What does it mean the norm symbol applied to a concept?

Question

I'm reading this paper: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.14 or http://www.jmlr.org/papers/volume10/kontorovich09a/kontorovich09a.pdf

On page 1101 are introduced 2 functions without a precise definition. It is said that they are functions of the type $\textrm{f}:\mathbb{A} \rightarrow \mathbb{N}$ where $\mathbb{A}$ is a set. For example $\mathbb{A}$ can be a concept or an instance. For regular languages an instance can be a string on the alphabet and the concept a specific DFA (between all DFA on that alphabet) (alternately the concept can be defined as a subset of the set of sets made of all subsets on all the possible instances (namely a subset of $2^X$ where $X$ is the instance space namely all the possible instances). The instances on the subset are the $accepted$ instances, the others are $rejected$ ).

(http://openscholarship.wustl.edu/cgi/viewcontent.cgi?article=1655&context=cse_research for precise definitions of instance and concept, look at definitions)

The 2 functions are : $\vert \bullet \vert$ that works on a instance (for example a string). Does mean this function? Can be the length of that instance?

The other function is : $\lVert \bullet \rVert$ and works on a concept (namely yours input is a concept). But I'm not able to imagine what this function means in this context and in the paper isn't explicated.