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 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 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| .

We show that there are infinitely many binary strings z, such that the sum of the on-line decision complexity of predicting the even bits of z given the previous uneven bits, and the decision complexity of predicting the uneven bits given the previous event bits, exceeds the Kolmogorov complexity of z by a linear term in the length of z.

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