Traditional computational models of geographic phenomena offer no room for imperfection. Underlying this tradition is the simplifying assumption that reality is certain, crisp, unambiguous, independent of context, and capable of quantitative representation. This paper reports on initial work which explicitly recognises that most geographic information is intrinsically imperfect. Based on an ontology of imperfection the paper explores a formal model of imperfect geographic information using multi-valued logic. The development of Java software able to assist with a geodemographic retail site assessment application is used to illustrate the utility of a formal approach. Computer, Environment and Urban Systems v25 pp. 89-103. 2.1 Error, imprecision and vagueness Knowledge about reality is gained through observations. Observations are therefore first class objects in our account, rather than the underlying objects that are observed (Worboys, 1998a). Observations are imperfect in the sense that they can never fully or correctly reflect all aspects of reality. Imperfection is therefore the root of our ontology, as the concept refers generally to the inevitable deviations from perfection when observing reality. Imperfection can be thought of as comprising two distinct orthogonal concepts: error and imprecision. Error, or inaccuracy, concerns a lack of correlation of an observation with reality; imprecision concerns a lack of specificity in representation. Observations will usually be inaccurate and imprecise, but error and imprecision are orthogonal concepts since the level of accuracy of an observation is not implied by the level of precision, nor vice versa. Intuitively, the statement "York is in England" is at the same time more accurate and less precise than the statement "York is in Lancashire". The general definitions of accuracy and precision above correspond closely to the more specialised statistical definitions of the terms in common usage (see Drummond, 1995). Any observation of reality will be subject to imprecision: Veregin (1999) discusses some of the different causes and types of imprecision. Granularity is closely related, but not identical to imprecision. Granularity refers to the existence of clumps or grains in information, in the sense that individual elements in the grain cannot be distinguished or discerned apart. Granulation is therefore the result of distinct entities becoming indiscernible due to the imprecision in an observation. Observations or representations of coarser granularity offer less detail, for example where the clumping of information into pixels in remotely sensed images may prevent sub-pixel entities being distinguished (Fisher, 1997). Vagueness, however, is a special type of imprecision which concerns the existence of indeterminate borderline cases. "Yorkshire is in England" is not a vague statement (both Yorkshire and England have clearly defined national or international boundaries), but is an imprecise statement. Although intuitively more precise, "Yorkshire is in the East of ...

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Abstract. In spatial reasoning the qualitative description of relations between spatial regions is of practical importance and has been widely studied. Examples of such relations are that two regions may meet only at their boundaries or that one region is a proper part of another. This paper shows how systems of relations between regions can be extended from precisely known regions to approximate ones. One way of approximating regions with respect to a partition of the plane is that provided by rough set theory for approximating subsets of a set. Relations between regions approximated in this way can be described by an extension of the RCC5 system of relations for precise regions. Two techniques for extending RC-C5 are presented, and the equivalence between them is proved. A more elaborate approximation technique for regions (boundary sensitive approximation) takes account of some of the topological structure of regions. Using this technique, an extension to the RCC8 system of spatial relations is presented.