In modern computer science and statistics, the complexity index of a function denotes the level of informational content, which in turn affects the difficulty of learning the function from examples. This is different from computational complexity, which is the difficulty to compute a function. Complexity indices characterize the entire class of functions to which the one we are interested in belongs. Focusing on Boolean functions, the detail of a class  of Boolean functions c essentially denotes how deeply the class is articulated.

Technical definition
To identify this index we must first define a sentry function of .
Let us focus for a moment on a single function c, call it a concept defined on a set  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 . 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  to each concept of  in such a way that:
 the sentry points are external to the concept c to be sentineled and internal to at least one other including it,
 each concept  including c has at least one of the sentry points of c either in the gap between c and , or outside  and distinct from the sentry points of , and
 they constitute a minimal set with these properties.

The technical definition coming from  is rooted in the inclusion of an augmented concept  made up of c plus its sentry points by another  in the same class.

Definition of sentry function 
For a concept class  on a space , a sentry function is a total function  satisfying the following conditions:
 Sentinels are outside the sentineled concept ( for all ).
 Sentinels are inside the invading concept (Having introduced the sets , an invading concept  is such that  and . Denoting  the set of concepts invading c, we must have that if , then ).
  is a minimal set with the above properties (No  exists satisfying (1) and (2) and having the property that  for every ).
 Sentinels are honest guardians. It may be that  but  so that . This however must be a consequence of the fact that all points of  are involved in really sentineling c against other concepts in  and not just in avoiding inclusion of  by . Thus if we remove  remains unchanged (Whenever  and  are such that  and , then the restriction of  to  is a sentry function on this set).

 is the frontier of c upon .

With reference to the picture on the right,  is a candidate frontier of  against . All points are in the gap between a  and . They  avoid inclusion of  in , provided that these points are not used by the latter for sentineling itself against other concepts. Vice versa we expect that  uses  and  as its own sentinels,  uses   and  and  uses   and  analogously. Point  is not allowed as a  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 .

Definition of detail 
The frontier size of the most expensive concept to be sentineled with the least efficient sentineling function, i.e. the quantity

,

is called detail of .   spans also over sentry functions on subsets of  sentineling in this case the intersections of the concepts with these subsets. Actually, proper subsets of  may host sentineling tasks that prove harder than those emerging with  itself.

The detail  is a complexity measure of concept classes dual to the VC dimension . The former uses points to separate sets of concepts, the latter concepts for partitioning sets of points. In particular the following inequality holds 

See also  Rademacher complexity for a recently introduced class complexity index.

Example: continuous spaces 
Class C of circles in  has detail , as shown in the picture on left below. Similarly, for the class of segments on , as shown in the picture on right.

Example: discrete spaces 
The class  on  whose concepts are illustrated in the following scheme, where "+" denotes an element  belonging to , "-" an element outside , and ⃝ a sentry point:

This class has . As usual we may have different sentineling functions. A worst case , as illustrated, is: . However a cheaper one is :

References 
 
 

Computational complexity theory
Algorithmic inference