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 Boolean functions c essentially denotes how deeply the class is articulated.

Technical definition

To identify this index we must first define a sentry function. Let us focus for a moment on a single function c, call it a concept of elements that we may figure as points in a Euclidean space. In this framework, the above function associates to c. 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 in such a way that:

1. the sentry points are external to the concept c to be sentineled and internal to at least one other including it,

2. including c has at least one of the sentry points of c either in the gap between c, and

3. they constitute a minimal set with these properties.

The technical definition coming from made up of c in the same class.

Definition of sentry function

, a sentry function is a total function satisfying the following conditions:

1. <big>)</big>.

2. the set of concepts invading c<big>)</big>.

3. <big>)</big>.

4. are involved in really sentineling c is a sentry function on this set<big>)</big>.

is the frontier of c.

, provided that these points are not used by the latter for sentineling itself against other concepts. Vice versa.

Definition of detail

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

1. ,

is called detail itself.

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, as shown in the picture on right.

Example: discrete spaces

, and⃝a sentry point:. As usual we may have different sentineling functions. A worst case :

References