Marked Systems and Circular Splicing

Abstract

In 1987 Head introduced a new operation on strings, called splicing, as a formal model of certain cut and paste biochemical transformation processes of an initial collection of DNA strands under the simultaneous influence of enzymes [11]. Two strands of DNA are cut at specified substrings (sites) by restriction enzymes that recognize a pattern inside the molecule and then the fragments are pasted by ligase enzymes. Since 1987, his basic idea has been formalized in terms of generative mechanisms for formal languages, called splicing systems or, more recently, H systems. An H system is defined by giving an initial language I (initial set of DNA molecules) and a set of special words or rules R (enzymes). The set I is then transformed by repeated applications of the splicing operation. DNA occurs in both linear and circular form and, correspondingly, there are three definitions of linear splicing systems and three definitions of circular splicing systems, given by Head, Paun and Pixton respectively. Circular splicing deals with circular strings, a notion which has been intensively examined in formal language theory (see [1,2,9,14,19]). We recall that for a given word w ∈ A * , a circular word ∼ w is the equivalence class of w with respect to the conjugacy relation ∼ defined by xy ∼ yx, for x, y ∈ A * (see [14]). Let ∼ A * be the set of all circular words over A, i.e., the quotient of A * with respect to ∼. A subset C of ∼ A * is a circular language and every language L ⊆ A * such that ∼ L = { ∼ w | w ∈ L} = C is a linearization of C. Therefore, a circular language C is a regular (resp. context-free) circular language if C has a regular (resp. context-free) linearization [13]. We set Reg ∼ = {C ⊆ ∼ A * | ∃L ∈ Reg : ∼ L = C}. Our results deal with Paun circular splicing systems, defined below.Paun's definition [13,18]. A Paun circular splicing system is a triple SC P A = (A, I, R), where A is a finite alphabet, I is the initial circular language, with I ⊆ ∼ A * and R is the (finite) set of rules, with R ⊆ A * #A * $A * #A * and #, $ ∈ A. Then, given a rule r = u 1 #u 2 $u 3 #u 4 and two circular words ∼ u 2 hu 1 , ∼ u 4 ku 3 , the rule r cuts and linearizes the two circular strings, obtaining u 2 hu 1 , u 4 ku 3 , pastes them and circularizes, obtaining ∼ u 2 hu 1 u 4 ku 3 . We say that * Proceedings of the conference Automata: from Mathematics to Applications (AutoMathA 2007). Partially supported by MIUR Project "Automi e Linguaggi Formali: aspetti matematici e applicativi" (2005), by 60% Project "Linguaggi formali e codici: problemi classici e modelli innovativi" (University of Salerno, 2005) and by 60% Project "Linguaggi formali e codici a lunghezza variabile: proprietà strutturali e nuovi modelli di rappresentazione" (University of Salerno, 2006).

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

“…Furthermore, we have already mentioned marked systems in Section 2.3. In [12,13] the authors proved that, given a marked system, we can decide whether the corresponding generated language is a regular circular language. We should notice that marked systems and monotone complete systems represent opposite situations with respect to regularity.…”

“…CSSH systems satisfying additional hypotheses, named marked systems, were introduced in [12,13] with a characterization of the corresponding generated regular circular languages. This characterization was reviewed in a graph theoretical setting in [7].…”

A characterization of regular circular languages generated by marked splicing systems

“…Furthermore, the main results in this paper have been recently reviewed in a graph theoretical setting and extended to marked systems with self-splicing [8]. Finally, it should be noted that Propositions 4, 6 and Property 1 in [12] have been slightly corrected in this paper.…”