Rosalba Zizza

Separating some splicing models

Abstract In this paper we show that the three main definitions of the splicing operation known in the literature, i.e., the Head [Bull. Math. Biology 49 (1987) 737–759], Paun [Theoret. Comput. Sci. 168 (1996) 321–326] and Pixton [Discrete Appl. Math. 69 (1996) 101–124] definitions, give rise to different subclasses of regular languages, when a finite set of rules is iterated on a finite set of axioms. More precisely, we show that the family of regular languages generated by finite splicing, as defined in the early paper by Head, is strictly included in the family defined later by Paun, which is in turn strictly included in the splicing family defined by Pixton. We describe instance languages in the difference sets, and we prove that they cannot be generated by the smaller families.

The structure of reflexive regular splicing languages via Schützenberger constants

The splicing operation was introduced in 1987 by Head as a mathematical model of the recombination of DNA molecules under the influence of restriction and ligases enzymes. This operation allows us to define a computing (language generating) device, called a splicing system. Other variants of this original definition were also proposed by Paun and Pixton respectively. The computational power of splicing systems has been thoroughly investigated. Nevertheless, an interesting problem is still open, namely the characterization of the class of regular languages generated by finite splicing systems. In this paper, we will solve the problem for a special class of finite splicing systems, termed reflexive splicing systems, according to each of the definitions of splicing given by Paun and Pixton. This special class of systems contains, in perticular, finite Head splicing systems. The notion of a constant, given by Schutzenberger, once again intervenes.

Decision problems for linear and circular splicing systems

We will consider here the splicing systems, generative devices inspired by cut and paste phenomena on DNA molecules under the action of restriction and ligase enzymes. A DNA strand can be viewed as a string over a four letter alphabet (the four deoxyribonucleotides), therefore we can model DNA computation within the framework of formal language theory. In spite of a vast literature on splicing systems, briefly surveyed here, a few problems related to their computational power are still open. We intend to evidence how classical techniques and concepts in automata theory are a legitimate tool for investigating some of these problems.

DNA and Circular Splicing

Circular splicing has been very recently introduced to model a specific recombinant behaviour of circular DNA, carrying on the investigation initiated with linear splicing. In this paper we restrict ourselves to the relationship between circular regular languages and circular splicing languages. We provide partial results towards a characterization of the class of circular regular languages generated by finite circular splicing systems. We consider a class of languages X* closed under conjugacy relation and with X a regular languages, called here star languages. Using automata theory and combinatorial techniques on words, we show that for a subclass of star languages the corresponding circular languages are circular (Paun) splicing languages. In particular, star languages with X being a finite set or X* being a free monoid belong to this subclass.

On the power of circular splicing

Splicing systems are generative devices of formal languages, introduced by Head in 1987 to model biological phenomena on linear and circular DNA molecules. Via automata properties we show that it is decidable whether a regular language L on a one-letter alphabet is generated by a finite (Paun) circular splicing system: L has this property if and only if there is a unique final state qn on the closed path in the transition diagram of the minimal finite state automaton A recognizing L and qn is idempotent (i.e., δ(qn, an) = qn). This result is obtained by an already known characterization of the unary languages L generated by a finite (Paun) circular splicing system and, in turn, allows us to simplify the description of the structure of L. This description is here extended to the larger class of the uniform languages, i.e., the circularizations of languages with the form AJ = ∪j∈J Aj, J being a subset of the set N of the nonnegative integers. Finally, we exhibit a regular circular language, namely ∼((A2)* ∪ (A3)*), that cannot be generated by any finite circular splicing system.

Regular languages generated by reflexive finite splicing systems

Splicing systems are a generative device inspired by a cut and paste phenomenon on DNA molecules, introduced by Head in 1987 and subsequently defined with slight variations also by Paun and Pixton respectively [8, 13, 17].We will face the problem of characterizing the class of regular languages generated by finite splicing systems. We will solve this problem for the special class of the reflexive finite splicing systems introduced in [9, 10]. As a byproduct, we give a characterization of the regular languages generated by finite Head splicing systems. As in already known results, the notion of constant, given by Schutzenberger in [19], intervenes.

A characterization of (regular) circular languages generated by monotone complete splicing systems

Circular splicing systems are a formal model of a generative mechanism of circular words, inspired by a recombinant behaviour of circular DNA. Some unanswered questions are related to the computational power of such systems, and finding a characterization of the class of circular languages generated by circular splicing systems is still an open problem. In this paper we solve this problem for monotone complete systems, which are finite circular splicing systems with rules of a simpler form. We show that a circular language L is generated by a monotone complete system if and only if the set Lin(L) of all words corresponding to L is a pure unitary language generated by a set closed under the conjugacy relation. The class of pure unitary languages was introduced by A. Ehrenfeucht, D. Haussler, G. Rozenberg in 1983, as a subclass of the class of context-free languages, together with a characterization of regular pure unitary languages by means of a decidable property. As a direct consequence, we characterize (regular) circular languages generated by monotone complete systems. We can also decide whether the language generated by a monotone complete system is regular. Finally, we point out that monotone complete systems have the same computational power as finite simple systems, an easy type of circular splicing system defined in the literature from the very beginning, when only one rule of a specific type is allowed. From our results on monotone complete systems, it follows that finite simple systems generate a class of languages containing non-regular languages, showing the incorrectness of a longstanding result on simple systems.

A characterization of regular circular languages generated by marked splicing systems

Splicing systems are generative devices of formal languages, introduced by Head in 1987 to model biological phenomena on linear and circular DNA molecules. A splicing system is defined by giving an initial set I and a set R of rules. Some unanswered questions are related to the computational power of circular splicing systems. In particular, a still open question is to find a characterization of circular languages generated by finite circular splicing systems (i.e., circular splicing systems with both I and R finite sets). In this paper we introduce a special class of the latter systems named marked systems. We prove that a marked system S generates a regular circular language if and only if S satisfies a special (decidable) property. As a consequence, we are able to characterize the structure of these regular circular languages.

On the regularity of circular splicing languages: a survey and new developments

Circular splicing has been introduced to model a specific recombinant behaviour of circular DNA, continuing the investigation initiated with linear splicing. In this paper we focus on the relationship between regular circular languages and languages generated by finite circular splicing systems. We survey the known results towards a characterization of the intersection between these two classes and provide new contributions on the open problem of finding this characterization. First, we exhibit a non-regular circular language generated by a circular simple system thus disproving a known result in this area. Then we give new results related to a restrictive class of circular splicing systems, the marked systems. Precisely, we review in a graph theoretical setting the recently obtained characterization of marked systems generating regular circular languages. In particular, we define a slight variant of marked systems, that is the g-marked systems, and we introduce the graph associated with a g-marked system. We show that a g-marked system generates a regular circular language if and only if its associated graph is a cograph. Furthermore, we prove that the class of g-marked systems generating regular circular languages is closed under a complement operation applied to systems. We also prove that marked systems with self-splicing generate only regular circular languages.

Chorale Music Splicing System: An Algorithmic Music Composer Inspired by Molecular Splicing

Splicing systems are a formal model of a generative mechanism of words (strings of characters), inspired by a recombinant behavior of DNA. They are defined by a finite alphabet \(\mathcal{A}\), an initial set \(\mathcal{I}\) of words and a set \(\mathcal{R}\) of rules. Many of the studies about splicing systems focused on the properties of the generated languages and their theoretical computational power.