  In this paper, we investigate the computability of thermodynamic invariants
at zero temperature for one-dimensional subshifts of finite type. In
particular, we prove that the residual entropy (i.e., the joint ground state
entropy) is an upper semi-computable function on the space of continuous
potentials, but it is not computable. Next, we consider locally constant
potentials for which the zero-temperature measure is known to exist. We
characterize the computability of the zero-temperature measure and its entropy
for potentials that are constant on cylinders of a given length k. In
particular, we show the existence of an open and dense set of locally constant
potentials for which the zero-temperature measure can be computationally
identified as an elementary periodic point measure. Finally, we show that our
methods do not generalize to treat the case when k is not given
