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By Andrea Milani; Giovanni B Valsecchi; David Vokrouhlický; International Astronomical Union. Symposium

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Additional info for Near Earth objects, our celestial neighbors : opportunity and risk : proceedings of the 236th Symposium of the International Astronomical Union held in Prague, Czech Republic August 14-18, 2006

Example text

I ∈ ns , labels (i, Ψ ) = , σ ∈ labelp (i, Ψ ), and Ψ = adde ((i, nc + 1), σ, addc ( , updates (i ← ( , Q), Ψ ))) where Q = labelp (i, Ψ )∪{σ}, and is non-deterministically chosen from Λc . Although not explicitly present, we assume here that both the client node nc + 1 (with some label taken from Λc ) and the edge (i, nc + 1) with label σ are in the upward closure of Ψ . We add them to Ψ since their presence is a precondition for the firing of r. A. Abdulla, G. Delzanno, and A. Rezine two cases.

Almost everything in the continuous world is already approximate and the infinite precision used to simulate TMs is unrealistic from the points of view of modeling, measurement and computation. Thus the expectations were lowered once more, from exact semi-decidability to approximate computation of reachable sets and then from unbounded time horizon to a finite one. Approximating tubes of trajectories of continuous dynamical systems turned out to be an interesting research domain involving graph algorithms, computational geometry in high dimension, numerical analysis and other 1 Such systems are very attractive for computer scientists like us because they can be studied with elementary linear algebra and there is no need to understand those horrible differential equations.

Here algorithmic information theory comes into play: We reject the hypothesis if we observe a simple event that has negligible probability according to this hypothesis. For example, if coin tossing produces thousand tails, this event is simple and has negligible probability, so we don’t believe the coin is fair. Both conditions (“simple” and “negligible probability”) are important: the event “the first bit is a tail” is simple but has probability 1/2, so it does not discredit the coin. On the other hand, every sequence of outcomes has negligible probability 2−1000 , but if it is not simple, its appearance does not discredits the fair coin assumption.

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