In this paper, we investigate the existence of a Friedberg numbering in fragments of Peano Arithmetic and initial segments of Gödel’s constructible hierarchy Lα, where α is Σ1 admissible. We prove that (1) Over P− +BΣ2, the existence of a Friedberg numbering is equivalent to IΣ2, and (2) For Lα, there is a Friedberg numbering if and only if the tame Σ2 projectum of α equals the Σ2 cofinality of α.

Novel digital technologies or simply unprecedented ways of using known technologies, or sometimes just immense increases in performance of proven technologies form the space needed to drive the digital economy. Here, motivated entrepreneurs with ingenious or sometimes downright strikingly simple ideas will find the soil to thrive on. If then potent and ideally patient risk capital providers com...

Identification of programs for computable functions from their graphs by algorithmic devices is a well studied problem in learning theory. Freivalds and Chen consider identification of ‘minimal’ and ‘nearly minimal’ programs for functions from their graphs. Freivalds showed that there exists a Gödel numbering in which only finite classes of functions can be identified using minimal programs. To...

Petascale parallel computers with more than a million processing cores are expected to be available in a couple of years. Although MPI is the dominant programming interface today for large-scale systems that at the highest end already have close to 300,000 processors, a challenging question to both researchers and users is whether MPI will scale to processor and core counts in the millions. In ...