The computational complexity of two simple string problems on compressed input strings is considered: the querying problem (What is the symbol at a given position in a given input string?) and the embedding problem (Can the first input string be embedded into the second input string?). Straight-line programs are used for text compression. It is shown that the querying problem becomes P-complete for compressed strings, while the embedding problem becomes hard for the complexity class $\Theta^{p}_{2}$.