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See also the debates on the role of logic in AI in Computational Intelligence, vol 3, ... complexity, AI researchers have been developing explicit representations of.

In: Foundations of Computer Science, C Freksa, M Jantzen, R Valk (Eds.), Lecture Notes in Computer Science 1337, 379-387, Berlin: Springer-Verlag.

Spatial and Temporal Structures in Cognitive Processes

*

Christian Freksa# University of Hamburg

Abstract. The structures of space and time are identified as essential for the realization of cognitive systems. It is suggested that the omnipresence of space and time may have been responsible for neglecting these dimensions in knowledge processing in the past. The evolving interest in space and time in cognitive science and some of the current conceptions of space and time are briefly reviewed. It is argued that space and time not only structure cognitive representations and processes but also provide useful information for knowledge processing. Various ways of structuring space and time are discussed and the merits of different languages for describing space and time are addressed. In particular, qualitative and quantitative descriptions are related to local and global reference frames and crisp qualities are related to fuzzy quantities. The importance of selecting an appropriate level of interpretation for a given description is stressed. Examples of interpreting spatial and temporal object descriptions in various ways are presented.

The Ubiquity of Space and Time in Cognitive Systems Space and time are everywhere – particularly in cognitive systems and around them. This situation – as trivial as it sounds – may be responsible for the fact that the relevance of space and time to cognition has been neglected in modeling cognitive representations and processes for a long time. Perception, the origin of all cognition, takes place in spatially extended regions and requires time to be carried out; memory requires spatial extension and the processes of storage and retrieval require some time; processing perceived or recorded information takes place in space and requires time; actions carried out on the basis of computation require both space and time. From an information processing perspective, something that is everywhere tends to be not very interesting: at first glance it appears unspecific and therefore not informative. Thus, it is not surprising that many knowledge representation approaches in artificial intelligence abstracted from time and space while truth and falsity were of central interest in these approaches1. Locations – in particular memory locations – were considered equivalent to one another and the times of occurrence of events were considered arbitrary – and therefore not relevant – in many models of the world. * # 1

Support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged. [email protected] FB 18, Vogt-Kölln-Str. 30, 22527 Hamburg, Germany. See also the debates on the role of logic in AI in Computational Intelligence, vol 3, no 3, August 1987 pp 149-237 and in KI, vol 6, no 3, September 1992.

But at second glance, the situation looks quite different: each location in space and each moment in time can be considered unique, and therefore very informative. However – unique entities are not interesting from an information processing point of view, as they are unpredictable; they are too specific to be useful for generalization. Fortunately, at third glance we observe that space and time have rather regular structures. And structure means predictability. In other words, space and time bear the potential of being interpreted in very specific ways due to the specificity of their parts and of being interpreted in more general ways due to their regular structures; thus they behave in a predictable way and can be exploited by information processes in general and by cognitive processes in particular. The structures of space and time serve as reference frames for our understanding of the world. Animals and people go to familiar places for security and sovereignty; they exploit the periodicity of events to predict new situations. We describe nonperceivable abstract dimensions in terms of the concrete dimensions space and time [cf. Freksa & Habel 1990];2 in this way we can exploit our familiarity with those dimensions and convey dependencies in other domains. During the last few years, the central roles of space and time for cognitive systems have been increasingly recognized. In artificial intelligence, a great interest has developed to understand and to model structures and uses of cognitive space and time. The work in this field is carried out in cooperation with other disciplines of cognitive science in an effort to jointly solve the puzzle of space, time, and their representation and use in cognitive systems.

