Banishing Robust Turing Completeness

This paper proves that “promise classes” are so fragilely structured that they do not robustly (i.e. with respect to all oracles) possess Turinghard sets even in classes far larger than themselves. In particular, this paper shows that FewP does not robustly possess Turing hard sets for UP ∩ coUP and IP ∩ coIP does not robustly possess Turing hard sets for ZPP. It follows that ZPP, R, coR, UP∩coUP, UP, FewP∩coFewP, FewP, and IP ∩ coIP do not robustly possess Turing complete sets. This both resolves open questions of whether promise classes lacking robust downward closure under Turing reductions (e.g., R, UP, FewP) might robustly have Turing complete sets, and extends the range of classes known not to robustly contain many-one complete sets.