Random equations in free groups

The unsolvability of some algorithmic problems is proved for equations in free groups and semigroups, namely, some simple properties of the solutions of the equations are determined and the absence of an algorithm permitting the determination of whether an arbitrary equation in a free group or semigroup has a solution with the properties introduced is proved.

Equations in free groups have become prominent recently in connection with the solution to the well-known Tarski conjecture. Results of Makanin and Rasborov show that solvability of systems of equations is decidable and there is a method for writing down in principle all solutions. However, no practical method is known; the best estimate for the complexity of the decision procedure is P -space. The special case of one-variable equations in free groups has been open for a number of years, although it is known that the solution sets admit simple descriptions. We use cancellation arguments to give a short and direct proof of this result and also to give a practical polynomial-time algorithm for finding solution sets. One-variable equations are the only general subclass of equations in free groups for which such results are known. We improve on previous attempts to use cancellation arguments by employing a new method of reduction motivated by techniques from formal language theory. Our paper is self-contained; we assume only knowedge of basic facts about free groups.

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.

We introduce the notion of a parametric equation in a free group; this is an equation containing natural parameters as exponents and a system of linear Diophantine equations relating these exponents. For these equations, we introduce elementary transformations that are necessary for the description of general solutions of ordinary equations in a free group. We prove that it is possible to linearize any relation among parameters that appears in the course of transformations of the given equation.

Finding all solutions of equations in free groups and monoids with involution

We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solution sets of quadratic equations in a free group.

Equations in free semigroups with involution and their relation to equations in free groups

Let w ( x ) w(x) be a one-variable equation in a free group F of finite rank. Lyndon has proved that it is possible to associate effectively to w ( x ) w(x) the set of its solutions, whereas Appel and Lorenc have provided a simpler representation of the set inferred. In this paper, we invert the problem and demonstrate that if the elements of any set S ⊂ F S \subset F are solutions of an equation w ( x ) w(x) , then w ( x ) w(x) belongs to the normal closure of finitely many short equations associated to S. A few consequences are given.

Let $w(x)$ be a one-variable equation in a free group

Let h1, h2,… be a sequence of elements in a free group and let H be the subgroup they generate. Let H′ be the subgroup generated by w1, w2, …, where each wi is a word in hi and possibly other hj, such that the associated directed graph has the finite paths property. We show that rank H′≥ rank H. As a corollary we get that [Formula: see text], where [Formula: see text] is the subgroup generated by the roots of the elements in H. If H0 is finitely generated and the sequence of subgroups H0, H1, H2, … satisfies [Formula: see text] then the sequence stabilizes, i.e. for some m, Hi=Hi+1 for every i≥ m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in "Lyndon equation" [Formula: see text] the exponents ai satisfy gcd (a1,…,an)≠1 then the inner rank is ⌊ n/2⌋. The proofs are mostly elementary.

We give slightly simplified version of the proof of Makanin's theorem on decidability of solvability problem for equations in free groups. We also provide an analysis of the Makanin's proof and point out why the estimates on the complexity of Makanin's algorithm and on the length of a minimal solution of an equation in a free group are not primitive recursive.

We give a brief account of some of the traditional ways that genetic algo- rithms have been applied, and explain how our approach to the use of genetic algorithms for solving problems in combinatorial group theory differs. We find that, in our situation, there seems to be a correlation between successful genetic algorithms and the existence of good non-genetic, sometimes deterministic, algorithms. We use a class of equations in free groups as a test bench. In particular, it allows us to trace the convergence of co-evolution of the population of fitness functions to a deterministic solution.

The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups and monoids with involution in the presence of rational constraints. This became possible due to the recently invented recompression technique of the second author.

We discuss the use of evolutionary algorithms for solving problems in combinatorial group theory, using a class of equations in free groups as a test bench. We find that, in this context, there seems to be a correlation between successful evolutionary algorithms and the existence of good deterministic algorithms. We also trace the convergence of co-evolution of the population of fitness functions to a deterministic solution.

Implicit function theorem over free groups