Complexity Hierarchies Beyond Elementary

We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a non-elementary complexity, which occur naturally in logic, combinatorics, formal languages, verification, etc., with complexities ranging from simple towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep updating the catalogue of problems from Section 6 in future revision

Some Thoughts on Hypercomputation

Hypercomputation is a relatively new branch of computer science that emerged from the idea that the Church--Turing Thesis, which is supposed to describe what is computable and what is noncomputable, cannot possible be true. Because of its apparent validity, the Church--Turing Thesis has been used to investigate the possible limits of intelligence of any imaginable life form, and, consequently, the limits of information processing, since living beings are, among others, information processors. However, in the light of hypercomputation, which seems to be feasibly in our universe, one cannot impose arbitrary limits to what intelligence can achieve unless there are specific physical laws that prohibit the realization of something. In addition, hypercomputation allows us to ponder about aspects of communication between intelligent beings that have not been considered befor

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Wittgenstein on Pseudo-Irrationals, Diagonal Numbers and Decidability

In his early philosophy as well as in his middle period, Wittgenstein holds a purely syntactic view of logic and mathematics. However, his syntactic foundation of logic and mathematics is opposed to the axiomatic approach of modern mathematical logic. The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and operations; its aim is to reduce mathematics to operations. This paper illustrates Wittgenstein’s approach by examining his discussion of irrational numbers

Cosmic Acceleration from Causal Backreaction with Recursive Nonlinearities

We revisit the causal backreaction paradigm, in which the need for Dark Energy is eliminated via the generation of an apparent cosmic acceleration from the causal flow of inhomogeneity information coming in towards each observer from distant structure-forming regions. This second-generation formalism incorporates "recursive nonlinearities": the process by which already-established metric perturbations will then act to slow down all future flows of inhomogeneity information. Here, the long-range effects of causal backreaction are now damped, weakening its impact for models that were previously best-fit cosmologies. Nevertheless, we find that causal backreaction can be recovered as a replacement for Dark Energy via the adoption of larger values for the dimensionless `strength' of the clustering evolution functions being modeled -- a change justified by the hierarchical nature of clustering and virialization in the universe, occurring on multiple cosmic length scales simultaneously. With this, and with one new model parameter representing the slowdown of clustering due to astrophysical feedback processes, an alternative cosmic concordance can once again be achieved for a matter-only universe in which the apparent acceleration is generated entirely by causal backreaction effects. One drawback is a new degeneracy which broadens our predicted range for the observed jerk parameter $j_{0}^{\mathrm{Obs}}$, thus removing what had appeared to be a clear signature for distinguishing causal backreaction from Cosmological Constant $\Lambda$CDM. As for the long-term fate of the universe, incorporating recursive nonlinearities appears to make the possibility of an `eternal' acceleration due to causal backreaction far less likely; though this does not take into account gravitational nonlinearities or the large-scale breakdown of cosmological isotropy, effects not easily modeled within this formalism.Comment: 53 pages, 7 figures, 3 tables. This paper is an advancement of previous research on Causal Backreaction; the earlier work is available at arXiv:1109.4686 and arXiv:1109.515