Compression of Data Streams Down to Their Information Content

According to the Kolmogorov complexity, every finite binary string is compressible to a shortest code-its information content-from which it is effectively recoverable. We investigate the extent to which this holds for the infinite binary sequences (streams). We devise a new coding method that uniformly codes every stream X into an algorithmically random stream Y, in such a way that the first n bits of X are recoverable from the first I(X |n) bits of Y, where I is any partial computable information content measure that is defined on all prefixes of X, and where X |n is the initial segment of X of length n. As a consequence, if g is any computable upper bound on the initial segment prefix-free complexity of X, then X is computable from an algorithmically random Y with oracle-use at most g. Alternatively (making no use of such a computable bound g), one can achieve an the oracle-use bounded above by K(X |n) + log n. This provides a strong analogue of Shannon's source coding theorem for the algorithmic information theory.