Elvira Mayordomo

Almost every set in exponential time is P-bi-immune

A set A is P-bi-immune if neither A nor its complement has an infinite subset in P. We investigate here the abundance of P-bi-immune languages in linear-exponential time (E). Using resource-bounded measure, we prove that the class of P-bi-immune sets has measure 1 in E. This implies that “almost” every language in E is P-bi-immune. A bit further, we show that every p-random (pseudorandom) language is E-bi-immune. Regarding the existence of P-bi-immune sets in NP, we show that if NP does not have measure 0 in E, then NP contains a P-bi-immune set. Another consequence is that the class of ≤ 1-tt P -complete languages for E has measure 0 in E. In contrast, using the approach of resource-bounded category, it is shown that in E, the class of P-bi-immune languages lacks the property of Baire (the Baire category analogue of Lebesgue measurability).

Finite-state dimension

Classical Hausdorff dimension (sometimes called fractal dimension) was recently effectivized using gales (betting strategies that generalize martingales), thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary (infinite) sequences. In this paper we use gales computed by multi-account finite-state gamblers to develop the finite-state dimensions of sets of binary sequences and individual binary sequences. The theorem of Eggleston (Quart. J. Math. Oxford Ser. 20 (1949) 31-36) relating Hausdorff dimension to entropy is shown to hold for finite-state dimension, both in the space of all sequences and in the space of all rational sequences (binary expansions of rational numbers). Every rational sequence has finite-state dimension 0, but every rational number in [0,1] is the finite-state dimension of a sequence in the low-level complexity class AC0. Our main theorem shows that the finite-state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information-lossless finite-state compressors.

Cook versus Karp-Levin: separating completeness notions if NP is not small

Under the hypothesis that NP does not have p-measure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is shown that there is a language that is ≤ T P -complete (“Cook complete”), but not ≤ m P -complete (“Karp-Levin complete”), for NP. This conclusion, widely believed to be true, is not known to follow from P ≠ NP or other traditional complexity-theoretic hypotheses.

Measure, Stochasticity, and the Density of Hard Languages

The main theorem of this paper is that, for every real number

Effective Strong Dimension in Algorithmic Information and Computational Complexity

\alpha<1

Dimensions of Points in Self-Similar Fractals

(e.g.,

Programs, Proofs, Processes

\alpha=0.99

Resource-Bounded Balanced Genericity, Stochasticity and Weak Randomness

), only a measure 0 subset of the languages decidable in exponential time are

Zeta-Dimension

\leq^{P}_{n^{\alpha} - tt}

Cook versus Karp-Levin: Separating completeness notions if NP is not small

-reducible to languages that are not exponentially dense. Thus every