Complexity index

Besides complexity intended as a difficulty to compute a function (see computational complexity), in modern computer science and in statistics another complexity index of a function stands for denoting its information content, in turn affecting the difficulty of learning the function from examples.
"Complexity indices" in this sense characterize the entire class of functions to which the one we are interested in belongs. Focusing on Boolean functions, the "detail" of a class formula_1 of Boolean functions "c" essentially denotes how deeply the class is articulated.
To identify this index we must first define a "sentry function" of formula_1.
Let us focus for a moment on a single function "c", call it a "concept" defined on a set formula_3 of elements that we may figure as points in a Euclidean space. In this framework, the above function associates to "c" a set of points that, since are defined to be external to the concept, prevent it from expanding into another function of formula_1. We may dually define these points in terms of sentinelling a given concept "c" from being fully enclosed (invaded) by another concept within the class. Therefore, we call these points either "sentinels" or "sentry points"; they are assigned by the sentry function formula_5 to each concept of formula_1 in such a way that:
The technical definition coming from is rooted in the inclusion of an augmented concept formula_11 made up of "c" plus its sentry points by another formula_12 in the same class.
Definition of sentry function.
For a concept class formula_1 on a space formula_14, a "sentry function" is a total function formula_15 satisfying the following conditions:
formula_25 is the "frontier" of "c" upon formula_5.
With reference to the picture on the right, formula_45 is a candidate frontier of formula_46 against formula_47. All points are in the gap between a formula_48 and formula_46. They avoid inclusion of formula_50 in formula_51, provided that these points are not used by the latter for sentineling itself against other concepts. "Vice versa" we expect that formula_37 uses formula_53 and formula_54 as its own sentinels, formula_38 uses formula_56 and formula_54 and formula_58 uses formula_53 and formula_56 analogously. Point formula_61 is not allowed as a formula_46 sentry point since, like any diplomatic seat, it should be located outside all other concepts just to ensure that it is not occupied in case of invasion by formula_46.
Definition of detail.
The frontier size of the most expensive concept to be sentineled with the least efficient sentineling function, i.e. the quantity