In the field of mutation analysis, mutation is the systematic generation of mutated programs (i.e., mutants) from an original program. The concept of mutation has been widely applied to various testing problems, including test set selection, fault localization, and program repair. However, surprisingly little focus has been given to the theoretical foundation of mutation-based testing methods, making it difficult to understand, organize, and describe various mutation-based testing methods. This paper aims to consider a theoretical framework for understanding mutation-based testing methods. While there is a solid testing framework for general testing, this is incongruent with mutation-based testing methods, because it focuses on the correctness of a program for a test, while the essence of mutation-based testing concerns the differences between programs (including mutants) for a test. In this paper, we begin the construction of our framework by defining a novel testing factor, called a test differentiator, to transform the paradigm of testing from the notion of correctness to the notion of difference. We formally define behavioral differences of programs for a set of tests as a mathematical vector, called a d-vector. We explore the multi-dimensional space represented by d-vectors, and provide a graphical model for describing the space. Based on our framework and formalization, we interpret existing mutation-based fault localization methods and mutant set minimization as applications, and identify novel implications for future work.
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