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Representing and Reasoning with Qualitative Spatial Relations About Regions

  • Anthony G. Cohn
  • Brandon Bennett
  • John Gooday
  • Nicholas M. Gotts
Chapter

Abstract

Qualitative Reasoning (QR) has now become a mature subfield of AI as its tenth annual international workshop, several books (e.g. (Weld and de Kleer, 1990; Faltings and Struss, 1992)) and a wealth of conference and journal publications testify. QR tries to make explicit our everyday commonsense knowledge about the physical world and also the underlying abstractions used by scientists and engineers when they create models. Given this kind of knowledge and appropriate reasoning methods, a computer could make predictions and diagnoses and explain the behavior of physical systems in a qualitative manner, even when a precise quantitative description is not available or is computationally intractable. Note that a representation is not normally deemed to be qualitative by the QR community simply because it is symbolic and utilizes discrete quantity spaces but because the distinctions made in these discretizations are relevant to high-level descriptions of the system or behavior being modeled.

Keywords

Convex Hull Geographical Information System Proper Part Intended Interpretation Spatial Entity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 18.
    This name is recent and is not used in many of our earlier papers.Google Scholar
  2. 19.
    Mereology’ is a term (first used by Legniewski) to describe the formal theory of part, whole and related concepts.Google Scholar
  3. 20.
    Ladkin (1986) has investigated temporal non convex interval logics. The spatial logic we present below will also allow non-convex spatial entities.Google Scholar
  4. 21.
    This problem has already been noted in a temporal context (Galton, 1990).Google Scholar
  5. 22.
    Alternatively, non empty regular closed sets of connected Ta-spaces have been proved to be models for the RCC axiom set (Gotts, 1996a).Google Scholar
  6. 23.
    The argument sorts for space are Region and Period, respectively, while the result sort is Spatial U NULL. Period is a sort denoting temporal intervals.Google Scholar
  7. 24.
    Note that this definition of overlap ensures that connection and overlap are different: if two regions overlap then they share a common region, while this need not be the case for connecting regions, which need only `touch’.Google Scholar
  8. 25.
    Actually, sometimes by `RCC8’, we will denote the logical theory (i.e. all the axioms and the definitions of the relations) and sometimes just the set of 8 relation names without necessarily presupposing the logical theory; context should make clear which we intend.Google Scholar
  9. 26.
    Quasi, because the lack of a null region means the functions do not form a Boolean algebra.Google Scholar
  10. 27.
    For notational convenience we will sometimes write x = y rather than EQ(x, y); technically the latter is preferable, since EQ is a relation defined in terms of C rather than true logical equality. However, for readability’s sake we will ignore this distinction here.Google Scholar
  11. 28.
    An interesting question arises: what is so special about RCC8? One answer might be that it is essentially the system that arises (in 1D) if one takes Allen’s calculus and ignores the before/after ordering: the thirteen relations collapse to eight, which mirror those of RCC8. However, note that Allen’s calculus assumes that all intervals are one piece and further relationships would exist if this were not the case (Ladkin, 1986). The 4-intersection model of Egenhofer and Franzosa (1991) also gives rise to exactly eight analogous relations under certain assumptions (such as zero co-dimension). In fact (Dornheim, 1995) shows that the interpretation of the RCC8 relations is slightly more general, but not in any practically interesting sense.Google Scholar
  12. 29.
    In fact, this predicate formally requires the introduction of natural numbers; however, the usage made of SEPNUM in defining a doughnut could always be cashed out in terms of predicates that do not require these additional primitives.Google Scholar
  13. 30.
    The corresponding definition in (Gotts, 1994b) is faulty.Google Scholar
  14. 31.
    Note, however, that this task becomes almost trivial once the conv(x) primitive is introduced in Section 4.5.Google Scholar
  15. 32.
    In cases where reasoning about dimensionality becomes important, the RCC system is not very powerful. To remedy this we have proposed a new primitive INCH (x,y), whose intended interpretation is that spatial entity x includes a chunk of y, where the included chunk is of the same dimension as x. The two entities may be of differing (though uniform) dimension. Thus if x is line crossing a 2D region y, then INCH(x,y) is true, but not vice versa. It is easy to define C(x,y) in terms of INCH, but not vice versa, so the previous RCC system can be defined as a sub theory. An initial exposition of this theory can be found in (Gotts, 1996b). Interestingly, a similar proposal was subsequently made independently by (Galton, 1996).Google Scholar
  16. 33.
