Three equivalent ordinal notation systems in cubical Agda

We present three ordinal notation systems representing ordinals below epsilon zero in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. We show how ordinal arithmetic can be developed for these systems, and how they admit a transfinite induction principle. We prove that all three notation systems are equivalent, so that we can transport results between them using the univalence principle. All our constructions have been implemented in cubical Agda

Diffusion-induced vortex filament instability in 3-dimensional excitable media

We studied the stability of linear vortex filaments in 3-dimensional (3D) excitable media, using both analytical and numerical methods. We found an intrinsic 3D instability of vortex filaments that is diffusion-induced, and is due to the slower diffusion of the inhibitor. This instability can result either in a single helical filament or in chaotic scroll breakup, depending on the specific kinetic model. When the 2-dimensional dynamics were in the chaotic regime, filament instability occurred via on-off intermittency, a failure of chaos synchronization in the third dimension.Comment: 5 pages, 5 figures, to appear in PRL (September, 1999

The Generic Model of Computation

Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state machines. This approach has recently been extended to suggest a formalization of the notion of effective computation over arbitrary countable domains. The central notions are summarized herein.Comment: In Proceedings DCM 2011, arXiv:1207.682

Konputazio-makinak eta adimena: '"Computing Machinery and Intelligence'". 1950. Mind 49 (236): 433-460.

Itzulpen hau A. M. Turingek, 1950ean, Mind aldizkarian '"Computing Machinery and Intelligence'" izenburupean argitaratutako bertsioarena da. Itzulpenaren lehenagoko bertsioa hemen argitaratu zen, 2003an: Gogoa III-2, 171-196 . [Bertsio berri hau Larraitz Zubeldiak eta Kepa Kortak orraztu dute]

Alan Turing Institute

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