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By Eyke Hüllermeier

Making use of alternative frameworks of approximate reasoning and reasoning below uncertainty, particularly probabilistic and fuzzy set-based options, this publication develops formal versions of the above inference precept, that's basic to CBR. The case-based approximate reasoning tools therefore acquired particularly emphasize the heuristic nature of case-based inference and features of uncertainty in CBR.

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19) given by n 1 xı . 20) f : (x1 , . . , xn ) → n ı=1 31 A t-norm is called Archimedean if (x, x) < x for all 0 < x < 1. 3 The concept of similarity Still, for a t-norm it is not difficult to show that the global measure σ : (a, a ) → σ1 (a1 , a1 ) is 39 σ2 (a2 , a2 ) ... σn (an , an ) -transitive whenever the same holds true for the local measures σı (1 ≤ ı ≤ n). Simple examples of aggregation operators which have been studied extensively in the literature on fuzzy sets are the minimum and the maximum.

This type of inference represents an alternative to the indirect (model-based) approach which derives an estimation h∗ of the complete functional relationship in a first step (induction) and evaluates this estimation at the points of interest afterwards (deduction). In connection with lazy (instance-based) learning, we shall also speak about “extrapolation” of (the information coming from) observed cases. In fact, typically the known values of the function f (·) are extrapolated – in a locally limited way – in order to estimate unknown values.

That is, π arranges the attributes according to the degree of similarity. 22) can then be written as follows: n σ : (x, x ) → h(π(ı)) · (η({π(1) . . π(ı)}) − η({π(1) . . 23) ı=1 df where η(∅) = 0. 23) includes several known aggregation operators as special cases. For instance, with η being the counting measure X → 1/n · |X| we obtain the arithmetic mean. More generally, let η be the additive measure with η({ı}) = αı for all 1 ≤ ı ≤ n, where 0 ≤ αı ≤ 1 and α1 + . . + αn = 1. 23) is then given by the weighted arithmetic mean n σ : (x, x ) → αı · σı (aı , aı ).

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