# A second step towards circuit complexity-theoretic analogs of Rice's theorem

## Abstract

Rice's Theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan [BS97] initiated the search for circuit 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.

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