering on the space of intervals. One is the familiar set inclusion relation  (and its inverse ) applied to intervals treated as sets (cf. relation P in Fig. 7), and the other, called here precedence relation and denoted by % (and its inverse k), is in a sense ‘‘orthogonal’’ to inclusion. Fig. 17 provides images of these ordering relations (both inclusion and precedence as well as their inverses), together with their definitions both in terms of relations between interval endpoints and as unions of basic interval relations. The figure also illustrates arithmetic definitions of these relations, both in endpoint and midpoint-radius coordinates. In formulae:
v  u; ^ u  v() v p u and upv()j
v  ujp^ u v()v  u:

(9)

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Fig. 17. Interval ordering relations of inclusion (a) and precedence (b) types: images in an MR-diagram and definitions using conjunction diagrams and unions of BIRs.

^ v
 u;
u%v() u p v and upv()j^v  ujp ukv()v%u:

(10)

It is easy to show that these relations fulfill the conditions for partial order relations (reflexivity, antisymmetry, and transitivity) within the space IR. The diagonal border lines belong to both adjoining regions. From the diagrams it follows that when two intervals u and v do not lie on the same diagonal line, exactly one of the four ordering relations f; ; %; kg must hold between them. For different intervals arranged on the same diagonal line, exactly two relations hold—one of the inclusion type, one of the precedence type. Obviously, all four relations hold when u ¼ v. There is a close correspondence between the in-between relation and the ordering relations. Namely, when uwv for a certain ordering relation w 2 f; ; %; kg, every interval x for which uwxwv lies in-between u and v. The reverse is also true. Figs. 16 and 17 demonstrate that diagrammatically (cf. also Definition 6). It follows from the diagrams that whenever uwv, the whole lozenge hhu; vii (containing all x’s lying in-between u and v) lies both within the image ðuwÞ and within the coimage ðwvÞ, that is, for all x 2 hhu; vii we have uwxwv and similarly for the reverse implication. As one of these ordering relations must hold between any two intervals u and v, for every two distinct intervals there are intervals lying in-between them. Because the interval space IR is partially ordered by the above relations, it is possible to define, according to Definition 1, intervals within the space IR i.e., intervals of intervals. Such intervals were indeed investigated under the name of twins [15], or metaintervals. There are two kinds of metaintervals— intervals with base sets ðIR; Þ or ðIR; %Þ, respectively.10 Therefore, lozenge can be also defined as the set of intervals belonging to some metainterval. As we can easily see in Fig. 16(a), some lozenges can be defined using either the inclusion relation or the precedence relation (usually using two different pairs of intervals, like hha; bii ¼ hhc; dii, except for thin lozenges like hhg; hii), while other lozenges 10

One may consider metaintervals in the dual base sets ðIR; Þ or ðIR; kÞ, but they do not introduce any new insights here, being the same objects taken in reverse.

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Fig. 18. Examples of two non-convex sets of intervals (after [11], with permission).

can be defined only by the precedence relation (using a single pair of intervals, like hhe; f ii). The latter are those ‘‘cut off’’ by the Om axis. Lozenges of the first kind afford a onedimensional representation, as intervals with uncertain endpoints. That does not work well, however, for the second kind of lozenges, see Fig. 16(b). 6.1.1. Convex interval relations Definition 7 (Convex interval sets: algebraic). A set S of intervals is said to be convex if, for every pair of intervals u; v 2 S, also w 2 S for every interval w lying between u and v. It is easy to formulate the definition in diagrammatic terms too, cf. Definitions 5 and 6. Definition 8 (Convex interval sets: diagrammatic). A set S of intervals is said to be convex if any lozenge with opposite corners in S is fully contained in S. It is easy to show that intersection of convex sets is also convex, while this property does not hold for the set union. As lozenges used to define convex interval sets are different objects than line segments used for that purpose in ordinary (linear) spaces, shapes of convex (and non-convex) interval sets differ from those in the more ordinary spaces. In Fig. 18, two examples of some non-convex interval sets are shown. Note that under the ordinary definition of convexity using line segments, both these sets would be convex. When looking at the examples in Fig. 7, one may notice that the relations Y and V have a different ‘‘look’’ than P and Q: the former seem more complicated, their definitions require logical formulae that contain negation or are disjunctive, and their W-diagrams and lattice diagrams look differently too—one may say, they look less compact. The observation is indeed significant: the P and Q relations belong to the important subclass of convex relations,11 whereas the relations Y and V are not convex. The convexity of arrangement interval relations can be defined as: Definition 9 (Convex interval relations). An arrangement interval relation is said to be convex if the image (and coimage) of any single interval under this relation is a convex interval set. The most common characterization of CIRs is the term characterization: ½Tc  An arrangement interval relation is in the class CIR if and only if it can be defined as a conjunction of simple terms involving only relations in the set fp; X; o; 4; ¼g defined in 11

