33 research outputs found
On the power of counting the total number of computation paths of NPTMs
Complexity classes defined by modifying the acceptance condition of NP
computations have been extensively studied. For example, the class UP, which
contains decision problems solvable by non-deterministic polynomial-time Turing
machines (NPTMs) with at most one accepting path -- equivalently NP problems
with at most one solution -- has played a significant role in cryptography,
since P=/=UP is equivalent to the existence of one-way functions. In this
paper, we define and examine variants of several such classes where the
acceptance condition concerns the total number of computation paths of an NPTM,
instead of the number of accepting ones. This direction reflects the
relationship between the counting classes #P and TotP, which are the classes of
functions that count the number of accepting paths and the total number of
paths of NPTMs, respectively. The former is the well-studied class of counting
versions of NP problems, introduced by Valiant (1979). The latter contains all
self-reducible counting problems in #P whose decision version is in P, among
them prominent #P-complete problems such as Non-negative Permanent, #PerfMatch,
and #Dnf-Sat, thus playing a significant role in the study of approximable
counting problems.
We show that almost all classes introduced in this work coincide with their
'# accepting paths'-definable counterparts. As a result, we present a novel
family of complete problems for the classes parity-P, Modkp, SPP, WPP, C=P, and
PP that are defined via TotP-complete problems under parsimonious reductions.Comment: 19 pages, 1 figur
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
Query Order and the Polynomial Hierarchy
Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] initiated the field of
query order, which studies the ways in which computational power is affected by
the order in which information sources are accessed. The present paper studies,
for the first time, query order as it applies to the levels of the polynomial
hierarchy. We prove that the levels of the polynomial hierarchy are
order-oblivious. Yet, we also show that these ordered query classes form new
levels in the polynomial hierarchy unless the polynomial hierarchy collapses.
We prove that all leaf language classes - and thus essentially all standard
complexity classes - inherit all order-obliviousness results that hold for P.Comment: 14 page