TIME AND SPACE IN COGNITIVE SYSTEMS

ion from location yields shape information. Abstraction from orientation yields shape information (provided there is a way of describing shape in an orientation-invariant manner). ion from orientation yields shape information (provided there is a way of describing shape in an orientation-invariant manner). Abstraction from shape yields a representation of location and possibly approximate size or maximal size (provided there is a way of describing location in the absence of shape). ion from shape yields a representation of location and possibly approximate size or maximal size (provided there is a way of describing location in the absence of shape). Abstraction from distance yields a representation of direction/orientation.ion from distance yields a representation of direction/orientation. It is worthwhile to reiterate here the central advantage of qualitative reasoning: a question like “Are A and B equal? ” as such is meaningless. It only acquires meaning if it is seen in relation to a task or problem. This relation defines the criteria necessary for the decision. Consequently, only necessary decisions are made. 3.2 Geometry: from topological to metrical relations Mathematics defines one particular, well-studied system of progressive abstraction from aspects of space [Klein 1939]. If, as is commonly assumed, geometry is the underlying model of ‘real spaces’ and also of ‘mental spaces’, we can order different levels of abstraction. Full metrical specifications are the most restrictive. Its properties are invariant with respect to a certain set of transformations: if we rotate or translate a spatial arrangement, its metrical description will remain the same. If we abstract from these properties, the resulting properties of the space are invariant with respect to a larger set of transformations. Continuing this relaxation process, we finally arrive at topology: now we can stretch and squeeze the spatial arrangement whichever way we want4, its topological relations will remain the same. In Psychology, the view that spatial cognitive development starts with topology and gradually progresses towards full metrical knowledge has been widely accepted since [Piaget & Inhelder 1956]’s 4Of course there are limits here too, but for practical purposes, they are not of too much interest. topological analysis of the development of spatial representation.5 This ‘sequence’ is also assumed to hold in adults when they learn new environments. Later studies [Lynch 1960, Siegel & White 1975] have elaborated on this view, defining the elements of more and more refined knowledge: a transition from knowledge of single landmarks (outstanding objects or places) via knowledge of routes between landmarks (specifying the topological relation of ‘connectedness’) to map-like ‘survey’ knowledge (more or less distorted metrical relations). Some approaches in AI have explicitly modelled a corresponding hierarchy of kinds of spatial knowledge, using ‘route’ and ‘survey’ knowledge (where the transition between the two forms of knowledge poses the same problems as in Psychology, however) [Kuipers & Levitt 1988]. Most approaches focus on a subset of spatial aspects. In the domain of Qualitative Spatial Reasoning (for a review, see [Freksa & Rohrig 1993]), a lot of effort has been spent on trying to find out ‘how far one can get’ with concepts that are as simple as possible. A few examples shall serve as an illustration: Randell,Cohn and Cui [Randell et al. 1992] focus on topology and derive all their relations from a single primitive ‘is connected to’. They are able to model processes like phagocytosis and exocytosis – a unicellular organism’s way of surrounding, engulfing and then digesting food particles and the expulsion of waste material from cells initiated by white blood cells, respectively. (It is obvious that for questions like these, aspects like inclusion are important, while direction and distance can usually be ignored.) Schlieder started from the question of how to describe a view (like a view from a mountain top, a tower, etc.). Obviously, the arrangement of the landmarks under consideration plays a decisive role here – ‘left’ or ‘right’ makes a difference. He is exploring how much expressive power is added to topology by knowing the arrangement ( the mathematical ‘sense’) of landmarks [Schlieder 1993]. Freksa and Zimmermann [Freksa 1992b, Freksa & Zimmermann 1992, Zimmermann & Freksa 1995] consider questions of route-planning, including finding shortcuts, finding home, etc. Here, orientation comes into play: ‘Where is the church with respect to the filling-station? ’. These examples should illustrate our central point again: for different questions, different aspects of space are required (i.e. different aspects of space can be abstracted from). ‘Intelligence’ and ‘knowledge’ are, again, all about finding the right level of abstraction in a given situation. 3.3 Conceptual neighbourhoods If space is important in a given task, it is because things that are ‘close’ have more to do with each other than things that are ‘far’. Space is an organisation principle. So if we are provided with this organisation, we can employ processes more effectively. As an example, consider planning a train journey. We can do this with the help of train timetables, which represent routes one-dimensionally. But it is far easier to do it with the 5It should be noted, however, that their notion of ‘topology’ is not exactly the same as that of Mathematics [Mandler 1988]. help of a map, which represents the same routes two-dimensionally. Now it is obvious that the two representations are equivalent (in [Palmer 78]’s sense). The difference is that the 2D representation is ‘intrinsic’ in the sense that it represents the constraints of the represented space implicitly. In the 1D representation, we have to note them and/or think of them explicitly: If our destination is north of our starting point, we have to know that considering a journey via a city south of it will (usually) not make much sense. And we as cognitive systems obviously have processes which can use this intrinsic 2D representation (cf. research on imagery). Neighbourhoods reflect this organisation: 1. If an object occupies a given region, neighbouring regions are prime candidates for being occupied by the same object. 2. If a region is occupied by a given object in a given state of affairs, only neighbouring regions are candidates for becoming occupied by that object in the next state after a (small) movement of that object. 3. When an error occurs in locating objects at a given place, neighbouring places are the prime candidates for carrying the object. For temporal events, 1. , 2. and 3. can be applied accordingly. Now this idea can be transferred to the domain of relations: certain spatial/temporal relations are conceptually closer to one another than others. Relations which differ by a minimal difference are conceptual neighbours.6 The special properties of conceptual neighbours are much like the ones of spatial neighbours. Conceptual neighbourhood allows us to order concepts of time and space. This will prove particularly useful for a flexible use of concepts, as illustrated in the next section.