Fixed-rate encoding of individual sequences with side information

For every infinite sequence
x
and a given side-information sequence
y
, we define a quality
H(x|y)
called the finite-state conditional complexity of
x
given
y
. It is shown that
H(x|y)
is the smallest asymptotically attainable fixed-rate at which
x
can be transmitted with negligibly small distortion, given
y
. Moreover, it is demonstrated that in order to achieve an arbitrary small distortion for all sequences such that
H(x|y)
is less than the allowable transmission rate it is not necessary for the encoder to have access to the side-information sequence
y
(provided it is available to the decoder). This result is a generalization of the classical Slepian-Wolf result for cases where the probabilistic characterization of
x
and
y
is not known, or does not exist.