1,397 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
An excursion to the Kolmogorov random strings
AbstractWe study the sets of resource-bounded Kolmogorov random strings:Rt={x|Ct(n)(x)⩾|x|} fort(n)=2nk. We show that the class of sets that Turing reduce toRthas measure 0 inEXPwith respect to the resource-bounded measure introduced by Lutz. From this we conclude thatRtis not Turing-complete forEXP. This contrasts with the resource-unbounded setting. ThereRis Turing-complete forco-RE. We show that the class of sets to whichRtbounded truth-table reduces, hasp2-measure 0 (therefore, measure 0 inEXP). This answers an open question of Lutz, giving a natural example of a language that is not weakly complete forEXPand that reduces to a measure 0 class inEXP. It follows that the sets that are ⩽pbbt-hard forEXPhavep2-measure 0
-Generic Computability, Turing Reducibility and Asymptotic Density
Generic computability has been studied in group theory and we now study it in
the context of classical computability theory. A set A of natural numbers is
generically computable if there is a partial computable function f whose domain
has density 1 and which agrees with the characteristic function of A on its
domain. A set A is coarsely computable if there is a computable set C such that
the symmetric difference of A and C has density 0. We prove that there is a
c.e. set which is generically computable but not coarsely computable and vice
versa. We show that every nonzero Turing degree contains a set which is not
coarsely computable. We prove that there is a c.e. set of density 1 which has
no computable subset of density 1. As a corollary, there is a generically
computable set A such that no generic algorithm for A has computable domain. We
define a general notion of generic reducibility in the spirt of Turing
reducibility and show that there is a natural order-preserving embedding of the
Turing degrees into the generic degrees which is not surjective
Responsibility and blame: a structural-model approach
Causality is typically treated an all-or-nothing concept; either A is a cause
of B or it is not. We extend the definition of causality introduced by Halpern
and Pearl [2001] to take into account the degree of responsibility of A for B.
For example, if someone wins an election 11--0, then each person who votes for
him is less responsible for the victory than if he had won 6--5. We then define
a notion of degree of blame, which takes into account an agent's epistemic
state. Roughly speaking, the degree of blame of A for B is the expected degree
of responsibility of A for B, taken over the epistemic state of an agent
Regularization for Cox's proportional hazards model with NP-dimensionality
High throughput genetic sequencing arrays with thousands of measurements per
sample and a great amount of related censored clinical data have increased
demanding need for better measurement specific model selection. In this paper
we establish strong oracle properties of nonconcave penalized methods for
nonpolynomial (NP) dimensional data with censoring in the framework of Cox's
proportional hazards model. A class of folded-concave penalties are employed
and both LASSO and SCAD are discussed specifically. We unveil the question
under which dimensionality and correlation restrictions can an oracle estimator
be constructed and grasped. It is demonstrated that nonconcave penalties lead
to significant reduction of the "irrepresentable condition" needed for LASSO
model selection consistency. The large deviation result for martingales,
bearing interests of its own, is developed for characterizing the strong oracle
property. Moreover, the nonconcave regularized estimator, is shown to achieve
asymptotically the information bound of the oracle estimator. A coordinate-wise
algorithm is developed for finding the grid of solution paths for penalized
hazard regression problems, and its performance is evaluated on simulated and
gene association study examples.Comment: Published in at http://dx.doi.org/10.1214/11-AOS911 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Genericity and measure for exponential time
AbstractRecently, Lutz [14, 15] introduced a polynomial time bounded version of Lebesgue measure. He and others (see e.g. [11, 13–18, 20]) used this concept to investigate the quantitative structure of Exponential Time (E = DTIME(2lin)). Previously, Ambos-Spies et al. [2, 3] introduced polynomial time bounded genericity concepts and used them for the investigation of structural properties of NP (under appropriate assumptions) and E. Here we relate these concepts to each other. We show that, for any c ⩾ 1, the class of nc-generic sets has p-measure 1. This allows us to simplify and extend certain p-measure 1-results. To illustrate the power of generic sets we take the Small Span Theorem of Juedes and Lutz [11] as an example and prove a generalization for bounded query reductions
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