When a software measure is not a measure

A paper by A.C. Melton et al. (1990) 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, the author shows that Melton et al.´s examples are not measures of the proposed intuitive document ordering after all. However, by dropping the restriction to real-valued functions, he shows that it is possible to define a measure which characterises Melton et al.´s order relation; this provides a considerable strengthening of their results. More generally, it is shown 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 work by E.J. Weyuker et al. (1988) on axioms for software complexity measures