I'm reading this paper: http://citeseerx.ist.psu.edu/viewdoc/summary?doi= 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.

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The only property that the functions $|\cdot|$ and $\|\cdot\|$ need to satisfy is that for every $n$ there is a finite number of $x$'s in the domain such that $|x|=n$ or $\|x\|=n$. The difference between the two is the domain: $|\cdot|$ is defined on $\mathcal{X}$ while $\|\cdot\|$ is defined on $\mathcal{C}$.

These functions are instantiated on page 1115: for $\mathcal{X} = \Sigma^*$, $|\cdot|$ is the length of the word; and for $\mathcal{C}$ being the set of all DFAs on $\Sigma$, $\|\cdot\|$ is the number of states.

  • $\begingroup$ Thank you very much! Maybe the authors could think of putting these two functions into definitions before introducing them! $\endgroup$ – Umbert Jul 5 '17 at 20:39

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