On the decidability of semigroup freeness

A ug 2 00 8 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 pr...

Decidability and Tameness in the Theory of Finite Semigroups

Matrix Semigroup Freeness Problems in $\mathrm{SL}(2,\mathbb{Z})$

In this paper we study decidability and complexity of decision problems on matrices from the special linear group SL(2,Z). In particular, we study the freeness problem: given a finite set of matrices G generating a multiplicative semigroup S, decide whether each element of S has at most one factorization over G. In other words, is G a code? We show that the problem of deciding whether a matrix ...

The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable

We prove that a semigroup generated by a reversible two-state Mealy automaton is either finite or free of rank 2. This fact leads to the decidability of finiteness for groups generated by two-state or two-letter invertible-reversible Mealy automata and to the decidability of freeness for semigroups generated by two-letter invertible-reversible Mealy automata.

Computational problems in matrix semigroups

This thesis deals with computational problems that are defined on matrix semigroups, which play a pivotal role in Mathematics and Computer Science in such areas as control theory, dynamical systems, hybrid systems, computational geometry and both classical and quantum computing to name but a few. Properties that researchers wish to study in such fields often turn out to be questions regarding t...