Solutions Sets to Systems of Equations in Hyperbolic Groups Are EDT0L in PSPACE

Laura Ciobanu, Murray Elder, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
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 NSPACE(n 2 log n) for the torsion-free case and 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, Jeż,
more » ... kert and others on 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. In this paper we consider systems of equations and inequations in hyperbolic groups, building on and generalising work done in the area of solving equations over various groups and monoids in PSPACE. Starting with work of Plandowski [38], many prominent researchers have given PSPACE algorithms [7, 14, 16, 17, 18, 30, 31] to find (all) solutions to systems of equations over free monoids, free groups, partially commutative monoids and groups, and virtually free groups (that is, groups which have a free subgroup of finite index). The satisfiability of equations over torsion-free hyperbolic groups is decidable by the work of Rips and Sela [39] , who reduced the problem in hyperbolic groups to solving equations in free groups, and then calling on Makanin's algorithm [36] . Kufleitner proved PSPACE for decidability in the torsion-free case [34] , without an explicit complexity bound, by following Rips-Sela and then using Plandowski's result [38] . Dahmani and Guirardel radically extended Rips and Sela's work to all hyperbolic groups (with torsion), by reducing systems of equations to systems over virtually free groups, which they then reduced to systems of twisted equations over free monoids [11] . In terms of describing solution sets, Grigorchuck and Lysionok gave efficient algorithms for the special case of quadratic equations [24] . Here we combine Rips, Sela, Dahmani and Guirardel's approach with recent work of the authors with Diekert [7, 14, 15 ] to obtain the following results.
doi:10.4230/lipics.icalp.2019.110 dblp:conf/icalp/CiobanuE19 fatcat:mpfuafgod5atphn7rywzw2ogn4