A Second Step Towards Complexity-Theoretic Analogs of Rice's Theorem

TL;DR

This paper explores the complexity of circuit properties, showing certain hardness results are unlikely to be improved and establishing new bounds for specific circuit properties within complexity classes.

Contribution

It demonstrates that the UP-hardness of counting properties cannot generally be strengthened to SPP-hardness, but identifies SPP-hardness for P-constructibly bi-infinite properties.

Findings

UP-hardness cannot be improved to SPP-hardness without unlikely class containments

P-constructibly bi-infinite counting properties are SPP-hard

Lower bounds extend from unambiguous to constant-ambiguity nondeterminism

Abstract

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.