  We show that the full set of solutions to systems of equations and
inequations in a hyperbolic group, with or without torsion, as shortlex
geodesic words, is an EDT0L language whose specification can be computed in
$\mathsf{NSPACE}(n^2\log n)$ for the torsion-free case and
$\mathsf{NSPACE}(n^4\log n)$ in the torsion case. Our work combines deep
geometric results by Rips, Sela, Dahmani and Guirardel on decidability of
existential theories of hyperbolic groups, work of computer scientists
including Plandowski, Je\.z, Diekert and others on $\mathsf{PSPACE}$ algorithms
to solve equations in free monoids and groups using compression, and an
intricate language-theoretic analysis. The present work gives an essentially
optimal formal language description for all solutions in all hyperbolic groups,
and an explicit and surprising low space complexity to compute them.
