A second step towards 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 [4] 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. *

Lower bounds and the hardness of counting properties

“…We make use of the ambiguity-limited counting operator # g [19] in deÿning these classes. In the deÿnitions below, for simplicity of notation, we use # 1 when we actually mean # x:1 .…”

“…(2) [19] For any total function f : N → N, and for any complexity class C, # f · C is the set of all functions g : * → N such that there exist a language L ∈ C and a polynomial p such that the following hold for each x ∈ * : (a) g(x)6f(|x|), and (b) {y | |y| = p(|x|) ∧ x; y ∈ L} = g(x).…”

“…Hemaspaandra and Rothe [19] prove that every nontrivial counting property of circuits is UP O(1) -hard. They also prove that it is unlikely that the UP O(1) lower bound can be raised much higher: If every nontrivial counting property of circuits is SPP-hard, then SPP ⊆ P NP .…”

“…Can one improve the FewP-6 p tt -hardness lower bound of nontrivial counting properties of circuits? Hemaspaandra and Rothe [19] proved that raising the lower bound to SPP-hardness would imply an unexpected complexity class containment, and cannot be established via relativizable proof techniques. However, in light of the fact that the previous UP-hardness and UP O(1) -hardness results in fact achieve in each case not just hardness (i.e., 6 p T -hardness) but even 6 p 1-tt -hardness, it would be natural to hope that the FewP-6 p tt -hardness result of Theorem 9 can at least be improved to FewP-6 p 1-tt -hardness.…”

“…(Throughout this paper C-hard always means C-6 p T -hard, unless some other reduction is explicitly inserted as in, for example, C-6 p 1-tt -hardness.) Hemaspaandra and Rothe [19] improved the UP-hardness lower bound of Borchert and Stephan [5] to UP O(1) -hardness. That is, they proved that every nontrivial counting property of circuits is UP O(1) -hard.…”