Dimension in Complexity Classes
SIAM Journal on Computing
The dimensions of individual strings and sequences
Information and Computation
SIGACT news complexity theory column 48
ACM SIGACT News
Effective Strong Dimension in Algorithmic Information and Computational Complexity
SIAM Journal on Computing
IEEE Transactions on Information Theory
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We study the setting in which the bits of an unknown infinite binary sequence x are revealed sequentially to an observer. We show that very limited assumptions about x allow one to make successful predictions about unseen bits of x. First, we study the problem of successfully predicting a single 0 from among the bits of x. In our model we have only one chance to make a prediction, but may do so at a time of our choosing. This model is applicable to a variety of situations in which we want to perform an action of fixed duration, and need to predict a "safe" time-interval to perform it. Letting Nt denote the number of 1s among the first t bits of x, we say that x is "ε-weakly sparse" if liminf (Nt/t) ≤ ε. Our main result is a randomized algorithm that, given any ε-weakly sparse sequence x, predicts a 0 of x with success probability as close as desired to 1 -- ε. Thus we can perform this task with essentially the same success probability as under the much stronger assumption that each bit of x takes the value 1 independently with probability ε. We apply this result to show how to successfully predict a bit (0 or 1) under a broad class of possible assumptions on the sequence x. The assumptions are stated in terms of the behavior of a finite automaton M reading the bits of x. We also propose and solve a variant of the well-studied "ignorant forecasting" problem. For every ε 0, we give a randomized forecasting algorithm Sε that, given sequential access to a binary sequence x, makes a prediction of the form: "A p fraction of the next N bits will be 1s." (The algorithm gets to choose p, N, and the time of the prediction.) For any fixed sequence x, the forecast fraction p is accurate to within ±ε with probability 1 − ε.