Space and Time in Various Disciplines of Cognitive Science Space and time have become of central interest to several branches of cognitive science. Psychologists study the cognition of perceived and imagined visual space, the cognition of large scale space, i.e. space which cannot be perceived from a single view point [cf. Lynch 1960], and the cognition of the duration of events. The relation between subjectively perceived duration and physically measured time in actual experience and in successive recollection hint at complex structures in the cognitive organization of time. Of particular interest in the cognition of space and time are questions of spatial and temporal reference frames, the relation between visual and haptic space, and the role of spatial scale for spatial cognition. Psychological experiments in spatial and temporal cognition are carried out by relating performance of human subjects in spatial and temporal tasks to models of spatial and temporal representation. In the neurosciences, spatial and temporal cognition is investigated mainly by studying the effects of neurological deficits, for example deficits in the ability to correctly order sequences of events, deficits in the cognition of personal space, deficits in the cognition of locations, and deficits in the cognition of objects [cf. Andersen 1987]. Reference systems play a decisive role here too. The research in these areas is not restricted to the study of human spatial cognition, but extends to animals as well. 2

See also: Habel & Eschenbach, Abstract structures in spatial cognition – this volume.

Linguistics studies adequate and inadequate use of language to describe space and time, for example the use of prepositions to describe and distinguish spatial and temporal arrangements of physical objects or events. Seemingly inconsistent usage of prepositions may be explained by identifying suitable reference systems [RetzSchmidt 1988]. Philosophers have been asking questions about possible structures of space and time for more than 2500 years. An increasing number of interdisciplinary treatments is published like Landau and Jackendoff’s [1993] much-discussed article ”What and where in spatial language and spatial cognition”, to mention one example. Artificial intelligence has traditionally treated space and time mostly implicitly. Realizing that this may cause problems of restricted expressiveness and excessive complexity, AI researchers have been developing explicit representations of knowledge about space and time. Some of these representations may serve as operational models of spatial and temporal cognition and raise interesting questions both to empirical and to analytical approaches to cognitive science.

Abstract and Concrete Spaces Two disciplines dealing extensively with space and time are mathematics and physics. One way of axiomatizing space in mathematics is as a structure made up of a set of points. Euclidean geometry builds a system of concepts on the basis of points, lines, and planes: distance, area, volume, and angle. In the context of spatial reasoning, Schlieder [1996] distinguishes between topological information (e.g. information about connectivity), orientation or ordering (e.g. information about convexity), and metric information (e.g. information about distances and angles). Classical physics is concerned with concrete physical space. This space can be described in terms of orthogonal components to solve problems of classical mechanics. Physical space is positively extended, and movement can take place in any spatial direction. Unlike in classical physics, common sense notions of the world generally conceive time as directed (and therefore irreversible). Physical space and its elements are related to other physical quantities in many ways: movement relates time and (spatial) distance, atomic structures relate mass and spatial extension, gravity relates weight and mass, etc. Cognitive systems appear to employ different conceptualizations of space. Central questions are: What are basic entities of the cognitive spatial structures? How are these entities related to one another? Which reference frames are employed? How are the entities and their relations cognitively processed? Which aspects of space are processed separate from others and which aspects are processed in an integrated manner? What is the role of spatial structures in generating and processing spatial metaphors?

Conceptions of Space and Time When we speak about space, we refer to notions of location, orientation, shape, size (height, width, length and their combination), connection, distance, neighborhood, etc. When we speak about time, we refer to notions of duration, precedence, concurrency, simultaneity, consequence, etc. Some of the notions have well-defined

