6 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
On the existence of complete disjoint NP-pairs
Disjoint NP-pairs are an interesting model of computation with important applications in cryptography and proof complexity. The question whether there exists a complete disjoint NP-pair was posed by Razborov in 1994 and is one of the most important problems in the field. In this paper we prove that there exists a many-one hard disjoint NP-pair which is computed with access to a very weak oracle (a tally NP-oracle). In addition, we exhibit candidates for complete NP-pairs and apply our results to a recent line of research on the construction of hard tautologies from pseudorandom generators
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
RESOURCE BOUNDED IMMUNITY AND SIMPLICITY ∗ Extended Abstract
Abstract 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 k-immunity and k-simplicity and their extensions: feasible k-immunity and feasible k-simplicity. Finally, we propose the k-immune hypothesis as a working hypothesis that ensures the existence of simple sets in NP