Called also continuous endpoint relations, see e.g. [33].

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(a)

(b) Fig. 19. Relations defined by single-element conjunctions (a), and decomposition of example convex relations (b).

the base set between interval endpoints, (i.e., the order relation, its inverse and their negations, plus equality) [29]. There are two basic diagrammatic characterizations of CIRs, using the W-diagram and L-diagram representations. An arrangement interval relation is in the class CIR if and only if: ½Wc  Its image (or coimage) in the W-diagram includes together with any two intervals also the lozenge defined by these intervals [11,12,17], or: [Lc ] It is a union of all relations belonging to some interval (including thin intervals) over a lattice LBIR of basic interval relations [12,17,20]. The ½Wc  characterization is a rephrased (into diagrammatic terms) form of Definition 9. Equivalence of these characterizations (and yet another one, called primary characterization, based directly on the in-between relation) was proven with the help of diagrammatic representations in [3,12]. Part of the proof leads to another characterization of convex relations, namely the conjunction characterization, which may be also considered as a diagrammatic analogue of the ½Tc  characterization: ½Cc  An arrangement interval relation (other than the total relation ‘‘?’’) is in the class CIR if and only if it can be obtained as an intersection of relations from the set of singleelement conjunction relations [3]. All the 20 relations representable by single-element conjunctions are shown in Fig. 19(a). The set contains four basic relations. In Fig. 19(b), our two exemplary convex relations P and Q are represented as intersections of such relations. At most three relations are needed to represent any convex relation (except the total relation) in this way. The Q relation is an example of a relation requiring three single-element conjunction relations to define, see Fig. 19(b). Other such relations are the two basic relations and , see their definitions in Table 1. The relations in Fig. 19(a) are grouped into nine pairs of mutually inverse

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Fig. 20. Counting the number of CIRs as a sum of the number of intervals starting with the given node of the LBIR lattice.

relations, except the first two relations which are symmetric. Thus, if in the representation of convex relations we were allowed to use also inverse besides intersection, only 11 constituent relations would suffice, see [12]. Using the [Lc ] characterization, one may represent every convex relation as a union of elements of an interval over the LBIR lattice. For example, the convex relations in Fig. 7 can be written as: P ¼ [ and Q ¼ [ . The total relation is also convex, and its defining interval is obviously the whole lattice: ? ¼ [ . Decompositions of convex relations given by the characterizations [Lc ] and [Cc ] are unique. The uniqueness of the lattice interval characterization leads to an easy method of counting the number of CIRs. Namely, for every element of the LBIR lattice, one can easily count the number of intervals in the lattice starting from that element: it is the number of nodes lying to the right of the given node (plus that node) along the edges of the lattice. Summing up these numbers, we get the total numbers of CIRs, as shown in Fig. 20. Of a special interest are also two other subsets of convex relations. The first subset—the (principal) filters of the lattice, Fig. 21(a)—contains all relations whose intervals end with the relation , while the second subset—the (principal) ideals of the lattice, Fig. 21(b)— contains all relations whose intervals start from the relation , i.e. FB ¼ [½B; ; IB ¼ [½ ; B, for all B 2 BIR. There are 25 ¼ 2  13  1 relations in these two families, because the total relation belongs to both families: ? ¼ F4 ¼ lo . The ideals and filters constitute their own lattices, isomorphic to the lattice LBIR of basic relations, as shown in Fig. 21 [3,11,12]. These two subsets are involved in yet another ideal-filter characterization of convex relations: ½IFc  An arrangement interval relation C is in the class CIR if and only if it can be obtained as an intersection of exactly two relations—a filter and an ideal corresponding to the endpoints of its defining interval c in the lattice LBIR [3,11,12], namely: C ¼ [½c; c ¼ [ð½c;

\½

; cÞ ¼ Fc \ Ic .

