Freeness Problem Research Articles

The symmetric Post Correspondence Problem, and errata for the freeness problem for matrix semigroups

In this paper, we define the symmetric Post Correspondence Problem (PCP) and prove that it is undecidable. As an application, we show that the original proof of undecidability of the freeness problem for [Formula: see text] integer matrix semigroups works for the symmetric PCP, but not for the PCP in general.

Shift dynamics of the groups of Fibonacci type

Abstract We study the shift dynamics of the groups G = G n ⁢ ( x 0 ⁢ x m ⁢ x k - 1 ) G=G_{n}(x_{0}x_{m}x_{k}^{-1}) of Fibonacci type introduced by Johnson and Mawdesley. The main result concerns the order of the shift automorphism of 𝐺 and determining whether it is an outer automorphism, and we find the latter occurs if and only if 𝐺 is not perfect. A result of Bogley provides that the aspherical presentations determine groups admitting a free shift action by Z n \mathbb{Z}_{n} on the nonidentity elements of 𝐺, from which it follows that the shift is an outer automorphism of order 𝑛 when 𝐺 is nontrivial. The focus of this paper is therefore on the non-aspherical cases, which include for example the Fibonacci and Sieradski groups. With few exceptions, the fixed-point and freeness problems for the shift automorphism are solved, in some cases using computational and topological methods.

Freeness Problem for Matrix Semigroups of Parikh Matrices

Since the undecidability of the mortality problem for 3 × 3 matrices over integers was proved using the Post Correspondence Problem, various studies on decision problems of matrix semigroups have emerged. The freeness problem in particular has received much attention but decidability remains open even for 2 × 2 upper triangular matrices over nonnegative integers. Parikh matrices are upper triangular matrices introduced as a generalization of Parikh vectors and have become useful tools in studying of subword occurrences. In this work, we focus on semigroups of Parikh matrices and study the freeness problem in this context.

On the freeness problem for truncated current algebras

Molev (in: Doebner, Scherer, Nattermann (eds) Group 21, physical applications and mathematical aspects of geometry, groups, and algebras, World Scientific, Singapore, vol 1, pp 172–176, 1997) constructed generators of the center of the universal enveloping algebra $$U(\mathfrak{g}_m(n))$$ for the truncated current Lie algebra $$\mathfrak{g}_m(n) = \mathfrak{gl}_n \otimes \mathbb{C}[x]/ (x^m)$$ . Such generators allow to define algebraic varieties associated to the center and the Gelfand–Tsetlin subalgebra. In this paper we prove that the Gelfand–Tsetlin variety is equidimensional of dimension $$mn(n-1)/2$$ if and only if $$n=1,2$$ , implying that $$U(\mathfrak{g}_m(2))$$ is free over the Gelfand–Tsetlin subalgebra.

Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness

In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a positive relation, i.e. a relation p = 1 such that p only contains positive generators. Besides its obvious relation to the freeness of the group, the absence of positive relations has previously been studied and is connected to the triviality of some stabilizers of the boundary. We show that the emptiness of the set of positive relations is equivalent to the dynamical property that all (directed positive) orbital graphs centered at non-singular points are acyclic. Gillibert showed that the finiteness problem for automaton semigroups is undecidable. In the second part of the paper, we show that this undecidability result also holds if the input is restricted to be bi-reversible and invertible (but, in general, not complete). As an immediate consequence, we obtain that the finiteness problem for automaton subsemigroups of semigroups generated by invertible, yet partial automata, so called automaton-inverse semigroups, is also undecidable. Erratum: Contrary to a statement in a previous version of the paper, our approach does not show that that the freeness problem for automaton semigroups is undecidable. We discuss this in an erratum at the end of the paper.

Relations in the semigroup of 2 × 2 upper-triangular matrices

Let [Formula: see text] with [Formula: see text] be [Formula: see text] upper-triangular matrices with rational entries. In the multiplicative semigroup generated by these matrices, we check if there are relations of the form [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] We give algorithms to find relations of the previous form. Our results are extensions of some theorems obtained by Charlier and Honkala in [The freeness problem over matrix semigroups and bounded languages, Inf. Comput. 237 (2014) 243–256]. Our paper is at the interface between algebra, number theory and theoretical computer science. While the main results concern decidability and semigroup theory, the methods for obtaining them come from number theory.

