When a software measure is not a measure

A recent interesting paper by Melton et al. [1] discussed finding measures which preserve intuitive orderings on software documents. Informally, if ≤ is such an ordering, then they argue that a measure M is a real-valued function defined on documents such that M(F) ≤ M(F′) whenever F ≤ F′. However, in measurement theory, this is only a necessary condition for a measure M. The representation condition for measurement additionally requires the converse; that F≤F′ whenever M(F) ≤ M(F′). Using the measurement theory definition of a measure, we show that Melton et al.'s examples, like McCabe's cyclomatic complexity [2] are not measures of the proposed intuitive document ordering after all. However, by dropping the restriction to real-valued functions, we show that it is possible to define a measure which characterises Melton et al.'s order relation; this provides a considerable strengthening of the results in Reference 1. More generally, we show that there is no single real-valued measure which can characterise any intuitive notion of ‘complexity’ of programs. The power of measurement theory is further illustrated in a critical analysis of some recent work by Weyuker [3] et al. on axioms for software complexity measures.