We could have used the normalized information distance between our description and an optimal description as a measure of sufeit, trying to be consistant with the definitions of miscoding and inaccuracy. Since we do not want to favour one of the possible descriptions of $r$ over the other, we have to compute the minimum distance with respect to all possible minimal descriptions of $r$:
[
\sigma (d, r) = { \underset{ d^\star \in d^\star_r } \min} \frac{ \max{ K \left( d \mid d^\star \right), K \left( d^\star \mid d \right) } } { \max{ K \left( d \right), K \left( d^\star \right) } }
]
This definition has the potential to replace not only our current definition of surfeit, but also the entire concept of inaccuracy, since when the distance between our description and an optimal description is zero, inaccuracy will also be zero. However, this approach is difficult to implement in practice, since it requires to know all possible descriptions of $r$. Even if we fix one of the poosibles styles of $e$ admited by the oracle, that description is unknown.
We could have used the normalized information distance between our description and an optimal description as a measure of sufeit, trying to be consistant with the definitions of miscoding and inaccuracy. Since we do not want to favour one of the possible descriptions of $r$ over the other, we have to compute the minimum distance with respect to all possible minimal descriptions of $r$: [ \sigma (d, r) = { \underset{ d^\star \in d^\star_r } \min} \frac{ \max{ K \left( d \mid d^\star \right), K \left( d^\star \mid d \right) } } { \max{ K \left( d \right), K \left( d^\star \right) } } ] This definition has the potential to replace not only our current definition of surfeit, but also the entire concept of inaccuracy, since when the distance between our description and an optimal description is zero, inaccuracy will also be zero. However, this approach is difficult to implement in practice, since it requires to know all possible descriptions of $r$. Even if we fix one of the poosibles styles of $e$ admited by the oracle, that description is unknown.