In Chapter 10 of Computational Complexity by Christos Papadimitriou, it is noted that reduction between problems of functional complexity classes are defined as follows:
Function problem A reduces to function problem B if the following holds: There are string functions $R$ and $S$, both computable in logarithmic space, such that for any strings $x$ and $z$ the following holds: if $x$ is an instance of A, then $R(x)$ is an instance of B. Furthermore, if $z$ is a correct output of $R(x)$, then $S(z)$ is a correct output of x
Procedurally, this definition follows from the usual notion of reductions in the space of decision problems. However, my uncertainty lies in the 'power' given to the reduction functions $R$ and $S$ . Naturally, I would assume polynomial time computable constraints as an obvious analog to the space of decision problems, however, this is not the case. More specifically, it is true that
$$ \text{SPACE}(f(n)) \subseteq \text{EXP}(f(n)) $$
which implies that
$$ \text{SPACE}(\log n) \subseteq \text{TIME}(\text{poly}(n)) $$
However, the relation in the reverse direction is unknown. This seems to imply that the reduction in functional space is given potentially less power than in decision space. Such discrepancies arise in more obvious cases such as NL where reductions are allowed only logarithmic space to prevent universal equivalence between problems up to reduction. In this case, however, I am struggling to justify the difference in reductive power between functional problems and decisions problems. Would appreciate any and all insight.