The decomposition of the two example relations P and Q into intersections of filter and ideal relations is shown, using several representations, in Fig. 21(c). This decomposition is also unique. Note that the first row of Fig. 21(c) contains intersections of interval relations, while the second row contains intersections of intervals over LBIR that define the relations above.

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(a)

(b)

(c) Fig. 21. Lattices of filters (a) and ideals (b) of the LBIR lattice, and decomposition of two example convex relations into an intersection of appropriate filter and ideal relations (c).

6.1.2. Pointisable interval relations Pointisable interval relations (PIRs) constitute a wider subset of arrangement interval relations than convex relations (there is more than twice as many PIRs as CIRs), but they have the same nice properties as the latter concerning tractability of algorithms for solving networks of constraints on intervals, see e.g. [29,32]. They constitute a simple extension of the class of convex interval relations. An example of a pointisable non-convex relation is given by the Y relation in Fig. 7. To characterize PIRs, it is convenient to introduce a set of four full-line relations fF i g4i¼1 . They are called full-line relations because their images (or coimages) constitute the longest possible straight lines in the W-diagram. As such, they are all one-dimensional; they are convex as well. Representations of these four relations and their complements (which are not convex) are shown in Fig. 22. The set F ¼ of these relations consists of two basic and two non-basic relations with The relations (as well as their complements) are symmetric (as their graphical symbols indicate), while the relations (as well as their complements) constitute a pair of mutually inverse relations.

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(•)

(a)

(•)

(b) Fig. 22. The set F of four full-line relations (a) and their complements (b) (after [14], with permission).

The full-line relations constitute also the set of border relations important for defining boundaries of solution sets of linear interval equations, see [26]. The term characterization of pointisable relations is a simple extension of the term characterization of convex relations, only with the inequality added to the repertoire of admissible ordering relations: ½Tp  An interval relation is in the class PIR if and only if it can be defined as a conjunction of simple terms involving only relations in the set fp; X; o; 4; ¼; ag defined in the base set between interval endpoints, (i.e., the order relation, its inverse, equality, and their negations) [29]. The three diagrammatic characterizations listed below were first defined in [14]. The equivalence of all four characterizations is stated by the appropriate theorem proved with the help of diagrams in [3,14]. An interval relation R is in the class PIR if and only if: ½Wp1  For every two intervals u, v belonging to the image (or coimage) of the relation R, the set difference between the lozenge hhu; vii defined by these intervals and the image (respectively, coimage) of R belongs to the image (respectively, coimage) of the union [f of some subset f  F (possibly empty) of the set of four full-line relations of Fig. 22, such that the image (respectively coimage) of [f and the image (respectively coimage) of R are disjoint. ½Wp2  An image (or coimage) of R is obtained from the image (respectively, coimage) of some convex relation C by deleting from it a subset (possibly empty) of the set of four fullline relations, see Fig. 22. ½Lp1  It is the union of all relations belonging to some interval over one of the 16 sublattices Lf , obtained from the lattice LBIR of basic interval relations by deleting from it all the nodes corresponding to any subset f (including the empty subset, when we obtain simply the lattice LBIR itself) of the set F of four full-line relations, see Fig. 22a. The listing of all 16 sublattices Lf mentioned in the characterization ½Lp  above can be found in [3]. A simpler version of this characterization is obtained by interchanging the order of deleting the full-line relations and taking the interval from the lattice. It can be considered a direct translation of the characterization ½Wp2  into the context of the lattice model: ½Lp2  An interval relation is in the class PIR if and only if it is the union of all relations belonging to some interval over the lattice LBIR of basic interval relations with all the nodes

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(c)

(d)

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(e) Fig. 23. Some pointisable and non-pointisable relations explained with the characterizations ½Wp1  (a–d) and ½Wp2  (e) (see text).

