Equations over free inverse monoids with idempotent variables

We introduce the notion of idempotent variables for studying equations in inverse monoids. It is proved that it is decidable in singly exponential time (DEXPTIME) whether a system of equations in idempotent variables over a free inverse monoid has a solution. The result is proved by a direct reduction to solve language equations with one-sided concatenation and a known complexity result by Baader and Narendran: Unification of concept terms in description logics, 2001. We also show that the problem becomes DEXPTIME hard , as soon as the quotient group of the free inverse monoid has rank at least two. Decidability for systems of typed equations over a free inverse monoid with one irreducible variable and at least one unbalanced equation is proved with the same complexity for the upper bound. Our results improve known complexity bounds by Deis, Meakin, and Senizergues: Equations in free inverse monoids, 2007. Our results also apply to larger families of equations where no decidability has been previously known.Comment: 28 pages. The conference version of this paper appeared in the proceedings of 10th International Computer Science Symposium in Russia, CSR 2015, Listvyanka, Russia, July 13-17, 2015. Springer LNCS 9139, pp. 173-188 (2015

Equations Over Free Inverse Monoids with Idempotent Variables

We introduce the notion of idempotent variables for studying equations in inverse monoids. It is proved that it is decidable in singly exponential time (DEX-PTIME) whether a system of equations in idempotent variables over a free inverse monoid has a solution. Moreover the problem becomes hard for DEXPTIME, as soon as the quotient group of the free inverse monoid has rank at least two. The upper bound is proved by a direct reduction to solve language equations with one-sided concatenation and a known complexity result by Baader and Narendran (J. Symb. Comput. 31, 277-305 2001). For the lower bound we show hardness for a restricted class of language equations. Decidability for systems of typed equations over a free inverse monoid with one irreducible variable and at least one unbalanced equation is proved with the same complexity upper-bound. Our results improve known complexity bounds by Deis et al. (IJAC 17, 761-795 2007). Our results also apply to larger families of equations where no decidability has been previously known. The lower bound confirms a conjecture made in the conference version of this paper