Satisfiability of equations in free groups is in PSPACE

We prove that the computational complexity of the problem of deciding if an equation in a free group has a solution is PSPACE. The problem was proved decidable in 1982 by Makanin, w h o s e algorithm was proved later to be non primitive recursive: this was the best upper bound known for this problem. Our proof consists in reducing equations in free groups to equations in free semigroups with antiinvolution, and presenting an algorithm for deciding equations in free semigroups with antiinvolution.

[1]  G. Makanin.  DECIDABILITY OF THE UNIVERSAL AND POSITIVE THEORIES OF A FREE GROUP , 1985 .

[2]  Ju I Hmelevskiĭ.  SYSTEMS OF EQUATIONS IN A FREE GROUP. I , 1971 .

[3]  Valery Durnev.  Studying Algorithmic Problems for Free Semi-groups and Groups , 1997, LFCS.

[4]  Leszek Pacholski,et al.  Complexity of Makanin's algorithm , 1996, JACM.

[5]  Wojciech Plandowski,et al.  Satisfiability of word equations with constants is in PSPACE , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[6]  Claudio Gutiérrez.  Equations in Free Semigroups with Anti-involution and Their Relation to Equations in Free Groups , 2000, LATIN.

[7]  A. Razborov.  ON SYSTEMS OF EQUATIONS IN A FREE GROUP , 1985 .

[8]  Wojciech Plandowski,et al.  Application of Lempel-Ziv Encodings to the Solution of Words Equations , 1998, ICALP.

[9]  R. Lyndon.  Equations in free groups , 1960 .

[10]  Leszek Pacholski,et al.  Makanin's Algorithm is not Primitive Recursive , 1998, Theor. Comput. Sci..

[11]  Claudio Gutiérrez.  Satisfiability of word equations with constants is in exponential space , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[12]  G. Makanin.  EQUATIONS IN A FREE GROUP , 1983 .

[13]  Ju I Hmelevskiĭ.  SYSTEMS OF EQUATIONS IN A FREE GROUP. II , 1972 .

[14]  G S Makanin.  Equations in a free semigroup , 1981 .

[15]  Wojciech Plandowski.  Satisfiability of word equations with constants is in NEXPTIME , 1999, STOC '99.

[16]  G. Makanin.  The Problem of Solvability of Equations in a Free Semigroup , 1977 .