corresponding to some subset f (including an empty subset) of the set F of four full-line relations deleted from the interval. An alternative formulation of this characterization was fist published by Ligozat in [20]. The representation of pointisable relations as intervals according to ½Lp1  is not unique (as intervals over different lattices can result in the same relation), hence counting PIRs in the easy way like for CIRs in Fig. 20 is not directly possible. Similar problems occur also with the ½Lp2  characterization. The characterization ½Wp1  is illustrated in Fig. 23(a–d). The first relation in Fig. 23(a) is convex, hence pointisable—the difference between lozenges mentioned in the characterization and the image of the relation is always empty. When we add the basic relation of equality to it, there are lozenges (like the one marked in Fig. 23(b), with one defining vertex in the centre of the diagram) for which the difference (the bottom-left side of the lozenge) is not empty, and is included in the full-line relation . However, the intersection of with our relation is not empty (it consists of the central point), hence the relation is not pointisable. A similar situation occurs with the relations in Fig. 23(c) and (d), only now the first relation, though pointisable, is not convex. Fig. 23(e) illustrates the ½Wp2  characterization for the relation from Fig. 23(c). In analogy to the ½Cc  characterization of convex relations, the following characterization of pointisable relations can be also formulated: ½Cp  An arrangement interval relation (other than the total relation ‘‘?’’) is in the class PIR if and only if it can be obtained as an intersection of relations from the set of singleelement conjunction relations of Fig. 19 augmented with four single-element conjunction relations involving the inequality relation ‘‘a’’, given in Fig. 22(b) [3]. Similarly as for convex relations, when inversion is admissible in the decomposition formula, only three additional relations are needed (see [14]) making the total of 14 relations sufficient to construct all PIRs in this way. An example is shown in Fig. 24; as many as four constituent relations may be necessary in this case. This decomposition is unique, like for convex relations. The ideal-filter characterization ½IFc  of convex relations has its analogue here as well— but now the decomposition is carried out within one of the 16 sublattices using ideals and filters defined for this sublattice: ½IFp  An arrangement interval relation P is in the class PIR if and only if it can be obtained as an intersection of exactly two relations—a filter and an ideal corresponding to

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Fig. 24. A pointisable relation decomposed into an intersection of four one-element conjunctions involving two full-line relations.

the endpoints of its defining interval p in one of the 16 sublattices Lf of the lattice LBIR , namely: P ¼ [½p; p ¼ [ð½p;

\½

; pÞ ¼ Ff ;p \ If ;p ,

where Ff ;p is some filter and If ;p an ideal defined in the sublattice Lf . Obvious examples are omitted here for brevity. Like with ½Lp1 , and for the same reasons, the decomposition given above is not unique. It was initially claimed that the path-consistency algorithm suffices for finding a minimal labelling within this class of relations. However, a counter-example was found in [29, Section 3.2.1]. Hence, pointisable relations have a bit less favourable properties than convex relations with respect to the minimal labelling problem. Nevertheless, also here the path-consistency algorithm is sufficient to decide consistency of the network. 6.1.3. Pre-convex interval relations Pre-convex interval relations (PCIRs) were introduced in [33] (where the set of PCIRs was called the ORD-Horn subclass). It was also shown there that it is a maximal set of relations containing all basic relations in which the consistency and minimal labelling problems are tractable. As the PIR set is a (proper) subset of PCIR, the standard threenode path-consistency algorithm is not guaranteed to produce a minimal labelling for networks with PCIRs, like for PIRs. However, this algorithm is still sufficient to decide consistency. The term characterization provided in [33] is involved and hard to check. In [22] Ligozat provided a simpler characterization, based on convex relations, and renamed correspondingly this class as ‘‘pre-convex relations’’: ½TCpc  An arrangement interval relation is in the class PCIR if and only if its topological closure is a convex relation. In this context, topological closure is understood as adding to the relation all basic relations that constitute borders of the basic relations comprising the given relation. The relations that may serve as borders of other relations comprise all zero- and onedimensional basic relations, that is, the set BIR0;1 ¼ BIR0 [ BIR1 ¼ . Diagrammatic characterizations of the pre-convex class are thus straightforward, namely an interval relation R is in the class PCIR if and only if: ½Wpc1  For every two intervals u, v belonging to the image (or coimage) of the relation R, the set difference between the lozenge hhu; vii defined by these intervals and the image (respectively, coimage) of R belongs to the image (respectively, coimage) of the union of some subset j (possibly empty) of the set BIR0;1 of zero- and one-dimensional basic relations.

