Paul C. Bell

Paul C. Bell, George Kenison, Reino Niskanen et al.

Embeddings of word structures into matrix semigroups provide a natural bridge between combinatorics on words and linear algebra. However, low-dimensional matrix semigroups impose strong structural restrictions on possible embeddings. Certain finitely generated groups admit faithful representations in SL(2, C) and other similar matrix groups. On the other hand, it is known that the product of two free semigroups on two generators cannot be embedded into the 2x2 complex matrices. In this paper we study embeddings of word structures into low-dimensional matrix semigroups over the complex numbers and develop new techniques for constructing word representations of the Euclidean Bianchi groups. These representations provide a symbolic framework and a natural first step towards analysing fundamental decision problems in 2x2 matrix semigroups.

Paul C. Bell, Eva Foster, Daniel Reidenbach

We study the notion of irreducibility of semigroup morphisms. Given an alphabet $Σ$, a morphism $φ:Σ^+\rightarrowΣ^+$ is irreducible if any factorisation $φ=ψ_2\circψ_1$ can only be satisfied if $ψ_1$ or $ψ_2$ is a trivial morphism. Otherwise, $φ$ is reducible. We introduce the notion of irreducibility, characterise this property and study a number of fundamental questions on the concepts under consideration.

Paul C. Bell, Pavel Semukhin

We show the surprising result that the cutpoint isolation problem is decidable for Probabilistic Finite Automata (PFA) where input words are taken from a letter-bounded context-free language. A context-free language $\mathcal{L}$ is letter-bounded when $\mathcal{L} \subseteq a_1^*a_2^* \cdots a_\ell^*$ for some finite $\ell > 0$ where each letter is distinct. A cutpoint is isolated when it cannot be approached arbitrarily closely. The decidability of this problem is in marked contrast to the situation for the (strict) emptiness problem for PFA which is undecidable under the even more severe restrictions of PFA with polynomial ambiguity, commutative matrices and input over a letter-bounded language as well as to the injectivity problem which is undecidable for PFA over letter-bounded languages. We provide a constructive nondeterministic algorithm to solve the cutpoint isolation problem, which holds even when the PFA is exponentially ambiguous. We also show that the problem is at least NP-hard and use our decision procedure to solve several related problems.