14 research outputs found
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
If not empty, NP — P is topologically large
AbstractIn the classical Cantor topology or in the superset topology, NP and, consequently, classes included in NP are meagre. However, in a natural combination of the two topologies, we prove that NP — P, if not empty, is a second category class, while NP-complete sets form a first category class. These results are extended to different levels in the polynomial hierarchy and to the low and high hierarchies. P-immune sets in NP, NP-simple sets, P-bi-immune sets and NP-effectively simple sets are all second category (if not empty). It is shown that if C is any of the above second category classes, then for all B∈NP there exists an A∈C such that A is arbitrarily close to B infinitely often
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
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
Power of Counting by Nonuniform Families of Polynomial-Size Finite Automata
Lately, there have been intensive studies on strengths and limitations of
nonuniform families of promise decision problems solvable by various types of
polynomial-size finite automata families, where "polynomial-size" refers to the
polynomially-bounded state complexity of a finite automata family. In this line
of study, we further expand the scope of these studies to families of partial
counting and gap functions, defined in terms of nonuniform families of
polynomial-size nondeterministic finite automata, and their relevant families
of promise decision problems. Counting functions have an ability of counting
the number of accepting computation paths produced by nondeterministic finite
automata. With no unproven hardness assumption, we show numerous separations
and collapses of complexity classes of those partial counting and gap function
families and their induced promise decision problem families. We also
investigate their relationships to pushdown automata families of polynomial
stack-state complexity.Comment: (A4, 10pt, 21 pages) This paper corrects and extends a preliminary
report published in the Proceedings of the 24th International Symposium on
Fundamentals of Computation Theory (FCT 2023), Trier, Germany, September
18-24, 2023, Lecture Notes in Computer Science, vol. 14292, pp. 421-435,
Springer Cham, 202