The language compression problem asks for succinct descriptions of
the strings in a language A such that the strings can be efficiently
recovered from their description when given a membership oracle for
A. We study randomized and nondeterministic decompression schemes
and investigate how close we can get to the information theoretic lower
bound of log |A^{=n}| for the description length of strings
of length n.

Using nondeterminism alone, we can achieve the information theoretic
lower bound up to an additive term of O(sqrt{log |A^{=n}|} log n);
using both nondeterminism and randomness, we can make do with an
excess term of O(log^3 n). With randomness alone, we show a lower
bound of n - log |A^{=n}| - O(log n) on the description length of strings in A of length
n, and a lower bound of 2 log |A^{=n}| - O(1) on the
length of any program that distinguishes a given string of length n
in A from any other string. The latter lower bound is
tight up to an additive term of O(log n).

The key ingredient for our upper bounds is the relativizable hardness
versus randomness tradeoffs based on the Nisan-Wigderson pseudorandom
generator construction.