Immunity and Simplicity for Exact Counting and Other Counting Classes

Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some relativized world, PSPACE (in fact, ParityP) contains a set that is immune to the polynomial hierarchy (PH). In this paper, we study and settle the question of (relativized) separations with immunity for PH and the counting classes PP, C_{=}P, and ParityP in all possible pairwise combinations. Our main result is that there is an oracle A relative to which C_{=}P contains a set that is immune to BPP^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A} and ParityP^{A}. Strengthening results of Tor\'{a}n [J.ACM 38, 1991] and Green [IPL 37, 1991], we also show that, in suitable relativizations, NP contains a C_{=}P-immune set, and ParityP contains a PP^{PH}-immune set. This implies the existence of a C_{=}P^{B}-simple set for some oracle B, which extends results of Balc\'{a}zar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] and provides the first example of a simple set in a class not known to be contained in PH. Our proof technique requires a circuit lower bound for ``exact counting'' that is derived from Razborov's [Mat. Zametki 41, 1987] lower bound for majority.Comment: 20 page

Diagonalizations over polynomial time computable sets

AbstractA formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets (the class of polynomial time computable functions respectively), like, e.g., p-immunity. It is shown that there are sets—called p-generic— which have all properties enforceable by such diagonalizations. We study the behaviour and the complexity of p-generic sets. In particular, we show that the existence of p-generic sets in NP is oracle dependent, even if we assume P ≠ NP

Resource Bounded Immunity and Simplicity

Revisiting the thirty years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k-immunity and feasible k-immunity, and their simplicity notions. Finally, we propose the k-immune hypothesis as a working hypothesis that guarantees the existence of simple sets in NP.Comment: This is a complete version of the conference paper that appeared in the Proceedings of the 3rd IFIP International Conference on Theoretical Computer Science, Kluwer Academic Publishers, pp.81-95, Toulouse, France, August 23-26, 200

Immunity and Pseudorandomness of Context-Free Languages

We discuss the computational complexity of context-free languages, concentrating on two well-known structural properties---immunity and pseudorandomness. An infinite language is REG-immune (resp., CFL-immune) if it contains no infinite subset that is a regular (resp., context-free) language. We prove that (i) there is a context-free REG-immune language outside REG/n and (ii) there is a REG-bi-immune language that can be computed deterministically using logarithmic space. We also show that (iii) there is a CFL-simple set, where a CFL-simple language is an infinite context-free language whose complement is CFL-immune. Similar to the REG-immunity, a REG-primeimmune language has no polynomially dense subsets that are also regular. We further prove that (iv) there is a context-free language that is REG/n-bi-primeimmune. Concerning pseudorandomness of context-free languages, we show that (v) CFL contains REG/n-pseudorandom languages. Finally, we prove that (vi) against REG/n, there exists an almost 1-1 pseudorandom generator computable in nondeterministic pushdown automata equipped with a write-only output tape and (vii) against REG, there is no almost 1-1 weakly pseudorandom generator computable deterministically in linear time by a single-tape Turing machine.Comment: A4, 23 pages, 10 pt. A complete revision of the initial version that was posted in February 200

On the structure of intractable sets

There are two parts to this dissertation. The first part is motivated by nothing less than a reexamination of what it means for a set to be NP-complete. Are there sets in NP that in a mathematically meaningful sense should be considered to be complete for NP, but that are not NP-complete in the usual sense that every set in NP is ≤q[subscript]spmP-reducible to it? We define a noneffective binary relation that makes precise the notion that the complexity of A is polynomially related to the complexity of B, This relation yields new completeness and hardness notions for complexity classes, and we show that there are sets that are hard for NP that are not NP-hard in the usual sense. We also show that there are sets that must be considered to be complete for E that are not even ≤q[subscript]spTP-complete for E;In a certain way, hardness and completeness with respect to the relation we define is related to the notion of almost everywhere (a.e.) complexity, and so we initiate this study by first investigating this notion. We state and prove a deterministic time hierarchy theorem for a.e. complexity that is as tight as the Hartmanis-Stearns hierarchy theorem for infinitely often complexity. This result is a significant improvement over all previously known hierarchy theorems for a.e. complex sets. We derive similar, very tight, hierarchy theorems for sets that cannot be a.e. complex for syntactic reasons, but for which, intuitively, a.e. complex notions should exit. Similar results are applied to the study of P-printable sets and sets of low generalized Kolmogorov complexity;The second part of this study deals with relativization. Does the fact that DTIME(O (n)) ≠ NTIME(n) help in leading us to a proof that P ≠ NP? Does one imply the other? We seek evidence that this is a hard . We construct an oracle that answers this question in the affirmative, and we construct an oracle that answers this question in the negative. We conclude that the result that DTIME(O (n)) ≠ NTIME(n) does not imply P ≠ NP by recursive theoretic techniques;Finally, we study the relationships between P, NP, and the unambiguous and random time classes UP, and RP. Questions concerning these relationships are motivated by complexity issues to public-key cryptosystems. We prove that there exists a recursive oracle A such that P[superscript]A ≠ UP[superscript]A≠ NP[superscript]A, and such that the first inequality is strong, i.e., there exists a P[superscript]A-immune set in UP[superscript]A. Further, we constructed a recursive oracle B such that UP[superscript]B contains an RP[superscript]B-immune set. As a corollary we obtain P[superscript]B ≠ RB[superscript]B≠ NP[superscript]B and both inequalities are strong. By use of the techniques employed in the proof that P[superscript]A≠ UP[superscript]A≠ NP[superscript]A, we are also able to solve an open problem raised by Book, Long and Selman

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