Abstract
An analogical proportion is a quaternary relation that is to be read “a is to b as c is to d”, verifying some symmetry and permutation properties. As can be seen, it involves a pair of pairs. Such a relation is at the basis of an approach to case-based reasoning called analogical extrapolation, which consists in retrieving three cases forming an analogical proportion with the target problem in the problem space and then in finding a solution to this problem by solving an analogical equation in the solution space. This paper studies how the notion of competence of pairs of source cases can be estimated and used in order to improve extrapolation. A preprocessing of the case base associates to each case pair a competence given by two scores: the support and the confidence of the case pair, computed on the basis of other case pairs forming an analogical proportion with it. An evaluation in a Boolean setting shows that using case pair competences improves significantly the result of the analogical extrapolation process.
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Notes
- 1.
D(0/1) indicates the disagreement between \({\mathtt{a}}\) and \({\mathtt{b}}\) (respectively between \({\mathtt{c}}\) and \({\mathtt{d}}\) and between \({\mathtt{a}}'\) and \({\mathtt{b}}'\)) when the former is equal to 0 and the latter is equal to 1. D(1/0) is the reverse disagreement.
- 2.
In Table 1, A(u, v, w) means that \({\mathtt{a}}={\mathtt{b}}=u\), \({\mathtt{c}}={\mathtt{d}}=v\) and \({\mathtt{a}}'={\mathtt{b}}'=w\).
- 3.
A generator \({\mathtt{CNF}}\), generating formulas in CNF (conjunctive normal form: conjunction of disjunctions of literals) could also have been considered. However, this does not add anything new since it is dual with the \({\mathtt{DNF}}\) generator for two reasons. First, the drawn inferences are code-independent, meaning that replacing the attributes by their negations does not change the result of the inference, in particular, for \({\mathtt{a}}, {\mathtt{b}}, {\mathtt{c}}, {\mathtt{d}}\in \mathbb {B}\), \({\mathtt{a}}{\mathtt{:}}\,{{\mathtt{b}}}{\mathtt{:\,\!:}}\,{{\mathtt{c}}{\mathtt{:}}\,{{\mathtt{d}}}}\) iff \(\lnot {\mathtt{a}}{\mathtt{:}}\,{\lnot {\mathtt{b}}}{\mathtt{:\,\!:}}\,{\lnot {\mathtt{c}}{\mathtt{:}}\,{\lnot {\mathtt{d}}}}\). Second, if \({\mathtt{f}}\) is obtained from the \({\mathtt{DNF}}\) generator then \(\lnot {\mathtt{f}}\) can be put easily in a function g written in CNF using De Morgan laws, and the distribution of g obtained this way would be the same as the distribution from a \({\mathtt{CNF}}\) generator with the same parameters.
- 4.
Reflexivity and symmetry are direct consequences of the postulates with the same names. By contrast, there exist analogical proportions for which transitivity does not hold [15].
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Lieber, J., Nauer, E., Prade, H. (2019). Improving Analogical Extrapolation Using Case Pair Competence. In: Bach, K., Marling, C. (eds) Case-Based Reasoning Research and Development. ICCBR 2019. Lecture Notes in Computer Science(), vol 11680. Springer, Cham. https://doi.org/10.1007/978-3-030-29249-2_17
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