  Hausdorff $\Phi$-dimension is a notion of Hausdorff dimension developed using
a restricted class of coverings of a set. We introduce a constructive analogue
of $\Phi$-dimension using the notion of constructive $\Phi$-$s$-supergales. We
prove a Point-to-Set Principle for $\Phi$-dimension, through which we get
Point-to-Set Principles for Hausdorff dimension, continued-fraction dimension
and dimension of Cantor coverings as special cases. We also provide a
Kolmogorov complexity characterization of constructive $\Phi$-dimension.
  A class of covering sets $\Phi$ is said to be "faithful" to Hausdorff
dimension if the $\Phi$-dimension and Hausdorff dimension coincide for every
set. Similarly, $\Phi$ is said to be "faithful" to constructive dimension if
the constructive $\Phi$-dimension and constructive dimension coincide for every
set. Using the Point-to-Set Principle for Cantor coverings and a new technique
for the construction of sequences satisfying a certain Kolmogorov complexity
condition, we show that the notions of ``faithfulness'' of Cantor coverings at
the Hausdorff and constructive levels are equivalent.
  We adapt the result by Albeverio, Ivanenko, Lebid, and Torbin to derive the
necessary and sufficient conditions for the constructive dimension faithfulness
of the coverings generated by the Cantor series expansion, based on the terms
of the expansion.
