Deciding Regularity of Hairpin Completions of Regular Languages in Polynomial Time

The hairpin completion is an operation on formal languages that has been inspired by the hairpin formation in DNA biochemistry and by DNA computing. In this paper we investigate the hairpin completion of regular languages. It is well known that hairpin completions of regular languages are linear context-free and not necessarily regular. As regularity of a (linear) context-free language is not decidable, the question arose whether regularity of a hairpin completion of regular languages is decidable. We prove that this problem is decidable and we provide a polynomial time algorithm. Furthermore, we prove that the hairpin completion of regular languages is an unambiguous linear context-free language and, as such, it has an effectively computable growth function. Moreover, we show that the growth of the hairpin completion is exponential if and only if the growth of the underlying languages is exponential and, in case the hairpin completion is regular, then the hairpin completion and the underlying languages have the same growth indicator

Dissecting Power of a Finite Intersection of Context Free Languages

Let $\exp^{k,\alpha}$ denote a tetration function defined as follows: $\exp^{1,\alpha}=2^{\alpha}$ and $\exp^{k+1,\alpha}=2^{\exp^{k,\alpha}}$, where $k,\alpha$ are positive integers. Let $\Delta_n$ denote an alphabet with $n$ letters. If $L\subseteq\Delta_n^*$ is an infinite language such that for each $u\in L$ there is $v\in L$ with $\vert u\vert<\vert v\vert\leq \exp^{k,\alpha}\vert u\vert$ then we call $L$ a language with the \emph{growth bounded by} $(k,\alpha)$-tetration. Given two infinite languages $L_1,L_2\in \Delta_n^*$, we say that $L_1$ \emph{dissects} $L_2$ if $\vert L_1\cap L_2\vert=\infty$ and $\vert(\Delta_n^*\setminus L_1)\cap L_2\vert=\infty$. Given a context free language $L$, let $\kappa(L)$ denote the size of the smallest context free grammar $G$ that generates $L$. We define the size of a grammar to be the total number of symbols on the right sides of all production rules. Given positive integers $n,k$ with $k\geq 2$, we show that there are context free languages $L_1,L_2,\dots, L_{3k-3}\subseteq \Delta^*_n$ with $\kappa(L_i)\leq 40 k$ such that if $\alpha$ is a positive integer and $L\subseteq\Delta_n^*$ is an infinite language with the growth bounded by $(k,\alpha)$-tetration then there is a regular language $M$ such that $M\cap\left(\bigcap_{i=1}^{3k-3}L_i\right)$ dissects $L$ and the minimal deterministic finite automaton accepting $M$ has at most $k+\alpha+3$ states

Geodesic growth in virtually abelian groups

We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and that the geodesic growth series is holonomic, and rational in the polynomial growth case. In addition, we show that the language of geodesics is blind multicounter.Comment: 23 pages, 1 figure, improved readabilit

On the Commutative Equivalence of Context-Free Languages

The problem of the commutative equivalence of context-free and regular languages is studied. In particular conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated

Groups and semigroups with a one-counter word problem

We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian