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.
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Notes
This name is recent and is not used in many of our earlier papers.
Mereology’ is a term (first used by Legniewski) to describe the formal theory of part, whole and related concepts.
Ladkin (1986) has investigated temporal non convex interval logics. The spatial logic we present below will also allow non-convex spatial entities.
This problem has already been noted in a temporal context (Galton, 1990).
Alternatively, non empty regular closed sets of connected Ta-spaces have been proved to be models for the RCC axiom set (Gotts, 1996a).
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.
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’.
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.
Quasi, because the lack of a null region means the functions do not form a Boolean algebra.
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.
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.
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.
The corresponding definition in (Gotts, 1994b) is faulty.
Note, however, that this task becomes almost trivial once the conv(x) primitive is introduced in Section 4.5.
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).
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).
As as simple corollary to (Grzegorczyk, 1951) we can show that RCC8 is undecidable (Gotts, 1996d).
Bennett, 1994a) discusses various other uses and aspects of composition tables.
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.
A newer, more principled implementation based on the transition calculus is described in (Gooday and Cohn, 1996a).
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.
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.
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).
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.
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.
See also their chapter in this book and (Varzi, 1996c; Varzi, 1996b).
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).
In his chapter of this book, Frank discusses the general question of ontologies from a consumer’s viewpoint.
p(U) means the power set of U.
Note that we use the connectives and r to emphasize that the logic is intuitionistic. 98 This explains the term entailment constraint.
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.
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).
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.
Note that we have addressed only the question of modelling indeterminate boundaries rather than indeterminate position.
We will use upper-case italic letters for variables ranging over OCregions. These are optionally crisp regions, which may be crisp or not.
Each cluster of Fig.24 represents one of these conceptual neighborhoods.
Clementini and DiFelice, 1994) have also produced a very similar analysis.
Asher and Vieu (1995) have provided a formal semantics for Clarke’s system.
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Cohn, A.G., Bennett, B., Gooday, J., Gotts, N.M. (1997). Representing and Reasoning with Qualitative Spatial Relations About Regions. In: Stock, O. (eds) Spatial and Temporal Reasoning. Springer, Dordrecht. https://doi.org/10.1007/978-0-585-28322-7_4
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