Freeness properties of weighted and probabilistic automata over bounded languages

There has been much research into freeness properties of finitely generated matrix semigroups under various constraints. Most freeness problems are undecidable starting from dimension three, even for upper-triangular integer matrices. A recent paper investigated freeness properties of bounded languages of matrices, which are matrices from a set M1⁎M2⁎⋯Mk⁎⊆Fn×n for some semiring F and fixed k∈N>0.We consider a notion of freeness and ambiguity for scalar reachability problems in matrix semigroups and bounded languages of matrices. Scalar reachability concerns the set {ρTMτ|M∈S}, where ρ,τ∈Fn are vectors and S⊆Fn×n is a finitely generated matrix semigroup. Ambiguity and freeness problems are defined in terms of the uniqueness of factorizations for each scalar. We show various undecidability results and connections to weighted and probabilistic finite automata.

Vector Ambiguity and Freeness Problems in SL(2, ℤ)

We study the vector ambiguity problem and the vector freeness problem in SL(2,Z). Given a finitely generated n×n matrix semigroup S and an n-dimensional vector x, the vector ambiguity problem is to decide whether for every target vector y=Mx, where M∈S, M is unique. We also consider the vector freeness problem which is to show that every matrix M which is transforming x to Mx has a unique factorization with respect to the generator of S. We show that both problems are NP-complete in SL(2,Z), which is the set of 2×2 integer matrices with determinant 1. Moreover, we generalize the vector ambiguity problem and extend to the finite and k-vector ambiguity problems where we consider the degree of vector ambiguity of matrix semigroups.

Remarks concerning the freeness problem over morphism and matrix semigroups

We study the freeness problem over morphism and matrix semigroups. We show that the freeness problem is undecidable for morphisms over a three-letter alphabet. We show that there is a commutative semiring R such that the freeness problem is undecidable for upper-triangular 2×2 matrices having entries in R.

The freeness problem over matrix semigroups and bounded languages

We study the freeness problem for matrix semigroups. We show that the freeness problem is decidable for upper-triangular 2×2 matrices with rational entries when the products are restricted to certain bounded languages. We also show that this problem becomes undecidable for sufficiently large matrices.

On the decidability of semigroup freeness

This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoid. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over three-by-three integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is three-fold: (i) to present general results concerning freeness problems, (ii) to study the decidability of freeness problems over various particular semigroups (special attention is devoted to multiplicative matrix semigroups), and (iii) to propose precise, challenging open questions in order to promote the study of the topic.

UNDECIDABILITY BOUNDS FOR INTEGER MATRICES USING CLAUS INSTANCES

There are several known undecidable problems for 3 × 3 integer matrices the proof of which use a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the number of matrices for the mortality, zero in the left upper corner, vector reachability, matrix reachability, scalar reachability and freeness problems. Also, we give a short proof for a strengthened result due to Bell and Potapov stating that the membership problem is undecidable for finitely generated matrix semigroups R ⊆ ℤ 4×4 whether or not kI4 ∈ R for any given |k| > 1. These bounds are obtained by using the Claus instances of the PCP.

Presentations of homotopy skein modules of oriented 3-manifolds

A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is computed.

ON THE UNDECIDABILITY OF FREENESS OF MATRIX SEMIGROUPS

We slightly improve the result of Klarner, Birget and Satterfield, showing that the freeness of finitely presented multiplicative semigroups of 3×3 matrices over ℕ is undecidable even for triangular matrices. This is achieved by proving a new variant of Post correspondence problem. We also consider the freeness problem for 2×2 matrices. On the one hand, we show that it cannot be proved undecidable using the above methods which work in higher dimensions, and, on the other hand, we give some evidence that its decidability, if so, is also a challenging problem.