meanings in disciplines like physics, topology, geometry, and theoretical computer science; but here we are concerned with the question how humans think and talk about them, how they represent such notions to get around in their spatio-temporal environment, how they reason successfully about the environment, and how they solve problems based upon this reasoning. In AI, these questions were first addressed in the framework of naive physics research [cf. Hayes 1978]. There is a multitude of ways in which space and time can be conceptualized, each of which rests on implicit assumptions or explicit knowledge about the physical structure of the world. We will start with a common sense picture, which could be something like: space is ‘a collection of places which stand in unchanging relative position to one another and which may or may not have objects located at them’; time is ‘an ever growing arrow along which changes take place’. Implicit in these pictures are the assumptions that the time arrow grows even when no other changes are taking place, that space is there even when there are no objects to fill it, and that spatial relations and changes can be observed and described. As these assumptions cannot be redeemed in practice, it is more reasonable to assume that objects and events constitute space or time, respectively. Another distinction concerns the question whether space or time should be modeled by infinite sets of (extensionless) points or by finite intervals (or regions). If we talk about Rocquencourt being located South-West of Rostock, it is likely that we think of two geometric points (without spatial extension) on a map of Europe. If, in a different situation, we say that you have to follow a certain road through Rocquencourt to reach a particular destination, Rocquencourt will be considered to have a spatial extension. Also, it is not clear from the outset whether a discrete, a dense, or a continuous representation of time and space may be more adequate for human cognition or for solving a given task [Habel 1994]: if we want to reason about arbitrarily small changes, a dense representation seems to be a good choice; if we want to express that two objects touch each other and we do not want anything to get in between them, a discrete representation seems preferable; if on one level of consideration a touching relation and on another level arbitrarily small changes seem appropriate, yet another structure may be required. Nevertheless, a continuous structure (e.g. R2 ) is often assumed which provides a better correspondence with models from physics.

Description in Terms of Quantities and Qualities Space and time can be described in terms of external reference values or by reference to domain-internal entities [e.g. Zimmermann 1995]. For external reference, usually standardized quantities with regular spacing (scales) are used; this is done particularly when precise and objective descriptions are desired; the described situations can be reconstructed accurately (within the tolerance of the granularity of the scale) in a different setting. In contrast, domain-internal entities usually do not provide regularly spaced reference values but only reference values which happen to be prominent in the given domain. The internal reference values define regions which correspond to sets of quantitatively neighboring external values. The system of internally defined regions is domain-specific.

Which of the two ways of representing knowledge about a physical environment is more useful for a cognitive system? In our modern world of ever-growing standardization we have learned that common reference systems and precise quantities are extremely useful for a global exchange of information. From an external perspective, the signals generated in receptor cells of (natural and artificial) perception systems also provide quantitative information to the successive processing stages. But already in the most primitive decision stages, for example in simple threshold units, rich quantitative information is reduced to comparatively coarse qualitative information, when we consider the threshold as an internal reference value. We can learn from these considerations, that information reduction and abstraction may be worthwhile at any level of processing. As long as we stay within a given context, the transition from quantitative to qualitative descriptions does not imply a loss of precision; it merely means focusing on situation-relevant distinctions. By using relevant entities from within a given environment for reference, we obtain a customized system able to capture the distinctions relevant in the given domain. Customization as information processing strategy was to be considered expensive when information processing power was centralized; but with decentralized computing, as we find in biological and in advanced technical systems, locally customized information processing may simplify computation and decision-making considerably.

Local & Global Reference Systems and Conceptual Neighborhood Significant decisions frequently are not only of local relevance; thus it must be possible to communicate them to other environments. How can we do this if we have opted for qualitative local descriptions? To answer this question, we must first decide which are the relevant aspects that have to be communicated. Do we have to communicate precise quantitative values as, for example, in international trade or do qualitative values like trends and comparisons suffice? In cognitive systems, a qualitative description of a local decision frequently will suffice to ‘get the picture’ of the situation; the specific quantities taken into account may have no particular meaning in another local context. Qualitative descriptions can convey comparisons from one context to another, provided that the general structures of the two contexts agree. If the descriptions refer to the spatio-temporal structures of two different environments, this will be the case [cf. Freksa 1980]. Now consider qualitative spatio-temporal descriptions in a given environment. As they compare one entity to another entity with respect to a certain feature dimension, they form binary (or possibly higher-order) relations like John is taller than the arch or Ed arrived after dinner was ready. In concrete situations in which descriptions serve to solve certain tasks, it only makes sense to compare given entities to specific other entities. For example, comparing the size of a person to the height of an arch is meaningful, as persons do want to pass through arches and comparing the arrival time of a person to the completion time of a meal may be meaningful, as the meal may have been prepared for consumption by that person; on the other hand, it may not make sense to compare the height of a person to the size of a shoe, or the arrival time of a person at home with the manufacturing date of some tooth paste. For this reason,

