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witzvo comments on Open Problems Related to Solomonoff Induction - Less Wrong
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I'm not sure what you have in mind by basing Solomonoff induction on a second-order logic. Can you give a reference?
Here is what I'd suggest as a baseline. Use John Tromp's Binary Lambda Calculus with some variation of Levin Search to explore time/complexity bounded programs that output some initial sequence of data.