PSPACE-complete$_{FP}$ problems are the PSPACE problems such that every other PSPACE problem can be transformed to it with a polynomial time reduction, i.e. the reduction is an algorithm $\in$ FP. This class is known as PSPACE-complete.
Similarly, we can define :
PSPACE-complete$_{FPSPACE}$ problems, the PSPACE problems such that every other PSPACE problem can be transformed to it with a polynomial space reduction. This class is still PSPACE, minus trivial problems $\Sigma^*$ and $\emptyset$.
PSPACE-complete$_{FNP}$ problems, the PSPACE problems such that every other PSPACE problem can be transformed to it with a FNP reduction (the function problem extension of the decision problem class NP).
PSPACE-complete$_{FPH}$ problems, the PSPACE problems such that every other PSPACE problem can be transformed to it with a FPH reduction (the function problem extension of the decision problem class PH).
It seems to me that all these classes are well defined and that we have :
PSPACE-complete $\subset$ PSPACE-complete$_{FNP}$ $\subset$ PSPACE-complete$_{FPH}$ $\subset$ PSPACE
Additionally, if PSPACE-complete$_{FPH}$ $\neq$ PSPACE-complete then P $\neq$ NP.
Naturally, PSPACE-complete$_{FNP}$ $\neq$ PSPACE-complete works also but it's more difficult to prove.
What more can be found about the two last defined classes ?
(Initially asked on cs.stackexchange.com)
