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 in free inverse monoids

In this dissertation, we study the solvability of equations in the free inverse monoid generated by a set A. This monoid is denoted by FIM(A). Our study is accomplished by using the following fact. A necessary condition for an equation to be consistent (i.e. have a solution) in FIM(A) is that the equation is consistent in the setting of the corresponding free group generated by A. This group is denoted by FG( A). If the equation is consistent in FG( A), we examine conditions on that equation to determine whether it is consistent in the setting of FIM(A ). It has been shown that the problem of determining consistency of systems of equations in FIM(A) is undecidable. We consider specific classes of equations and show that consistency is decidable. In particular, we examine multilinear, single variable and quadratic equations in FIM(A). We show that determining the consistency for systems of multilinear equations in FIM(A) is undecidable. In addition, we associate with each system of equations in FIM(A ), a single equation in a free inverse monoid generated by a set which contains A; this equation has the property that its solutions in FIM(A) are the same as the solutions to the original system of equations in FIM( A)

Equations in free inverse monoids

It is known that the problem of determining consistency of a finite system of equations in a free group or a free monoid is decidable, but the corresponding problem for systems of equations in a free inverse monoid of rank at least two is undecidable. Any solution to a system of equations in a free inverse monoid induces a solution to the corresponding system of equations in the associated free group in an obvious way, but solutions to systems of equations in free groups do not necessarily lift to solutions in free inverse monoids. In this paper we show that the problem of determining whether a solution to a finite system of equations in a free group can be extended to a solution of the corresponding system in the associated free inverse monoid is decidable. We are able to use this to solve the consistency problem for certain classes of single variable equations in free inverse monoids.