Resource-bounded Measure on Probabilistic Classes

We extend Lutz’s resource-bounded measure to probabilistic classes, and obtain notions of resource-bounded measure on probabilistic complexity classes such as BPE and BPEXP. Unlike former attempts, our resource bounded measure notions satisfy all three basic measure properties, that is every singleton {L} has measure zero, the whole space has measure one, and "enumerable infinite unions" of measure zero sets have measure zero

Resource-bounded Measure on Probabilistic Classes

We extend Lutz’s resource-bounded measure to probabilistic classes, and obtain notions of resource-bounded measure on probabilistic complexity classes such as BPE and BPEXP. Unlike former attempts, our resource bounded measure notions satisfy all three basic measure properties, that is every singleton {L} has measure zero, the whole space has measure one, and "enumerable infinite unions" of measure zero sets have measure zero

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

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

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

One-Way Functions and Balanced NP

The existence of cryptographically secure one-way functions is related to the measure of a subclass of NP. This subclass, called BNP (``balanced NP\u27\u27), contains 3SAT and other standard NP problems. The hypothesis that BNP is not a subset of P is equivalent to the P \u3c\u3e NP conjecture. A stronger hypothesis, that BNP is not a measure 0 subset of E_2 = DTIME(2^polynomial) is shown to have the following two consequences. 1. For every k, there is a polynomial time computable, honest function f that is (2^{n^k})/n^k-one-way with exponential security. (That is, no 2^{n^k}-time-bounded algorithm with n^k bits of nonuniform advice inverts f on more than an exponentially small set of inputs.) 2. If DTIME(2^n) ``separates all BPP pairs,\u27\u27 then there is a (polynomial time computable) pseudorandom generator that passes all probabilistic polynomial-time statistical tests. (This result is a partial converse of Yao, Boppana, and Hirschfeld\u27s theorem, that the existence of pseudorandom generators passing all polynomial-size circuit statistical tests implies that BPP\subset DTIME(2^{n^epsilon}) for all epsilon\u3e0.) Such consequences are not known to follow from the weaker hypothesis that P \u3c\u3e NP