# Possibility of hierarchy with $UP$ class?

I am not sure if this is a cheap query. However I am unable to find this myself. So I am posting here. The standard complexity class is built with $$NP$$ and $$coNP$$ and leads up to $$PSPACE$$. The reference https://link.springer.com/content/pdf/10.1007%2FBF01184809.pdf gives some classes for which we have oracles with respect to which $$UP$$ is not low.

1. Which classes are known to be low with $$UP$$ oracle ($$PSPACE$$ is one of them)?

2. Is there a hierarchy to concoct with $$UP$$ and $$coUP$$ analogous to the standard polynomial hierarchy and in particular is there an analog of $$PSPACE$$?

• I know that if ${\bf UP}$ is low for ${\bf \#P}$ then ${\bf UP} = {\bf CoUP}$. You can read it here = J. KOBLER, U. SCHONING, AND J. TORAN, On counting and approximation, Acta Inform. 26 (1989), 363-379. – Tayfun Pay Jul 8 '19 at 2:47
• @TayfunPay Very interesting. Then likely that there is an unambiguous version of things and a fewbiguous version of things. – 1.. Jul 8 '19 at 9:44