    Of course, this lattice allows certain kinds of reasoning involving subsumption and disjointness of relations to be performed efficiently as noted in (Randell and Cohn, 1992).Google Scholar
  17. 34.
    As as simple corollary to (Grzegorczyk, 1951) we can show that RCC8 is undecidable (Gotts, 1996d).Google Scholar
  18. 35.
    Bennett, 1994a) discusses various other uses and aspects of composition tables.Google Scholar
  19. 36.
    Note that the assumptions about what is continuous behavior are quite sophisticated here: imagine two regions, one that is two piece and has one component that is an NTPP of the other region and a second component which is DC from the other regions; thus the two regions are P0. If the component which was an NTPP disappeared (a puddle drying in the sun?), then there would be an instantaneous transition from PO to DC! However, we argue that becoming NULL is discontinuous.Google Scholar
  20. 37.
    A newer, more principled implementation based on the transition calculus is described in (Gooday and Cohn, 1996a).Google Scholar
  21. 38.
    An interesting open theoretical question is raised here: is this method for checking the logical consistency of a set of ground atoms with respect to the full first order RCC8 theory complete? We have not found any counterexample to this conjecture but equally have not been able to prove it.Google Scholar
  22. 39.
    Fig.11 reveals a subtle difficulty with our analysis of state transition. In the first transition on the second row the food particle crosses the boundary and touches the enzyme all in one step but in fact since the crossing of the boundary happens instantaneously it must precede the coming together of enzyme and food. The distinction between instantaneous and durative changes has been examined by Galton(1995c) and in Galton’s chapter in the present work.Google Scholar
  23. 40.
    As mentioned above when outlining how to define a doughnut, it is possible to describe some non-convex regions using C alone, but it is impossible to describe the holes themselves as regions. Moreover, not all kinds of concave shapes can be distinguished using C alone (for example, depressions in a surface cannot be distinguished).Google Scholar
  24. 41.
    One possible line of attack would be to introduce an alternative primitive, “region y is between regions x and z” (see Tarski’s axiomatisation of geometry which uses a point based betweenness primitive (Tarski, 1959)) and define cony in terms of this primitive. Linking this primitive to Tarski’s point based betweenness relation may provide a way to verify the completeness of the axiomatization.Google Scholar
  25. 42.
    It should be noted that these axoms are not all independent. It is quite easy to prove that axiom 28 is a consequence of axiom 33 and that axiom 30 is entailed by axiom 31; and it is probable that there are further dependencies.Google Scholar
  26. 43.
    See also their chapter in this book and (Varzi, 1996c; Varzi, 1996b).Google Scholar
  27. 44.
    It has been shown that using just the two primitives of conv(x) and C(x, y), any two 2D shapes not related by an affine transformation can be distinguished (Davis et al., 1997).Google Scholar
  28. 45.
    In his chapter of this book, Frank discusses the general question of ontologies from a consumer’s viewpoint.Google Scholar
  29. 46.
    p(U) means the power set of U.Google Scholar
  30. 47.
    Note that we use the connectives and r to emphasize that the logic is intuitionistic. 98 This explains the term entailment constraint.Google Scholar
  31. 49.
    We termed this representation the the `egg-yolk’ calculus, for obvious reasons, and will meet it again when describing an extension to RCC to handle regions with indeterminate boundaries below.Google Scholar
  32. 50.
    This system has been extended to allow for regions with holes, and relationships between regions of different dimensions, for example, see (Egenhofer et al., 1994; Egenhofer, 1994; Egenhofer and Franzosa, 1995).Google Scholar
  33. 51.
    We are sceptical about the merits of `fuzzy’ approaches to indeterminacy, believing that their use of real number indices of degrees of membership and truth are both counterintuitive and logically problematic. We have no space to argue this controversial viewpoint here; see (Elkan, 1994) and responses for arguments on both sides.Google Scholar
  34. 52.
    Note that we have addressed only the question of modelling indeterminate boundaries rather than indeterminate position.Google Scholar
  35. 53.
    We will use upper-case italic letters for variables ranging over OCregions. These are optionally crisp regions, which may be crisp or not.Google Scholar
  36. 54.
    Each cluster of Fig.24 represents one of these conceptual neighborhoods.Google Scholar
  37. 55.
    Clementini and DiFelice, 1994) have also produced a very similar analysis.Google Scholar
  38. 56.
    Asher and Vieu (1995) have provided a formal semantics for Clarke’s system.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Anthony G. Cohn
  • Brandon Bennett
  • John Gooday
  • Nicholas M. Gotts

There are no affiliations available

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