we frequently abbreviate binary spatial or temporal relations by unary relations (leaving the reference of the comparison implicit). Thus, to a person understanding the situation context, the absolute descriptions John is tall and Ed arrived late in effect may provide the same information as the previous statements in terms of explicit comparisons. As long as it is clear from the situation context, that there is only one meaningful reference object and dimension for the implicit comparison, the descriptions can be considered crisp (as the description either is fully true or fully false in the reference world). However, in more realistic settings, situations are not completely specified and it is therefore not completely clear which should be considered the single relevant reference object. In fact, usually even the producer of a description himself or herself will not be able to precisely indicate which is the unique correct reference object. For example, when I assert John is tall, I am not able to uniquely specify with respect to which reference value I consider John to be tall. Why is the description still meaningful and why is it possible to understand the meaning of such a description? In many cases, the reason that qualitative descriptions with indeterminate reference value work is that potential reference candidates provide a neighborhood of similar values, or – in terms of the terminology of qualitative reasoning – the values form a conceptual neighborhood [Freksa 1991]. Conceptual neighbors in spatial and temporal reasoning have the property that they change the result of a computation very little or not at all, in comparison to the original value. For example, in interpreting the description John is tall as an implicit comparison of the height of John with the height of other objects, it may not be critical whether I use as reference value the height of other people in the room, the average height of persons of the same category, or the median, provided that these values are in the same ballpark of values. On the other hand, if the context of the situation does not sufficiently specify which category of values provides an adequate reference for interpreting a given description, we may be in trouble. For example, if the description John is tall is generated in the context of talking about pre-school children, it may be inappropriate to interprete it with reference to the average height of the male population as a whole.

Crisp Qualities and Fuzzy Quantities We have discussed descriptions like John is tall in terms of qualitative descriptions. The theory of fuzzy sets characterizes descriptions of this type in terms of fuzzy possibility distributions [Zadeh 1978]. What is the relation between the fuzzy set interpretation and the interpretation in terms of qualitative attributes?3 The fuzzy set tall describes objects in terms of a range of (external objective) quantities. Having a range of quantities instead of a single quantity reflects the fact that the value is not precisely given with respect to the precision of the scale used. This is a simple granularity effect implicit also in crisp quantitative descriptions. For example, distance measures in full meters usually imply a range of 100 possible actual values in full centimeters or 1000 possible actual values in full millimeters, etc. The graded 3

See also: Hernández, Qualitative vs. quantitative representations of spatial distance – this volume.

membership values associated with different actual values in addition account for the fact that in many descriptions not all actual values are possible to the same degree. Nevertheless, the description in terms of quantities with graded membership in a fuzzy set is a quantitative description. The qualitative view directly accounts for the fact that the granulation of a scale in terms of meters, centimeters, or millimeters is somewhat arbitrary and does not directly reflect relevant differences in the domain described. Thus, the description John is tall can be viewed as a statement about the height category which John belongs to and whose distinction from other height categories is relevant in a certain context – for example when walking through an arch. In the qualitative view, the fact that the category tall can be instantiated by a whole range of actual height values is not of interest; the view abstracts from quantities. As fuzzy sets relate the qualitative linguistic terms to a quantitative interpretation, they provide an interface between the qualitative and the quantitative levels of description.

