Hairpin completions and reductions: semilinearity properties

International Journal of Foundations of Computer Science, 2016

Hairpin completion is a formal operation inspired from DNA biochemistry. It is known that the (one step) hairpin completion of a regular language is linear context-free, but not regular in general. Further, it is decidable whether the (one step) hairpin completion of a regular language is regular. However, it is an open question whether the iterated hairpin completion of a regular language is regular, even if it is a singleton. If the word is a non-crossing α-word, there are results, but for crossing words there are no results. In this paper, we give necessary and sufficient conditions that the iterated hairpin completion of a given crossing (2, 2)-α-word in [Formula: see text] is regular.

International Journal of Foundations of Computer Science, 2010

We consider several problems regarding the iterated or non-iterated hairpin completion of some subclasses of regular languages. Thus we obtain a characterization of the class of regular languages as the weak-code images of the k-hairpin completion of centerdisjoint k-locally testable languages in the strict sense. This result completes two results from [3] and [11]. Then we investigate some decision problems and closure properties of the family of the iterated hairpin completion of singleton languages. Finally, we discuss some algorithms regarding the possibility of computing the values of k such that the noniterated or iterated k-hairpin completion of a given regular language does not produce new words.

Theoretical Computer Science, 2001

The optimism about the possibilities of DNA computing is based on two central issues: the Watson-Crick complementarity and the massive parallelism of DNA strands. While the latter issue renders exhaustive searches possible and thus may settle problems previously considered intractable, the former issue is the cause behind the universality of many models of DNA computing. Moreover, complementarity can be viewed as a purely language-theoretic operation: undesirable circumstances in a string trigger a transition to the complementary string. This aspect of complementarity is investigated in the present paper, mainly from the point of view of L systems. New types of word sequences will be discovered. Sometimes the resulting decision problems are equivalent to well-known open problems from other areas.

Information and Computation, 2014

Pseudopalindromes are words that are fixed points for some antimorphic involution. In this paper we discuss a newer word operation, that of pseudopalindromic completion, in which symbols are added to either side of the word such that the new obtained words are pseudopalindromes. This notion represents a particular type of hairpin completion, where the length of the hairpin is at most one. We give precise descriptions of regular languages that are closed under this operation and show that the regularity of the closure under the operation is decidable.

2006

The concept of hairpin structures in formal languages is motivated from the biocomputing and bioinformatics fields. Hairpin (-free) DNA structures have numerous applications to DNA computing and molecular genetics in general. A word is called hairpin-free if it cannot be written in the form £ ¥ ¤ § ¦ © ¥ ¤ , with certain additional conditions, for an involution¨(a function¨with the property that § equals the identity function). A particular involution, the so-called Watson-Crick involution, can characterize binding of two DNA strands. We study algebraic and decision properties, finiteness and descriptional complexity of hairpin (-free) languages. We show an existence of polynomial-time algorithms deciding hairpin-freeness of regular and context-free sets. Two related DNA secondary structures are considered, taking into the account imperfect bonds (bulges, mismatches) and multiple hairpins. Finally, effective methods for design of long hairpin-free DNA words are given.

International Journal of Foundations of Computer Science, 2008

DNA strands that, mathematically speaking, are finite strings over the alphabet {A, G, C, T } are used in DNA computing to encode information. Due to the fact that A is Watson-Crick complementary to T and G to C, DNA single strands that are Watson-Crick complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: If some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper studies a

Let be a class of automata (in a precise sense to be defined) and be the class obtained by augmenting each automaton in with finitely many reversal-bounded counters. We show that if the languages defined by are effectively semilinear, then so are the languages defined by , and, hence, their emptiness problem is decidable. We give examples of how this result can be used to show the decidability of certain problems concerning the equivalence of morphisms on languages. We also prove a surprising undecidability result for commutation of languages: given a fixed two element code Ã, it is undecidable whether a given context-free language Ä commutes with Ã, i.e., ÄÃ ÃÄ.

Unconventional Computation and Natural Computation, 2019

We continue the investigation of three operations on words and languages with motivations coming from DNA biochemistry, namely unbounded and bounded hairpin completion and hairpin lengthening. We first show that each of these operations can be used for replacing the third step, the most laborious one, of the solution to the CNF-SAT reported in . As not all the bounded/unbounded hairpin completion or lengthening of semilinear languages remain semilinear, we study sufficient conditions for semilinear languages to preserve their semilinearity property after applying once either the bounded or unbounded hairpin completion, or lengthening. A similar approach is then started for the iterated variants of the three operations. A few open problems are finally discussed.

2012

Abstract Pseudopalindromes are words that are fixed points for some antimorphic involution. In this paper we discuss a newer word operation, that of pseudopalindromic completion, in which symbols are added to either side of the word such that the new obtained words are pseudopalindromes. This notion represents a particular type of hairpin completion, where the length of the hairpin is at most one.