Makanin's algorithm is not primitive recursive
Theoretical Computer Science
Satisfiability of word equations with constants is in PSPACE
Journal of the ACM (JACM)
The existential theory of equations with rational constraints in free groups is PSPACE-complete
Information and Computation
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CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
Rational subsets and submonoids of wreath products
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II
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It is shown that the existential theory of ${\mathbb G}$ with rational constraints, over an HNN-extension ${\mathbb G}=\langle {\mathbb H},t; t^{-1}at=\varphi(a) (a \in A) \rangle$ is decidable, provided that the same problem is decidable in the base group ${\mathbb H}$ and that A is a finite group. The positive theory of ${\mathbb G}$ is decidable, provided that the existential positive theory of ${\mathbb G}$ is decidable and that A and ϕ(A) are proper subgroups of the base group ${\mathbb H}$ with A ∩ϕ(A) finite. Analogous results are also shown for amalgamated products. As a corollary, the positive theory and the existential theory with rational constraints of any finitely generated virtually-free group is decidable