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Eugine_Nier comments on The Irrationality Game - Less Wrong
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I believe it would at the very least have to sacrifice at least all adaptability by doing so, as in only sending out nodes with all instructions in ROM and instructions to periodically rest all non-ROM memory and shelf-destruct if it notices any failures of its triple redundancy ROM. As well as an extremely strong directive against anything that would let nodes store long term state.
Remember, you're the one trying to prove impossibility of a task here. Your inability to imagine a solution to the problem is only very weak evidence.