Modal Logics for Qualitative Spatial Reasoning

Abstract

Spatial reasoning is essential for many AI applications. In most existing systems the representation is primarily numerical, so the information that can be handled is limited to precise quantitative data. However, for many purposes the ability to manipulate high-level qualitative spatial information in a exible way would be extremely useful. Such capabilities can be proveded by logical calculi; and indeed 1st-order theories of certain spatial relations have been given 20]. But computing inferences in 1st-order logic is generally intractable unless special (domain dependent) methods are known. 0-order modal logics provide an alternative representation which is more expressive than classical 0-order logic and yet often more amenable to automated deduction than 1st-order formalisms. These calculi are usually interpreted as propositional logics: non-logical constants are taken as denoting propositions. However, they can also be given a nominal interpretation in which the constants stand for some kind of object. I show how 0-order logics can be given a spatial interpretation: constants denote regions and logical operators correspond to operations on regions which are important for characterising spatial situations.Representing certain spatial concepts requires the introduction of modal operators, interpreted as functions generating regions related in speci c ways to those denoted by their arguments. A signi cant example is the convex-hull operator whose value is the smallest convex region containing its argument. I investigate how this this operator can be captured in a multi-modal logic.

Reasoning About Space: The Modal Way

“…( [1] for back-up to this section.) The general logic of this new language is known [4]: it is the system S4+S5, being S4 for 2, S5 for U , plus the 'bridge axiom' U P → 2P . Moreover, according to [19], we have natural extension of the McKinsey and Tarski theorem: the {2, U } modal theory of R, and indeed of every Euclidean space R n , is exactly S4+S5 plus the given connectedness axiom.…”

Qualitative Spatial Representation and Reasoning

“…The relationship between any two regions can be characterised by a 3 × 3 matrix 11 called the 9-intersection model, in which every entry in the matrix takes one of two values, denoting whether the intersection of the two point sets is empty or not; for example, the matrix in which every entry takes the non-empty value corresponds to the PO relation above. 12 Although it would seem that there are 2 9 = 512 possible matrices, after taking into account the physical reality of 2D space and some specific assumptions about the nature of regions, it turns out that the there are exactly 8 remaining matrices, which correspond to the RCC-8 relations. Note, however, that the 9-intersection model only considers one-piece regions without holes in two-dimensional space, while RCC-8 allows much more general domains.…”

“…However the 3 × 3 matrix allows more expressive sets of relations to be defined as noted below since it takes into account the relationship between the regions and its embedding space. 12 The RCC-8 relations have different names in the 9-intersection model, in fact English words such as "overlap" instead of PO.…”

“…In order to create a decidable reasoning procedure, Bennett developed encodings of the RCC-8 relations first in propositional intuitionistic logic [11] and later an advanced encoding in propositional modal logic [12]. The encoding does not reflect the full expressive power of the first order RCC-8 theory, but does enable a decision procedure to be built.…”

Order of Magnitude Qualitative Reasoning with Bidirectional Negligibility

“…For example, the Region Connection Calculus (RCC) [2,14] for managing qualitative spatial reasoning; the multimodal logics used in [3,17] to deal with qualitative spatio-temporal representations, and the use of branching temporal logics to describe the possible solutions of ordinary differential equations when we have a lack of complete information about a system [12]. On the other hand, logics dealing with order-of-magnitude reasoning have been developed in [5,6].…”