Lower bounds and the hardness of counting properties

  • Authors:

  • Lane A. Hemaspaandra;Mayur Thakur

  • Affiliations:

  • Department of Computer Science, University of Rochester, P.O. Box 270226, Rochester, NY;University of Missouria at Rolla and Department of Computer Science, University of Rochester, P.O. Box 270226, Rochester, NY

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2004

Quantified Score

Hi-index 5.23

Visualization

Abstract

Rice's Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan (Math. Logic Quart. 46 (4) (2000) 489-504) started the search for complexity-theoretic analogs of Rice's Theorem, and proved that every nontrivial counting property of boolean circuits is UP-hard. Hemaspaandra and Rothe (Theoret. Comput. Sci. 244 (1-2) (2000) 205-217) improved the UP-hardness lower bound to UPO(1)-hardness. The present paper raises the lower bound for nontrivial counting properties from UPO(1)-hardness to FewP-hardness, i.e., from constant-ambiguity nondeterminism to polynomial-ambiguity nondeterminism. Furthermore, we prove that no relativizable technique can raise this lower bound to FewP-≤1-ttp -hardness. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard, and for a broad class of leaf-language classes we prove a sufficient condition for the natural analog of Rice's Theorem to hold.