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We show that the full set of solutions to systems of equations andinequations in a hyperbolic group, with or without torsion, as shortlexgeodesic 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 deepgeometric results by Rips, Sela, Dahmani and Guirardel on decidability ofexistential theories of hyperbolic groups, work of computer scientistsincluding Plandowski, Je\.z, Diekert and others on $\mathsf{PSPACE}$ algorithmsto solve equations in free monoids and groups using compression, and anintricate language-theoretic analysis. The present work gives an essentiallyoptimal formal language description for all solutions in all hyperbolic groups,and an explicit and surprising low space complexity to compute them.

Solutions sets to systems of equations in hyperbolic groups are EDT0L in PSPACE