Rice's Theorem states that every nontrivial language property of the
recursively enumerable sets is undecidable. Borchert and Stephan initiated the
search for complexity-theoretic analogs of Rice's Theorem. In particular, they
proved that every nontrivial counting property of circuits is UP-hard, and that
a number of closely related problems are SPP-hard.
The present paper studies whether their UP-hardness result itself can be
improved to SPP-hardness. We show that their UP-hardness result cannot be
strengthened to SPP-hardness unless unlikely complexity class containments
hold. Nonetheless, we prove that every P-constructibly bi-infinite counting
property of circuits is SPP-hard. We also raise their general lower bound from
unambiguous nondeterminism to constant-ambiguity nondeterminism.Comment: 14 pages. To appear in Theoretical Computer Scienc