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**1 - 7**of**7**### Turing's titanic machine? -- Embodied and disembodied computing at the Turing Centenary

, 2012

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### New Challenges in the Axiomatization of Relativity Theory1

"... Abstract: Einstein’s theory of relativity not just had but still has a great impact on science. It has an impact even on military sciences, e.g., via GPS technology (which cannot exist without relativity theory). Any theory with such an impact is also interesting from the point of view of axiomatic ..."

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Abstract: Einstein’s theory of relativity not just had but still has a great impact on science. It has an impact even on military sciences, e.g., via GPS technology (which cannot exist without relativity theory). Any theory with such an impact is also interesting from the point of view of axiomatic foundations. The aim of this paper is to outline the new challenges of the axiomatic approach to relativity theory developed by our research group led by Hajnal Andréka and István Németi. 1. Cutting edge engineering based on the two theories of relativity Einstein’s formulated his theory of special relativity in 1905 a decade later he generalized special relativity and introduced his theory of general relativity in 1915. Even Einstein's special theory of relativity radically changed our way of thinking about space and time, because among other things it states that there are no such things as observer independent concepts of time and space. However, the two theories of relativity not just had a great impact on our way of thinking about space and time but on engineering sciences and even on our every day life. GPS technology is a famous cutting edge technology of today, which greatly depends on the

### Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 16 Visualizations of Relativity, Relativistic

"... sent visualizations of relativity. For example, we will present a movie showing what an astronaut would see while flying through a huge Kerr-Newmann worm-hole or any other kind of wormhole. We will also outline the ideas of relativistic hypercomputing, i.e., how Malament-Hogarth spacetimes can be us ..."

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sent visualizations of relativity. For example, we will present a movie showing what an astronaut would see while flying through a huge Kerr-Newmann worm-hole or any other kind of wormhole. We will also outline the ideas of relativistic hypercomputing, i.e., how Malament-Hogarth spacetimes can be used for de-signing artificial systems computing beyond the Turing barrier. Any spacetime admitting a CTC (closed timelike curve) is suitable for constructing such a hy-percomputer, but the existence of CTC's is not really needed for this. A much milder condition called Malament-Hogarth property is sufficient. We refer to [1], [2], and [3] for more detail. (The most satisfactory solution to the so called blue-shift problem is available in [4].) References [1] Dávid, Gy., Németi, I., Relativistic computers and the Turing barrier.

### accumulations; c.e. and d-c.e. real numbers; Signal machine.

, 2012

"... Abstract geometrical computation 7: Geometrical accumulations and computably enumerable real numbers ..."

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Abstract geometrical computation 7: Geometrical accumulations and computably enumerable real numbers

### Author manuscript, published in "UC '11, Turku: Finland (2011)" DOI: 10.1007/978-3-642-21341-0 Geometrical accumulations and

, 2012

"... Abstract. Abstract geometrical computation involves drawing colored line segments (traces of signals) according to rules: signals with similar color are parallel and when they intersect, they are replaced according to their colors. Time and space are continuous and accumulations can be devised to un ..."

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Abstract. Abstract geometrical computation involves drawing colored line segments (traces of signals) according to rules: signals with similar color are parallel and when they intersect, they are replaced according to their colors. Time and space are continuous and accumulations can be devised to unlimitedly accelerate a computation and provide, in a finite duration, exact analog values as limits. In the present paper, we show that starting with rational numbers for coordinates and speeds, the time of any accumulation is a c.e. (computably enumerable) real number and moreover, there is a signal machine and an initial configuration that accumulates at any c.e. time. Similarly, we show that the spatial positions of accumulations are exactly the d-c.e. (difference of computably enumerable) numbers. Moreover, there is a signal machine that can accumulate at any c.e. time or d-c.e. position. Key-words. Abstract geometrical computations; Computable analysis; Geometrical accumulations; c.e. and d-c.e. real numbers; Signal machine. 1