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

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

On the Power of Parity Polynomial Time

This paper proves that the complexity class Ef)P, parity polynomial time [PZ83], contains the class of languages accepted by NP machines with few accepting paths. Indeed, Ef)P contains a. broad class of languages accepted by path-restricted nondeterministic machines. In particular, Ef)P contains the polynomial accepting path versions of NP, of the counting hierarchy, and of ModmNP for m > 1. We further prove that the class of nondeterministic path-restricted languages is closed under bounded truth-table reductions

The Complexity of Computing the Size of an Interval

Given a p-order A over a universe of strings (i.e., a transitive, reflexive, antisymmetric relation such that if (x, y) is an element of A then |x| is polynomially bounded by |y|), an interval size function of A returns, for each string x in the universe, the number of strings in the interval between strings b(x) and t(x) (with respect to A), where b(x) and t(x) are functions that are polynomial-time computable in the length of x. By choosing sets of interval size functions based on feasibility requirements for their underlying p-orders, we obtain new characterizations of complexity classes. We prove that the set of all interval size functions whose underlying p-orders are polynomial-time decidable is exactly #P. We show that the interval size functions for orders with polynomial-time adjacency checks are closely related to the class FPSPACE(poly). Indeed, FPSPACE(poly) is exactly the class of all nonnegative functions that are an interval size function minus a polynomial-time computable function. We study two important functions in relation to interval size functions. The function #DIV maps each natural number n to the number of nontrivial divisors of n. We show that #DIV is an interval size function of a polynomial-time decidable partial p-order with polynomial-time adjacency checks. The function #MONSAT maps each monotone boolean formula F to the number of satisfying assignments of F. We show that #MONSAT is an interval size function of a polynomial-time decidable total p-order with polynomial-time adjacency checks. Finally, we explore the related notion of cluster computation.Comment: This revision fixes a problem in the proof of Theorem 9.

Complexity of certificates, heuristics, and counting types , with applications to cryptography and circuit theory

In dieser Habilitationsschrift werden Struktur und Eigenschaften von Komplexitätsklassen wie P und NP untersucht, vor allem im Hinblick auf: Zertifikatkomplexität, Einwegfunktionen, Heuristiken gegen NP-Vollständigkeit und Zählkomplexität. Zum letzten Punkt werden speziell untersucht: (a) die Komplexität von Zähleigenschaften von Schaltkreisen, (b) Separationen von Zählklassen mit Immunität und (c) die Komplexität des Zählens der Lösungen von ,,tally`` NP-Problemen

On the Size of Classes with Weak Membership Properties

It is shown that the following classes have measure 0 in E: the class of P-selective sets, the class of P-multiselective sets, the class of cheatable sets, the class of easily countable sets, the class of easily approximable sets, the class of near-testable sets, the class of nearly near-testable sets, the class of sets that are not P-bi-immune. These are corollaries of a more general result stating that the class of sets that are p-isomorphic to P-quasi-approximable sets has measure 0 in E. By considering the recent approach of Allender and Strauss for measuring in subexponential classes, we obtain similar results with respect to P for classes having weak logarithmic time membership properties

On Parity and Near-Testability: PA≠NTAP^{A} \neq NT^{A} With Probability 1

The class of near-testable sets, NT, was defined by Goldsmith, Joseph, and Young. They noted that $P \subseteq NT \subseteq PSPACE$, and asked whether P=NT. This note shows that NT shares the same $m$-degree as the parity-based complexity class $\bigoplus P$ (i.e., $NT\equiv^{p}_{m} \oplus P$) and uses this to prove that relative to a random oracle $A, P^{A} \neq NT^{A}$ with probability one. Indeed, with probability one, $NT^{A} - (NP^{A} \bigcup coNP^{A}) \neq 0$