Ricardo Ruiz | Searcharxiv

arXiv2024

You Can't Solve These Super Mario Bros. Levels: Undecidable Mario Games

We prove RE-completeness (and thus undecidability) of several 2D games in the Super Mario Bros. platform video game series: the New Super Mario Bros. series (original, Wii, U, and 2), and both Super Mario Maker games in all five game styles (Super Mario Bros. 1 and 3, Super Mario World, New Super Mario Bros. U, and Super Mario 3D World). These results hold even when we restrict to constant-size levels and screens, but they do require generalizing to allow arbitrarily many enemies at each location and onscreen, as well as allowing for exponentially large (or no) timer. Our New Super Mario Bros. constructions fit within one standard screen size. In our Super Mario Maker reductions, we work within the standard screen size and use the property that the game engine remembers offscreen objects that are global because they are supported by "global ground". To prove these Mario results, we build a new theory of counter gadgets in the motion-planning-through-gadgets framework, and provide a suite of simple gadgets for which reachability is RE-complete.

cs.CC↗

arXiv2015

Bit Patterned Magnetic Recording: Theory, Media Fabrication, and Recording Performance

Bit Patterned Media (BPM) for magnetic recording provide a route to densities $>1 Tb/in^2$ and circumvents many of the challenges associated with conventional granular media technology. Instead of recording a bit on an ensemble of random grains, BPM uses an array of lithographically defined isolated magnetic islands, each of which stores one bit. Fabrication of BPM is viewed as the greatest challenge for its commercialization. In this article we describe a BPM fabrication method which combines e-beam lithography, directed self-assembly of block copolymers, self-aligned double patterning, nanoimprint lithography, and ion milling to generate BPM based on CoCrPt alloys. This combination of fabrication technologies achieves feature sizes of $<10 nm$, significantly smaller than what conventional semiconductor nanofabrication methods can achieve. In contrast to earlier work which used hexagonal close-packed arrays of round islands, our latest approach creates BPM with rectangular bitcells, which are advantageous for integration with existing hard disk drive technology. The advantages of rectangular bits are analyzed from a theoretical and modeling point of view, and system integration requirements such as servo patterns, implementation of write synchronization, and providing for a stable head-disk interface are addressed in the context of experimental results. Optimization of magnetic alloy materials for thermal stability, writeability, and switching field distribution is discussed, and a new method for growing BPM islands on a patterned template is presented. New recording results at $1.6 Td/in^2$ (teradot/inch${}^2$, roughly equivalent to $1.3 Tb/in^2$) demonstrate a raw error rate $<10^{-2}$, which is consistent with the recording system requirements of modern hard drives. Extendibility of BPM to higher densities, and its eventual combination with energy assisted recording are explored.

cond-mat.other↗