13 research outputs found
Solutions of twisted word equations, EDT0L languages, and context-free groups
© Volker Diekert and Murray Elder; 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, F.4.2 Grammars and Other Rewriting Systems, F.4.3 Formal Languages. We prove that the full solution set of a twisted word equation with regular constraints is an EDT0L language. It follows that the set of solutions to equations with rational constraints in a contextfree group (= finitely generated virtually free group) in reduced normal forms is EDT0L. We can also decide whether or not the solution set is finite, which was an open problem. Moreover, this can all be done in PSPACE. Our results generalize the work by Lohrey and Sénizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to complexity and with respect to expressive power. Both papers show that satisfiability is decidable, but neither gave any concrete complexity bound. Our results concern all solutions, and give, in some sense, the "optimal" formal language characterization
Languages, groups and equations
The survey provides an overview of the work done in the last 10 years to
characterise solutions to equations in groups in terms of formal languages. We
begin with the work of Ciobanu, Diekert and Elder, who showed that solutions to
systems of equations in free groups in terms of reduced words are expressible
as EDT0L languages. We provide a sketch of their algorithm, and describe how
the free group results extend to hyperbolic groups. The characterisation of
solutions as EDT0L languages is very robust, and many group constructions
preserve this, as shown by Levine.
The most recent progress in the area has been made for groups without
negative curvature, such as virtually abelian, the integral Heisenberg group,
or the soluble Baumslag-Solitar groups, where the approaches to describing the
solutions are different from the negative curvature groups. In virtually
abelian groups the solutions sets are in fact rational, and one can obtain them
as -regular sets. In the Heisenberg group producing the solutions to a
single equation reduces to understanding the solutions to quadratic Diophantine
equations and uses number theoretic techniques. In the Baumslag-Solitar groups
the methods are combinatorial, and focus on the interplay of normal forms to
solve particular classes of equations.
In conclusion, EDT0L languages give an effective and simple combinatorial
characterisation of sets of seemingly high complexity in many important classes
of groups.Comment: 26 page
Solutions to twisted word equations and equations in virtually free groups
It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper, we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in [Formula: see text]. Within the same complexity bound we can decide whether the solution set is empty, finite, or infinite. In the second part of the paper we apply the results for twisted equations to obtain in [Formula: see text] an EDT0L description of the solution set of equations with rational constraints for finitely generated virtually free groups in standard normal forms with respect to a natural set of generators. If the rational constraints are given by a homomorphism into a fixed (or “small enough”) finite monoid, then our algorithms can be implemented in [Formula: see text], that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and Sénizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to both complexity and expressive power. Neither paper gave any concrete complexity bound and the results in these papers are stated for subsets of solutions only, whereas our results concern all solutions. </jats:p
Quadratic Diophantine equations, the Heisenberg group and formal languages
We express the solutions to quadratic equations with two variables in the
ring of integers using EDT0L languages. We use this to show that EDT0L
languages can be used to describe the solutions to one-variable equations in
the Heisenberg group. This is done by reducing the question of solving a
one-variable equation in the Heisenberg group to solving an equation in the
ring of integers, exploiting the strong link between the ring of integers and
nilpotent groups.Comment: 33 page
Post's correspondence problem for hyperbolic and virtually nilpotent groups
Post's Correspondence Problem (the PCP) is a classical decision problem in
theoretical computer science that asks whether for pairs of free monoid
morphisms there exists any non-trivial
such that .
Post's Correspondence Problem for a group takes pairs of group
homomorphisms instead, and similarly asks
whether there exists an such that holds for non-elementary
reasons. The restrictions imposed on in order to get non-elementary
solutions lead to several interpretations of the problem; we consider the
natural restriction asking that and prove that
the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic
, but decidable when is virtually nilpotent. We also study
this problem for group constructions such as subgroups, direct products and
finite extensions. This problem is equivalent to an interpretation due to
Myasnikov, Nikolev and Ushakov when one map is injective.Comment: 17 page
Hardness Results for Constant-Free Pattern Languages and Word Equations
We study constant-free versions of the inclusion problem of pattern languages and the satisfiability problem of word equations. The inclusion problem of pattern languages is known to be undecidable for both erasing and nonerasing pattern languages, but decidable for constant-free erasing pattern languages. We prove that it is undecidable for constant-free nonerasing pattern languages. The satisfiability problem of word equations is known to be in PSPACE and NP-hard. We prove that the nonperiodic satisfiability problem of constant-free word equations is NP-hard. Additionally, we prove a polynomial-time reduction from the satisfiability problem of word equations to the problem of deciding whether a given constant-free equation has a solution morphism ? such that ?(xy) ? ?(yx) for given variables x and y
47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
We
study constant-free versions of the inclusion problem of pattern
languages and the satisfiability problem of word equations. The
inclusion problem of pattern languages is known to be undecidable for
both erasing and nonerasing pattern languages, but decidable for
constant-free erasing pattern languages. We prove that it is undecidable
for constant-free nonerasing pattern languages. The satisfiability
problem of word equations is known to be in PSPACE and NP-hard. We prove
that the nonperiodic satisfiability problem of constant-free word
equations is NP-hard. Additionally, we prove a polynomial-time reduction
from the satisfiability problem of word equations to the problem of
deciding whether a given constant-free equation has a solution morphism α
such that α(xy) ≠ α(yx) for given variables x and y.
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Subgroup Membership in GL(2,Z)
It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time where all group elements are represented by so-called power words, i.e., words of the form p_1^{z_1} p_2^{z_2} ? p_k^{z_k}. Here the p_i are explicit words over the generating set of the group and all z_i are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group GL(2,?) can be decided in polynomial time when all matrix entries are given in binary notation