Short lists with short programs in short time

Given a machine U, a c-short program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any standard Turing machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contain a $${\operatorname{O}\bigl({\mathrm{log}}|x|\bigr)}$$O(log|x|)-short program for x. We also show that there exists a computable function that maps every x to a list of size |x|2 containing a $${\operatorname{O}\bigl(1\bigr)}$$O(1)-short program for x. This is essentially optimal because we prove that for each such function there is a c and infinitely many x for which the list has size at least c|x|2. Finally we show that for some standard machines, computable functions generating lists with 0-short programs must have infinitely often list sizes proportional to 2|x|.

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