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Softw. Eng. Methodol."],"published-print":{"date-parts":[[2013,10]]},"abstract":"<jats:p>An important research area of Spectrum-Based Fault Localization (SBFL) is the effectiveness of risk evaluation formulas. Most previous studies have adopted an empirical approach, which can hardly be considered as sufficiently comprehensive because of the huge number of combinations of various factors in SBFL. Though some studies aimed at overcoming the limitations of the empirical approach, none of them has provided a completely satisfactory solution. Therefore, we provide a theoretical investigation on the effectiveness of risk evaluation formulas. We define two types of relations between formulas, namely, equivalent and better. To identify the relations between formulas, we develop an innovative framework for the theoretical investigation. Our framework is based on the concept that the determinant for the effectiveness of a formula is the number of statements with risk values higher than the risk value of the faulty statement. We group all program statements into three disjoint sets with risk values higher than, equal to, and lower than the risk value of the faulty statement, respectively. For different formulas, the sizes of their sets are compared using the notion of subset. We use this framework to identify the maximal formulas which should be the only formulas to be used in SBFL.<\/jats:p>","DOI":"10.1145\/2522920.2522924","type":"journal-article","created":{"date-parts":[[2013,10,17]],"date-time":"2013-10-17T12:23:34Z","timestamp":1382012614000},"page":"1-40","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":296,"title":["A theoretical analysis of the risk evaluation formulas for spectrum-based fault localization"],"prefix":"10.1145","volume":"22","author":[{"given":"Xiaoyuan","family":"Xie","sequence":"first","affiliation":[{"name":"Swinburne University of Technology, Australia"}]},{"given":"Tsong Yueh","family":"Chen","sequence":"additional","affiliation":[{"name":"Swinburne University of Technology, Australia"}]},{"given":"Fei-Ching","family":"Kuo","sequence":"additional","affiliation":[{"name":"Swinburne University of Technology, Australia"}]},{"given":"Baowen","family":"Xu","sequence":"additional","affiliation":[{"name":"Nanjing University, Jiangsu, China"}]}],"member":"320","published-online":{"date-parts":[[2013,10,22]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jss.2009.06.035"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/PRDC.2006.18"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.5555\/1308173.1308264"},{"key":"e_1_2_1_4_1","volume-title":"Proceedings of the 6th International Symposium on Software Reliability Engineering. 143--151","author":"Agrawal H.","unstructured":"Agrawal , H. , Horgan , J. R. , London , S. , and Wong , W. E . 1995. Fault localization using execution slices and dataflow tests . In Proceedings of the 6th International Symposium on Software Reliability Engineering. 143--151 . Agrawal, H., Horgan, J. R., London, S., and Wong, W. E. 1995. Fault localization using execution slices and dataflow tests. In Proceedings of the 6th International Symposium on Software Reliability Engineering. 143--151."},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/1831708.1831717"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/ASE.2011.6100088"},{"key":"e_1_2_1_7_1","volume-title":"Proceedings of the 32nd IEEE\/IFIP International Conference on Dependable Systems and Networks. 595--604","author":"Chen M.","unstructured":"Chen , M. , Kiciman , E. , Fratkin , E. , Fox , A. , and Brewer , E . 2002. Pinpoint: Problem determination in large, dynamic internet services . In Proceedings of the 32nd IEEE\/IFIP International Conference on Dependable Systems and Networks. 595--604 . Chen, M., Kiciman, E., Fratkin, E., Fox, A., and Brewer, E. 2002. Pinpoint: Problem determination in large, dynamic internet services. In Proceedings of the 32nd IEEE\/IFIP International Conference on Dependable Systems and Networks. 595--604."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/0164-1212(89)90039-3"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/11531142_23"},{"key":"e_1_2_1_10_1","volume-title":"Proceedings of the 23rd International Conference on Software Engineering. 339--348","author":"Dickinson W.","unstructured":"Dickinson , W. , Leon , D. , and Podgurski , A . 2001. Finding failures by cluster analysis of execution profiles . In Proceedings of the 23rd International Conference on Software Engineering. 339--348 . Dickinson, W., Leon, D., and Podgurski, A. 2001. 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J., Naish, L., and Ramamohanarao, K. 2009b. The effectiveness of using non redundant test cases with program spectra for bug localization. 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