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(a)

(b) Fig. 25. Pre-convex relations which cannot be represented by a single conjunction: diagrammatic derivation of a disjunctive formula for the example relation (a), and for the Nebel–Bu¨rckert relation from [33] (b).

½Wpc2  The image (or coimage) of R is obtained from the image (respectively, coimage) of some convex relation C by deleting from it a subset j (possibly empty) of the set BIR0;1 of zero- and one-dimensional basic relations. ½Lpc1  It is the union of all relations belonging to some interval over one of the 128 sublattices Lj , obtained from the lattice LBIR of basic interval relations by deleting from it all the nodes corresponding to any subset j (including the empty subset, when we obtain simply the lattice LBIR itself) of the set BIR0;1 of zero- and one-dimensional basic relations. ½Lpc2  It is the union of all relations belonging to some interval over the lattice LBIR of basic interval relations with all the nodes corresponding to some subset (including the empty subset) of the set BIR0;1 of zero- and one-dimensional basic relations deleted from the interval. The above characterizations are very similar to those for pointisable relations, only now the relations that can be deleted (comprising the set BIR0;1 ) are the individual basic relations being components of the full-line relations used in characterizations of PIRs. Examples like that in Fig. 23 are easy to construct here, and are omitted for brevity (but see Fig. 26(b, c)). There is no decomposition characterization of pre-convex relations analogous to the ½Cp  characterization, i.e., as an intersection of several one-element conjunction relations, because some pre-convex relations cannot be represented by a single conjunction, see the examples in Fig. 25. However, a decomposition characterization analogous to the ½IFp  characterization is possible here, with filters and ideals obtained from one of the 128 sublattices mentioned in the ½Lpc1  characterization, or as filters and ideals of the LBIR

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(a)

(b)

(c)

(d)

Fig. 26. Comparison of convex (a), non-convex but pointisable (b), non-pointisable but pre-convex (c), and nonpre-convex (d) example relations.

lattice with some zero- and one-dimensional basic relations removed from them. Due to a large number of such filters and ideals, this characterization is not very practical, hence its more detailed description is omitted here. The uniqueness properties of various decompositions of PCIRs discussed above are analogous as for PIRs. A comparison of similar relations belonging to the classes considered here is shown in Fig. 26. 6.2. Quantitative reasoning with time intervals Using the original MR-diagram (Fig. 3), we can represent not only qualitative relations between intervals and their sets, but also more quantitative data as well. The simple illustration of such a possibility is given by the following ‘‘Meeting at Lunch’’ example (adapted from [18],12 see also [11,35]). Let us start with the following piece of narrative: I want to meet someone during my lunch break. I need at least 20 minutes for conversation with the person, and he/she must depart before 12:45 pm. My lunch break may not start earlier than 11:30 am, must end at l pm at the latest, and may last from half an hour to an hour. When can I arrange the meeting? The relations between sets of time intervals referred to in this text can be translated into appropriate regions in the MR-diagram, as shown in Fig. 27(a). Constructing the set of intervals that may occur during the lunch break (the larger shaded trapezium in Fig. 27(b), obtained by consulting the coimage W-diagram of Fig. 4(b)), and intersecting it with the set of possible meetings, we easily obtain the solution set (the smaller dark grey trapezium). Translating it back to a textual form, we get: The meeting may start between 11:30 am and 12:25 pm, and will end between 11:50 am and 12:45 pm, lasting from 20 minutes to an hour. 7. 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 12

Proposed by Rit as an example of the type of problems to be considered, but neither discussed nor solved in his paper [18].

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(a)

(b) Fig. 27. ‘‘Meeting at Lunch’’ problem: diagrammatic translation (a), and solution (b) (after [11], modified).

structure and the arrangement of intervals that belong to the given relation (as shown in the second column of the table). This is more informative than various alphabetic names used for them previously (see the last two columns in the table). The intended use of the symbols is mostly in textual formulae, where they contribute additional visual clues to the meaning of the formula, like many other graphical symbols for relations and operators in mathematics. They can also be used as labels in diagrams, in places where other representations of basic relations (especially the W-icons, see below) are not practical, like in Fig. 4. In these roles they have been used thoroughly in the paper, with good results, making the formulae involving interval relations much easier to comprehend ‘‘at a glance’’. The idea of these symbols works well for basic relations, but cannot in general be extended to arbitrary arrangement relations, especially those requiring disjunctive formulae to define. However, in certain cases similar symbols can be constructed for non-basic relations, like two full-line relations in Fig. 22. Also, in [3] similar symbols were devised for certain important non-arrangement relations. Conjunction diagrams. These diagrams actually represent interval relations indirectly, through depicting the structure of logical formulae (conjunctions of terms in this case) used for algebraic definition of certain arrangement interval relations, as defined in Table 1. Their advantage lies in presentation of the depicted conjunction which is visually more

