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

Subshifts, MSO Logic, and Collapsing Hierarchies

We use monadic second-order logic to define two-dimensional subshifts, or sets of colorings of the infinite plane. We present a natural family of quantifier alternation hierarchies, and show that they all collapse to the third level. In particular, this solves an open problem of [Jeandel & Theyssier 2013]. The results are in stark contrast with picture languages, where such hierarchies are usually infinite.Comment: 12 pages, 5 figures. To appear in conference proceedings of TCS 2014, published by Springe

The "paradox" of computability and a recursive relative version of the Busy Beaver function

In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy Beaver function which we will call Busy Beaver Plus function. Therefore, we will prove that the computable Busy Beaver Plus function defined on any Turing submachine is not computable by any program running on this submachine. We will thereby demonstrate the existence of a "paradox" of computability a la Skolem: a function is computable when "seen from the outside" the subsystem, but uncomputable when "seen from within" the same subsystem. Finally, we will raise the possibility of defining universal submachines, and a hierarchy of negative Turing degrees.Comment: 10 pages. 0 figures. Supported by the National Council for Scientific and Technological Development (CNPq), Brazil. Book chapter published in Information and Complexity, Mark Burgin and Cristian S. Calude (Editors), World Scientific Publishing, 2016, ISBN 978-981-3109-02-5, available at http://www.worldscientific.com/worldscibooks/10.1142/10017. arXiv admin note: substantial text overlap with arXiv:1612.0522

Complexity Bounds for Ordinal-Based Termination

`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program running times. Our main tool for this are length function theorems, which provide complexity bounds on the use of well quasi orders. We illustrate how to prove such theorems in the simple yet until now untreated case of ordinals. We show how to apply this new theorem to derive complexity bounds on programs when they are proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability Problems (RP 2014, 22-24 September 2014, Oxford

Synchronous Online Philosophy Courses: An Experiment in Progress

There are two main ways to teach a course online: synchronously or asynchronously. In an asynchronous course, students can log on at their convenience and do the course work. In a synchronous course, there is a requirement that all students be online at specific times, to allow for a shared course environment. In this article, the author discusses the strengths and weaknesses of synchronous online learning for the teaching of undergraduate philosophy courses. The author discusses specific strategies and technologies he uses in the teaching of online philosophy courses. In particular, the author discusses how he uses videoconferencing to create a classroom-like environment in an online class

Limits on Fundamental Limits to Computation

An indispensable part of our lives, computing has also become essential to industries and governments. Steady improvements in computer hardware have been supported by periodic doubling of transistor densities in integrated circuits over the last fifty years. Such Moore scaling now requires increasingly heroic efforts, stimulating research in alternative hardware and stirring controversy. To help evaluate emerging technologies and enrich our understanding of integrated-circuit scaling, we review fundamental limits to computation: in manufacturing, energy, physical space, design and verification effort, and algorithms. To outline what is achievable in principle and in practice, we recall how some limits were circumvented, compare loose and tight limits. We also point out that engineering difficulties encountered by emerging technologies may indicate yet-unknown limits.Comment: 15 pages, 4 figures, 1 tabl