Levels of Interpretation We have discussed a qualitative and a quantitative perspective on spatial and temporal representations and I have argued that the qualitative view uses meaningful reference values to establish the categories for the qualitative descriptions while the quantitative view relies on pre-established categories from a standard scale, whose categories are not specifically tuned to the application domain. In this comparison, the two approaches reflected by the two views are structurally not very different: both rely on reference values – the first from the domain, the second from an external scale. As a result of the adaptation to the specific domain, qualitative categories typically are coarser than their quantitative counterparts. So what else is behind the qualitative / quantitative distinction? Let us consider again the qualitative / quantitative interface manifested in the fuzzy set formalism. A problem with the use of fuzzy sets for modeling natural language or other cognitive systems is that the interpretation of qualitative concepts is carried out through the quantitative level linked via the interface to the qualitative concept while cognitive systems seem to be able to abstract from the specific quantitative level [Freksa 1994]. For example, from X is tall and Y is much taller than X we can infer Y is very tall, independent of a specific quantitative interpretation of the category tall. How is this possible? The answer is simple. The semantic correspondence between spatial and temporal categories should be established on the abstract qualitative level of interpretation rather than on a specific quantitative level. This requires that relations between spatial and temporal categories are established directly on the qualitative level, for example in terms of topological relations or neighborhood relations. In this way, the essence of descriptions can be transmitted and evaluated more easily. An interesting aspect of transferring spatial and temporal operations from the quantitative level to the qualitative level is that fuzzy relations on the quantitative level may map into crisp relations on the qualitative level. For example, to compute the relation between the spatial categories tall and medium-sized on the quantitative

level, fuzzy set operations have to be carried out. On the other hand, on the qualitative level, we may provide partial ordering, inclusion, exclusion, overlap information, etc. to characterize the categories and to infer appropriate candidate objects matching the description. From a philosophical point of view, there is an important difference between the quantitative and the qualitative levels of interpretation. While quantitative interpretations – for example in terms of fuzzy sets – implicitly refer to a dense structure of target candidate values, qualitative interpretations directly refer to singular discrete objects which actually exist in the target domain. For this reason, certain issues relevant to fuzzy reasoning and decision making do not arise in qualitative reasoning. This concerns particularly the problem of setting appropriate threshold values for description matching.

Horizontal Competition – Vertical Subsumption Creating and interpreting object descriptions with respect to a specific set of objects calls for the use of discriminating features rather than for a precise characterization in terms of a universally applicable reference system. This is particularly true when the description is needed for object identification where the reference context is available (as opposed to object reconstruction in the absence of a reference context). When the reference context is available for generating and interpreting object descriptions, this context can be employed in two interesting ways. First of all, the context can provide the best discriminating features for distinguishing the target object from competing objects; second, the set of competing objects can provide the appropriate granulation for qualitatively describing the target object in relation to the competing objects. For example, when a description is needed to distinguish one person from one other person in a given situation, a single feature, say height, and a coarse granulation into qualitative categories, say tall and short, may suffice to unambiguously identify the target; if, on the other hand, in the interpretation of a description there are many competing objects, the description tall may be interpreted on a much finer granulation level distinguishing the categories very tall, tall, mediumsized, short, very short, which are subsumed by the coarser categories [Freksa & Barkowsky 1996].

Conclusion and Outlook I have presented a view of cognitive representations of spatial and temporal structures and I have sketched a perspective for exploiting these structures in spatial and temporal reasoning. The discussion focused on the description and interpretation of actually existing concrete situations. The general arguments should carry over to non-spatial and non-temporal domains. However, difficulties can be expected in using the ideas for describing hypothetical worlds and impossible situations, as the approach relies heavily on the reference to spatially and/or temporally representable situations. By carrying the principle of context-sensitive interpretation of the semantics of spatial and temporal categories a step further, we may get a rather different

explanation of the meaning of semantic categories: whether a given category is applicable or not may not depend so much on their fulfilling certain absolute qualifying criteria as on the availability of potentially competing categories for description.

Acknowledgments I thank Bettina Berendt, Reinhard Moratz, Thomas Barkowsky, and Ralf Röhrig for valuable comments and suggestions.

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