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|.

中文翻译：

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短名单短节目短时间

给定机器 U，x 的 c-short 程序是一个字符串 p，使得 U(p) = x 并且 p 的长度以 c +（x 的最短程序的长度）为界。我们证明，对于任何标准图灵机，可以在多项式时间内计算输入 xa 多项式大小的列表，保证包含 $${\operatorname{O}\bigl({\mathrm{log}}|x|\ bigr)}$$O(log|x|)-x 的简短程序。我们还表明存在一个可计算的函数，该函数将每个 x 映射到一个大小为 |x|2 的列表，该列表包含一个 $${\operatorname{O}\bigl(1\bigr)}$$O(1)-short program for X。这本质上是最优的，因为我们证明对于每个这样的函数都有 ac 和无穷多个 x，其中列表的大小至少为 c|x|2。最后我们证明对于一些标准机器，