It Is NL-complete to Decide Whether a Hairpin Completion of Regular Languages Is Regular

The hairpin completion is an operation on formal languages which is inspired by the hairpin formation in biochemistry. Hairpin formations occur naturally within DNA-computing. It has been known that the hairpin completion of a regular language is linear context-free, but not regular, in general. However, for some time it is was open whether the regularity of the hairpin completion of a regular language is is decidable. In 2009 this decidability problem has been solved positively by providing a polynomial time algorithm. In this paper we improve the complexity bound by showing that the decision problem is actually NL-complete. This complexity bound holds for both, the one-sided and the two-sided hairpin completions

An investigation into inter- and intragenomic variations of graphic genomic signatures

We provide, on an extensive dataset and using several different distances, confirmation of the hypothesis that CGR patterns are preserved along a genomic DNA sequence, and are different for DNA sequences originating from genomes of different species. This finding lends support to the theory that CGRs of genomic sequences can act as graphic genomic signatures. In particular, we compare the CGR patterns of over five hundred different 150,000 bp genomic sequences originating from the genomes of six organisms, each belonging to one of the kingdoms of life: H. sapiens, S. cerevisiae, A. thaliana, P. falciparum, E. coli, and P. furiosus. We also provide preliminary evidence of this method's applicability to closely related species by comparing H. sapiens (chromosome 21) sequences and over one hundred and fifty genomic sequences, also 150,000 bp long, from P. troglodytes (Animalia; chromosome Y), for a total length of more than 101 million basepairs analyzed. We compute pairwise distances between CGRs of these genomic sequences using six different distances, and construct Molecular Distance Maps that visualize all sequences as points in a two-dimensional or three-dimensional space, to simultaneously display their interrelationships. Our analysis confirms that CGR patterns of DNA sequences from the same genome are in general quantitatively similar, while being different for DNA sequences from genomes of different species. Our analysis of the performance of the assessed distances uses three different quality measures and suggests that several distances outperform the Euclidean distance, which has so far been almost exclusively used for such studies. In particular we show that, for this dataset, DSSIM (Structural Dissimilarity Index) and the descriptor distance (introduced here) are best able to classify genomic sequences.Comment: 14 pages, 6 figures, 5 table

On the iterated hairpin completion

The hairpin completion is a natural operation on formal languages which has been inspired by biochemistry and DNA-computing. In this paper we solve two problems which were posed first in 2008 and 2009, respectively, and still left open: 1.) It is known that the iterated hairpin completion of a regular language is not context-free in general, but it was open whether the iterated hairpin completion of a singleton or finite language is regular or at least context-free. We will show that it can be non-context-free. (It is of course context-sensitive.) 2.) A restricted but also very natural variant of the hairpin completion is the bounded hairpin completion. It was unknown whether the iterated bounded hairpin completion of a regular language remains regular. We prove that this is indeed the case. Actually we derive a more general result. We will present a general representation of the iterated bounded hairpin completion for any language using basic operations. Thus, each language class closed under these basic operations is also closed under iterated bounded hairpin completion

Methods for relativizing properties of codes

The usual setting for information transmission systems assumes that all words over the source alphabet need to be encoded. The demands on encodings of messages with respect to decodability, error-detection, etc. are thus relative to the whole set of words. In reality, depending on the information source, far fewer messages are transmitted, all belonging to some specific language. Hence the original demands on encodings can be weakened, if only the words in that language are to be considered. This leads one to relativize the properties of encodings or codes to the language at hand. We analyse methods of relativization in this sense. It seems there are four equally convincing notions of relativization. We compare those. Each of them has their own merits for specific code properties. We clarify the differences between the four approaches. We also consider the decidability of relativized properties. If P is a property defining a class of codes and L is a language, one asks, for a given language C, whether C satisfies P relative to L. We show that in the realm of regular languages this question is mostly decidable

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

Complexity results and the growths of hairpin completions of regular languages

The hairpin completion is a natural operation on formal languages which has been inspired by molecular phenomena in biology and by DNA-computing. In 2009 we presented a (polynomial time) decision algorithm to decide regularity of the hairpin completion. In this paper we provide four new results: 1.) We show that the decision problem is NL-complete. 2.) There is a polynomial time decision algorithm which runs in time O(n8), this improves our previous results, which provided O(n^{20}). 3.) For the one-sided case (which is closer to DNA computing) the time is O(n2), only. 4.) The hairpin completion of a regular language is unambiguous linear context-free. This result allows to compute the growth (generating function) of the hairpin completion and to compare it with the growth of the underlying regular language