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clear and uniform than possible in the cumbersome one-dimensional string format, as shown with the definition and examples in the table and elsewhere in the paper. However, the usage of these diagrams is limited, as they become cumbersome for non-pointisable relations (Sections 6.1.1, 6.1.2 and 6.1.3) that require more than one conjunction to represent them. Yet, they can be used to better explain the structure of some sets of related relations (like the ordering relations in Fig. 17, single-element conjunctions in Fig. 19, filters and ideals in Fig. 21, or full-line relations in Fig. 22), providing additional cognitive dimension to their description. Observe in these examples the interesting correspondences between the structure of the relations (as revealed by their W-icons) and the structure of their conjunction diagrams. However, in other cases such a correspondence seems obscure and non-intuitive, like in Fig. 7 (for the relation V) or in Fig. 25. For certain relations, settheoretic operations on them, especially conjunction, can be also easily performed with conjunction diagrams, see examples in Figs. 19(b), 21 (c), and 24. In general, however, procedures for performing operations on relations represented by conjunction diagrams are cumbersome and impractical, hence they have not been listed in the paper. In conclusion, conjunction diagrams remain an auxiliary tool only, useful in certain contexts involving structurally simpler relations. MR-diagram. This is the basic diagram representing all the intervals of the interval space IR as points on the (half) plane, Fig. 3. This allows us to represent sets of intervals as regions or lines on the plane, while the representation of sets of intervals is practically impossible in one-dimensional representation (Fig. 1) for any but the simplest sets of intervals (like lozenges of the first type, Fig. 16(b)). For investigation of interval relations, the most important sets are images and coimages of relations. Representing them for arrangement relations leads to the construction of the W-diagram, see Section 4.3 and below. The representation of images of ordering relations and lozenges (metaregions) in Section 6.1 is essential for diagrammatic characterizations of classes of interval relations. The advantages of using the midpoint-radius coordinates in the MR-diagram versus the endpoint coordinates (the E-diagram occurring sometimes in the literature) is discussed in Section 3. The main advantages are visual stressing of proper symmetries of the interval space and the possibility of operating uniformly with both the midpoint-radius and endpoint coordinates, as well as one-dimensional representation of segments of the Om axis, see Fig. 3. Besides being a basis for the W-diagram, the MR-diagram can be directly used to solve certain types of interval problems in a quantitative way, like in the example in Section 6.2, Fig. 27. Its numerous applications in investigation of interval arithmetic and interval linear equations are published elsewhere [3,13,26–28]. Generally, its use is indispensable and convenient whenever sets of intervals or general properties of the interval space IR should be represented and investigated. W-diagram and W-icons. The W-diagram is the main tool for diagrammatic investigation of arrangement interval relations. Obtained from the MR-diagram by marking images (or coimages) of an arbitrary interval u under all 13 basic interval relations (Fig. 4), it reveals the structure and certain properties of the space of basic relations, and in consequence also of arrangement relations which are subsets of the set of basic relations. With appropriate simplification of the diagram, arrangement relations can be conveniently represented by W-icons, as explained in Fig. 5. They show, with grey and black coloured regions and lines, not only the constituent basic relations, but also useful information about their dimensionality, shapes, and significant (for their properties) arrangement of parts (see

