Abstract
Case-based reasoning usually exploits source cases (consisting of a source problem and its solution) individually, on the basis of the similarity between the target problem and a particular source problem. This corresponds to approximation. Then the solution of the source case has to be adapted to the target. We advocate in this paper that it is also worthwhile to consider source cases by two, or by three. Handling cases by two allows for a form of interpolation, when the target problem is between two similar source problems. When cases come by three, it offers a basis for extrapolation. Namely the solution of the target problem is obtained, when possible, as the fourth term of an analogical proportion linking the three source cases with the target, where the analogical proportion handles both similarity and dissimilarity between cases. Experiments show that interpolation and extrapolation techniques are of interest for reusing cases, either in an independent or in a combined way.
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- 1.
When these values are Boolean, this can also be written \((b_i\wedge {}c_i\mathrel {\Rightarrow }{}a_i)\wedge (a_i\mathrel {\Rightarrow }{}b_i\vee {}c_i)=1\).
- 2.
An affine function \(\mathtt{{f}}: \mathbb {B}^m\mathrel {\rightarrow }\mathbb {B}\) has the form \(\mathtt{{f}}(\mathtt{{x}}) = \mathtt{{x}}_{i_1}\mathrel {\oplus }\ldots \mathrel {\oplus }\mathtt{{x}}_{i_q}\mathrel {\oplus }{}c\) where \(\{i_1, \ldots , i_q\}\) is a subset of [1, m] and \(c\in \{0, 1\}\). An affine function \(\mathtt{{f}}: \mathbb {B}^m\mathrel {\rightarrow }\mathbb {B}^n\) is of the form \(\mathtt{{f}}(\mathtt{{x}})=(\mathtt{{f}}_1(\mathtt{{x}}), \ldots , \mathtt{{f}}_n(\mathtt{{x}}))\) where \(\mathtt{{f}}_j : \mathbb {B}^m\mathrel {\rightarrow }\mathbb {B}\) is an affine function.
- 3.
The number of iterations in the for loop is \(|\mathtt{{CB}}|-1\). In each iteration, the set \(\mathtt{{access}}(\mathtt{{key}}, \mathtt{{key\_table}})\) contains at most \((|\mathtt{{CB}}|-1)\) elements, though in practice, this set contains in general a much smaller number of elements. So, the number of O(1) operations of this online procedure in the worst case is not more than \((|\mathtt{{CB}}|-1)\times {}(|\mathtt{{CB}}|-1)\), hence a complexity in \(O(|\mathtt{{CB}}|^2)\).
References
Bergmann, R., Wilke, W.: Building and refining abstract planning cases by change of representation language. J. Artif. Intell. Res. 3, 53–118 (1995)
Billingsley, R., Prade, H., Richard, G., Williams, M.-A.: Towards analogy-based decision - a proposal. In: Christiansen, H., Jaudoin, H., Chountas, P., Andreasen, T., Legind Larsen, H. (eds.) FQAS 2017. LNCS (LNAI), vol. 10333, pp. 28–35. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59692-1_3
Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci. 47(3), 549–595 (1993)
Bounhas, M., Prade, H., Richard, G.: Analogy-based classifiers for nominal or numerical data. Int. J. Approx. Reasoning 91, 36–55 (2017)
Branting, L.K., Aha, D.W.: Stratified case-based reasoning: reusing hierarchical problem solving episodes. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI 1995), vol. 1, pp. 384–390, August 1995
Couceiro, M., Hug, N., Prade, H., Richard, G.: Analogy-preserving functions: a way to extend Boolean samples. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI 2017), pp. 1575–1581. Morgan Kaufmann, Inc. (2017)
Craw, S., Wiratunga, N., Rowe, R.C.: Learning adaptation knowledge to improve case-based reasoning. Artif. Intell. 170(16–17), 1175–1192 (2006)
d’Aquin, M., Badra, F., Lafrogne, S., Lieber, J., Napoli, A., Szathmary, L.: Case base mining for adaptation knowledge acquisition. In: Veloso, M.M. (ed.) Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI 2007), pp. 750–755. Morgan Kaufmann, Inc. (2007)
Dubois, D., Hüllermeier, E., Prade, H.: Flexible control of case-based prediction in the framework of possibility theory. In: Blanzieri, E., Portinale, L. (eds.) EWCBR 2000. LNCS, vol. 1898, pp. 61–73. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44527-7_7
Hanney, K., Keane, M.T.: Learning adaptation rules from a case-base. In: Smith, I., Faltings, B. (eds.) EWCBR 1996. LNCS, vol. 1168, pp. 179–192. Springer, Heidelberg (1996). https://doi.org/10.1007/BFb0020610
Jarmulak, J., Craw, S., Rowe, R.: Using case-base data to learn adaptation knowledge for design. In: Proceedings of the 17th International Joint Conference on Artificial Intelligence (IJCAI 2001), pp. 1011–1016. Morgan Kaufmann, Inc. (2001)
Lepage, Y., Denoual, É.: Purest ever example-based machine translation: detailed presentation and assessment. Mach. Transl. 19, 251–282 (2005)
McSherry, D.: Diversity-conscious retrieval. In: Craw, S., Preece, A. (eds.) ECCBR 2002. LNCS (LNAI), vol. 2416, pp. 219–233. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-46119-1_17
Miclet, L., Prade, H.: Handling analogical proportions in classical logic and fuzzy logics settings. In: Sossai, C., Chemello, G. (eds.) ECSQARU 2009. LNCS (LNAI), vol. 5590, pp. 638–650. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02906-6_55
Minor, M., Bergmann, R., Görg, S.: Case-based adaptation of workflows. Inf. Syst. 40, 142–152 (2014)
Prade, H., Richard, G.: From analogical proportion to logical proportions. Logica Universalis 7(4), 441–505 (2013)
Prade, H., Richard, G.: Analogical proportions and analogical reasoning - an introduction. In: Aha, D.W., Lieber, J. (eds.) ICCBR 2017. LNCS (LNAI), vol. 10339, pp. 16–32. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-61030-6_2
Richter, M.M., Weber, R.O.: Case-Based Reasoning: A Textbook. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40167-1
Schockaert, S., Prade, H.: Interpolation and extrapolation in conceptual spaces: a case study in the music domain. In: Rudolph, S., Gutierrez, C. (eds.) RR 2011. LNCS, vol. 6902, pp. 217–231. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23580-1_16
Schockaert, S., Prade, H.: Interpolative and extrapolative reasoning in propositional theories using qualitative knowledge about conceptual spaces. Artif. Intell. 202, 86–131 (2013)
Schockaert, S., Prade, H.: Interpolative reasoning with default rules. In: Rossi, F. (ed.) Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI 2013), Beijing, 3–9 August 2013, pp. 1090–1096 (2013)
Schockaert, S., Prade, H.: Completing symbolic rule bases using betweenness and analogical proportion. In: Prade, H., Richard, G. (eds.) Computational Approaches to Analogical Reasoning: Current Trends. SCI, vol. 548, pp. 195–215. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54516-0_8
Smyth, B.: Case-based design. Ph.D. thesis, Trinity College, University of Dublin (1996)
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Lieber, J., Nauer, E., Prade, H., Richard, G. (2018). Making the Best of Cases by Approximation, Interpolation and Extrapolation. In: Cox, M., Funk, P., Begum, S. (eds) Case-Based Reasoning Research and Development. ICCBR 2018. Lecture Notes in Computer Science(), vol 11156. Springer, Cham. https://doi.org/10.1007/978-3-030-01081-2_38
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