Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in (relatively) hyperbolic groups". We study there the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. In the second part, called "Finding relative hyperbolic structures", we provide a general algorithm that recognizes the class of groups that are hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in Israel J. Math, and Bull. London Math. Soc. respectivel

On Solving Word Equations Using SAT

We present Woorpje, a string solver for bounded word equations (i.e., equations where the length of each variable is upper bounded by a given integer). Our algorithm works by reformulating the satisfiability of bounded word equations as a reachability problem for nondeterministic finite automata, and then carefully encoding this as a propositional satisfiability problem, which we then solve using the well-known Glucose SAT-solver. This approach has the advantage of allowing for the natural inclusion of additional linear length constraints. Our solver obtains reliable and competitive results and, remarkably, discovered several cases where state-of-the-art solvers exhibit a faulty behaviour

Constraint Solving on Bounded String Variables

Abstract Constraints on strings of unknown length occur in a wide variety of real-world problems, such as test case generation, program analysis, model checking, and web security. We describe a set of con-straints sufficient to model many standard benchmark problems from these fields. For strings of an unknown length bounded by an integer, we describe propagators for these constraints. Finally, we provide an experi-mental comparison between a state-of-the-art dedicated string solver, CP approaches utilising fixed-length string solving, and our implementation extending an off-the-shelf CP solver.

Functions for the General Solution of Parametric Word Equations

: In this article we introduce the functions Fi (x 1 , x 2 ) l1,..., ls and Th (x 1 , x 2 , x 3 ) i l1,..., l2s (i = 1, 2, 3), of the word variables x i and of the natural number variables li, where s ³ 0. By means of these functions, we give exactly the general solution (i.e. the set of all the solutions) of the first basic parametric equation: x 1 x 2 x 3 x 4 = x 3 x 1 l x 2 x 5 , in a free monoid. 1. Introduction The following four parametric equations: x 1 x 2 x 3 x 4 = x 3 x 1 l x 2 x 5 , x 1 x 2 x 3 x 4 = x 2 x 3 l x 1 x 5 , x 1 x 2 2 x 3 x 4 = x 3 x 1 2 x 2 x 5 , x 1 x 2 l+1 x 3 x 4 = x 3 x 2 µ+1 x 1 x 5 , in a free monoid, are called basic equations. They arise in the graph of the prefixeequations in free monoid (cf. [2], [3]) and play an important role in the hierarchy of the parametric equations, in reason of the structures of their solutions. In particular, the general solution of any equation in a free monoid of the form F(x 1 , x 2 , x 3 ) x 4 = Y(x 1 , x 2 , ..