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e.g. the structural characterization of relation classes in Section 6, especially the comparison of classes in Fig. 26). From W-icons, one can also easily deduce the onedimensional arrangement of intervals satisfying the relation (see Fig. 5(b)). Important operations on arrangement relations (set-theoretic ones in Fig. 8, and composition in Fig. 9) can be also easily carried out with these icons, which makes it possible to use them in qualitative reasoning with relation networks, see Section 5. The special form of the W-diagram, involving the inclusion relations  and , the nonempty intersection relation [3,26] and their border relations (which happen to be the full-line relations of Fig. 22(a)), is also the basic tool for investigating and classifying solution sets of linear interval equations, as shown in [3,26]. Lattice diagram. This is an auxiliary diagrammatic construct depicting the neighbourhood structure of the space of basic interval relations, as reflected in the structure of region arrangement in the W-diagram, see Section 4.4 and Fig. 6. As has already been mentioned, Ligozat developed a similar diagram, topologically equivalent but with a different shape than the lattice diagram, on the basis of the so-called ‘‘chain calculus’’ model of basic interval relations [20]. The lattice diagram developed here depicts instead the symmetric structure of the space of relations (as dictated by the neighbourhood relation), and of certain other, isomorphic lattices of relations, like filters and ideals in Fig. 21 and their sublattices (not shown in this paper, but see [3,14]). An additional functionality of this diagram is achieved when W-icons of arrangement relations are used as nodes. The diagram is especially useful for listing different cases of special configurations of basic relations, used in equivalence proofs of various characterizations of relation classes in [3,12,14]. Enumeration of various configurations or relation classes can be also carried out with the help of this diagram, see the example in Fig. 20. The lattice diagram can be also used as a basis of icons for arrangement relations, as shown in Figs. 7 and 22. They are simpler than W-icons, but much less functional, because they retain only the neighbourhood relation between the constituent basic relations, omitting the information on the dimensionality and shape of the corresponding images. For example, it is practically impossible to deduce directly the one-dimensional arrangement of intervals fulfilling the relation from such icons or to perform diagrammatically the composition of relations with them. Hence they are little used in this paper, but see [9]. 8. 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 in [34]. An interesting question would be to link the structure of relations in these classes to their tractability properties concerning the relation networks discussed in Section 5. Another application of the discussed tools is representation and investigation of interval relations outside the class of arrangement relations, to which Allen’s classic analysis [8] is restricted. Some steps in this direction were already taken by this author in [3], but otherwise the area remains unexplored.

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An unexplored area is quantitative reasoning with sets of intervals and interval relations, as exemplified in the paper with the ‘‘Meeting at Lunch’’ problem in Section 6.2. The delineation of the class of problems that can be solved in this manner and the general methodology of solving them are the obvious targets for investigation. Some steps in this direction were made by Rit [18] in the context of task scheduling problems, but as far as this author knows, that research has not been continued any further. One should note that the target here is not so much actual solving of such problems with the MR-diagram, but rather the use of the diagram to investigate the nature of these problems in order to develop algorithms for solving them on a computer. One may then consider making the diagrammatic tools developed here an object of a separate study concerning their cognitive structure and usability properties, e.g., along the lines of Green’s cognitive dimensions methodology [36]. That may show ways to improve them, although the structure and appearance of these tools is mostly dictated by the mathematical structure of the represented objects, so that there is not much room left for purely graphical design tuning and modifications of them. And finally, of course, there is still the task of looking for new diagrammatic tools to represent and reason with other subjects and problem in the interval relations area. One of the important directions here is the development of the theory and appropriate diagrammatic tools to investigate relations in the extended algebra of directed intervals [24] (which include intervals with negative radius). Extension of the MR-diagram to accommodate these intervals is easy—the additional objects (called improper intervals) occupy the half plane below the Om axis. However, an appropriate extension of Allen’s theory of relations to this domain is not straightforward, as a preliminary analysis conducted by this author has revealed. 9. Conclusions The diagrammatic notation for interval analysis developed by the author has found already several applications in various branches of the field [3,12–14,26–28]. 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 the interval space), the lattice diagram, and various constructions involving these tools, like operations on relations (especially the diagrammatic composition of relations). In the paper all these tools have been introduced and explained, and their various applications illustrated. They involve studying properties of interval relations, qualitative reasoning with relation networks, characterization of convex, pointisable and pre-convex classes of relations, and quantitative reasoning with interval sets. The diagrammatic approach has allowed for more direct comprehension of properties and structures of abstract objects in the interval space and the interval relation space, facilitating their understanding, finding their new relationships and properties, and reasoning with the objects. It has also led to the discovery of some new phenomena, like various characterizations of the main classes of relations, especially those based on their decomposition into intersections of certain small number of simple relations. These results allow for asserting feasibility and usefulness of the developed diagrammatic tools in the research on interval relations.

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