This article presents an improved approach to assist diagnosis of failures in software (fault localisation) by ranking program statements or blocks in accordance with to how likely they are to be buggy. We present a very simple single-bug program to model the problem. By examining different possible execution paths through this model program over a number of test cases, the effectiveness of different proposed spectral ranking methods can be evaluated in idealised conditions. The results are remarkably consistent to those arrived at empirically using the Siemens test suite and Space benchmarks. The model also helps identify groups of metrics that are equivalent for ranking. Due to the simplicity of the model, an optimal ranking method can be devised. This new method out-performs previously proposed methods for the model program, the Siemens test suite and Space. It also helps provide insight into other ranking methods.

Numerous set similarity metrics have been used for ranking "suspiciousness" of code in spectral fault localization, which uses execution profiles of passed and failed test cases to help locate bugs. Research in data mining has identified several forms of possibly desirable symmetry in similarity metrics. Here we define several forms of "duals" of metrics, based on these forms of symmetries. Use of these duals, plus some other slight modifications, leads to several new similarity metrics. We show that versions of several previously proposed metrics are optimal, or nearly optimal, for locating single bugs. We also show that a form of duality exists between locating single bugs and locating "deterministic" bugs (execution of which always results in test case failure). Duals of the various single bug optimal metrics are optimal for locating such bugs. This more theoretical work leads to a conjecture about how different metrics could be chosen for different stages of software development.

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