311 research outputs found
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
Separating Cook Completeness from Karp-Levin Completeness Under a Worst-Case Hardness Hypothesis
We show that there is a language that is Turing complete for NP but not many-one complete for NP, under a worst-case hardness hypothesis. Our hypothesis asserts the existence of a non-deterministic, double-exponential time machine that runs in time O(2^2^n^c) (for some c > 1) accepting Sigma^* whose accepting computations cannot be computed by bounded-error, probabilistic machines running in time O(2^2^{beta * 2^n^c) (for some beta > 0). This is the first result that separates completeness notions for NP under a worst-case hardness hypothesis
Autoreducibility of NP-Complete Sets
We study the polynomial-time autoreducibility of NP-complete sets and obtain
separations under strong hypotheses for NP. Assuming there is a p-generic set
in NP, we show the following:
- For every , there is a -T-complete set for NP that is -T
autoreducible, but is not -tt autoreducible or -T autoreducible.
- For every , there is a -tt-complete set for NP that is -tt
autoreducible, but is not -tt autoreducible or -T autoreducible.
- There is a tt-complete set for NP that is tt-autoreducible, but is not
btt-autoreducible.
Under the stronger assumption that there is a p-generic set in NP
coNP, we show:
- For every , there is a -tt-complete set for NP that is -tt
autoreducible, but is not -T autoreducible.
Our proofs are based on constructions from separating NP-completeness
notions. For example, the construction of a 2-T-complete set for NP that is not
2-tt-complete also separates 2-T-autoreducibility from 2-tt-autoreducibility
Nonuniform Reductions and NP-Completeness
Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP0completeness, obtaining both separations and upper bounds for nonuniform completeness vs uniform complessness in NP.
Under various hypotheses, we obtain the following separations:
1. There is a set complete for NP under nonuniform many-one reductions, but not under uniform many-one reductions. This is true even with a single bit of nonuniform advice.
2. There is a set complete for NP under nonuniform many-one reductions with polynomial-size advice, but not under uniform Turing reductions. That is, polynomial nonuniformity is stronger than a polynomial number of queries.
3. For any fixed polynomial p(n), there is a set complete for NP under uniform 2-truth-table reductions, but not under nonuniform many-one reductions that use p(n) advice. That is, giving a uniform reduction a second query makes it more powerful than a nonuniform reduction with fixed polynomial advice.
4. There is a set complete for NP under nonuniform many-one reductions with polynomial ad- vice, but not under nonuniform many-one reductions with logarithmic advice. This hierarchy theorem also holds for other reducibilities, such as truth-table and Turing.
We also consider uniform upper bounds on nonuniform completeness. Hirahara (2015) showed that unconditionally every set that is complete for NP under nonuniform truth-table reductions that use logarithmic advice is also uniformly Turing-complete. We show that under a derandomization hypothesis, the same statement for truth-table reductions and truth-table completeness also holds
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
A parameterized halting problem, the linear time hierarchy, and the MRDP theorem
The complexity of the parameterized halting problem for nondeterministic Turing machines p-Halt is known to be related to the question of whether there are logics capturing various complexity classes [10]. Among others, if p-Halt is in para-AC0, the parameterized version of the circuit complexity class AC0, then AC0, or equivalently, (+, x)-invariant FO, has a logic. Although it is widely believed that p-Halt ∉. para-AC0, we show that the problem is hard to settle by establishing a connection to the question in classical complexity of whether NE ⊈ LINH. Here, LINH denotes the linear time hierarchy.
On the other hand, we suggest an approach toward proving NE ⊈ LINH using bounded arithmetic. More specifically, we demonstrate that if the much celebrated MRDP (for Matiyasevich-Robinson-Davis-Putnam) theorem can be proved in a certain fragment of arithmetic, then NE ⊈ LINH. Interestingly, central to this result is a para-AC0 lower bound for the parameterized model-checking problem for FO on arithmetical structures.Peer ReviewedPostprint (author's final draft
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
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