Makanin's Algorithm Research Articles

On PSPACE generation of a solution set of a word equation and its applications

There are three approaches to solve satisfiability problem for word equations:•Makanin's algorithm,•Plandowski's algorithm,•Jez's algorithm. The approach by Makanin and the approach by Jez can be extended to generate a representation of all solutions of a word equation. In case of Makanin's algorithm the representation is usually an infinite tree. In case of Jez's approach the representation is a finite graph.Until now there was no extension of Plandowski's algorithm to do the same. We fill this gap. We present extension of Plandowski's algorithm which generates a representation of all solutions of a word equation. The representation is a finite graph.This allows to decide in PSPACE whether a solution set of a word equation is finite. Our approach is generalized to solve in PSPACE two problems dealing with expressible relations, namely the emptiness of the relation and finiteness of the relation.

Computing JSJ decompositions of hyperbolic groups

We present an algorithm that computes Bowditch's canonical JSJ decomposition of a given one-ended hyperbolic group over its virtually cyclic subgroups. The algorithm works by identifying topological features in the boundary of the group. As a corollary we also show how to compute the JSJ decomposition of such a group over its virtually cyclic subgroups with infinite centre. We also give a new algorithm that determines whether or not a given one-ended hyperbolic group is virtually fuchsian. Our approach uses only the geometry of large balls in the Cayley graph and avoids Makanin's algorithm.

A short solution for the HDT0L sequence equivalence problem

We give a solution for the HDT0L sequence equivalence problem which uses Hilbert's Basis Theorem but avoids the use of Makanin's algorithm or Hall's results about metabelian groups.

Makanin's algorithm is not primitive recursive

We prove that, contrary to the common belief, the algorithm deciding satisfiability of equations in free groups, given by Makanin, is not primitive recursive.

Complexity of Makanin's algorithm

The exponent of periodicity is an important factor in estimates of complexity of word-unification algorithms. We prove that the exponent of periodicity of a minimal solution of a word equation is of order 2 1.07d , where d is the length of the equation. We also give a lower bound 2 0.29d so our upper bound is almost optimal and exponentially better than the original bound (6d) 22d4 + 2 . Consequently, our result implies an exponential improvement of known upper bounds on complexity of word-unification algorithms.

Word unification and transformation of generalized equations

Makanin's algorithm shows that it is decidable whether a word equation has a solution. The original description was hard to understand and not designed for implementation. Since words represent a fundamental data type, various authors have given improved descriptions. In this paper we present a version of the algorithm that probably cannot be further simplified without fundamentally new insights that exceed Makanin's original ideas. We give a transformation that is efficient and conceptually simple and applies to arbitrary generalized equations. No further subprocedure is needed for the generation of the search tree. Particular attention is then given to the proof that proper generalized equations are transformed into proper generalized equations. This point, which is important for the termination argument, was treated erroneously in other papers. We also show that a combination of the basic algorithm for string-unification and Makanin's algorithm offers a simple solution to the problem of terminating minimal and complete word unification.