Static Watson-Crick Linear Grammars and Its Computational Power

DNA computing, or more generally, molecular computing, is a recent development on computations using biological molecules, instead of the traditional silicon-chips. Some computational models which are based on different operations of DNA molecules have been developed by using the concept of formal language theory. The operations of DNA molecules inspire various types of formal language tools which include sticker systems, grammars and automata. Recently, the grammar counterparts of Watson-Crick automata known as Watson-Crick grammars which consist of regular, linear and context-free grammars, are defined as grammar models that generate double-stranded strings using the important feature of Watson-Crick complementarity rule. In this research, a new variant of static Watson-Crick linear grammar is introduced as an extension of static Watson-Crick regular grammar. A static Watson-Crick linear grammar is a grammar counterpart of sticker system that generates the double-stranded strings and uses rule as in linear grammar. The main result of the paper is to determine some computational properties of static Watson-Crick linear grammars. Next, the hierarchy between static Watson-Crick languages, Watson-Crick languages, Chomsky languages and families of languages generated by sticker systems are presented.

Static Watson-Crick Context-Free Grammars

Sticker systems and Watson-Crick automata are two modellings of DNA molecules in DNA computing. A sticker system is a computational model which is coded with single and double-stranded DNA molecules; while Watson-Crick automata is the automata counterpart of sticker system which represents the biological properties of DNA. Both of these models use the feature of Watson-Crick complementarity in DNA computing. Previously, the grammar counterpart of the Watson-Crick automata have been introduced, known as Watson-Crick grammars which are classified into three classes: Watson-Crick regular grammars, Watson-Crick linear grammars and Watson-Crick context-free grammars. In this research, a new variant of Watson-Crick grammar called a static Watson-Crick context-free grammar, which is a grammar counterpart of sticker systems that generates the double-stranded strings and uses rule as in context-free grammar, is introduced. The static Watson-Crick context-free grammar differs from a dynamic Watson-Crick context-free grammar in generating double-stranded strings, as well as for regular and linear grammars. The main result of the paper is to determine the generative powers of static Watson-Crick context-free grammars. Besides, the relationship of the families of languages generated by Chomsky grammars, sticker systems and Watson-Crick grammars are presented in terms of their hierarchy.

Computational Power of Static Watson-Crick Context-free Grammars

Sticker system is a computer model which is coded with single and double-stranded molecules of DNA; meanwhile, Watson-Crick automata is the automata counterpart of the sticker system representing the biological properties of DNA. Both are the modelings of DNA molecules in DNA computing which use the feature of Watson-Crick complementarity. Formerly, Watson-Crick grammars which are classified into three classes have been introduced [1]. In this research, a grammar counterpart of sticker systems that uses the rule as in context-free grammar is introduced, known as a static Watson-Crick context-free grammar. The research finding on the computational power of these grammar shows that the family of context-free languages is strictly included in the family of static Watson-Crick context-free languages; the static Watson-Crick context-free grammars can generate non context-free languages; the family of Watson-Crick context-free languages is included in the family of static Watson-Crick context-free languages which are presented in terms of their hierarchy.

Static Watson-Crick regular grammar

DNA computing, or more generally, molecular computing, is a recent development at the interface of computer science and molecular biology. In DNA computing, many computational models have been proposed in the framework of formal language theory and automata such as Watson-Crick grammars and sticker systems. A Watson-Crick grammar is a grammar model that generates double stranded strings, whereas a sticker system is a DNA computing model of the ligation and annealing operations over DNA strands using the Watson-Crick complementarity to form a complete double stranded DNA sequence. Most of the proposed DNA computing models make use of this concept, including the Watson-Crick grammars and sticker systems. Watson-Crick grammars and their variants can be explored using formal language theory which allows the development of new concepts of Watson-Crick grammars. In this research, a new variant of Watson-Crick grammar called a static Watson-Crick regular grammar is introduced as an analytical counterpart of sticker systems. The computation of a sticker system starts from a given set of incomplete double stranded sequence to form a complete double stranded sequence. Here, a static Watson-Crick regular grammar differs from a dynamic Watson-Crick regular grammar in generating double stranded strings: the latter grammar produces each strand string “independently” and only check for the Watson-Crick complementarity of a generated complete double stranded string at the end, while the former grammar generates both strand strings “dependently”, i.e., checking for the Watson-Crick complementarity for each complete substring. In this paper, computational properties of static Watson-Crick regular grammars are investigated to correlate with the Chomsky hierarchy and hierarchy of the families of dynamic Watson-Crick regular languages. The relationship between families of languages generated by static Watson-Crick regular grammars with several variants of sticker systems, Watson-Crick regular grammars and Chomsky grammars are presented by showing the hierarchy.

Sticker Systems Over Monoids

Molecular computing has gained many interests among researchers since Head introduced the first theoretical model for DNA based computation using the splicing operation in 1987. Another model for DNA computing was proposed by using the sticker operation which Adlemanused in his successful experiment for the computation of Hamiltonian paths in a graph: a double stranded DNA sequence is composed by prolonging to the left and to the right a sequence of (single or double) symbols by using given single stranded strings or even more complex dominoes with sticky ends, gluing these ends together with the sticky ends of the current sequence according to a complementarity relation. According to this sticker operation, a language generative mechanism, called a sticker system, can be defined: a set of (incomplete) double-stranded sequences (axioms) and a set of pairs of single or double-stranded complementary sequences are given. The initial sequences are prolonged to the left and to the right by using sequences from the latter set, respectively. The iterations of these prolongations produce “computations” of possibly arbitrary length. These processes stop when a complete double stranded sequence is obtained. Sticker systems will generate only regular languages without restrictions. Additional restrictions can be imposed on the matching pairs of strands to obtain more powerful languages. Several types of sticker systems are shown to have the same power as regular grammars; one type is found to represent all linear languages whereas another one is proved to be able to represent any recursively enumerable language. The main aim of this research is to introduce and study sticker systems over monoids in which with each sticker operation, an element of a monoid is associated and a complete double stranded sequence is considered to be valid if the computation of the associated elements of the monoid produces the neutral element. Moreover, the sticker system over monoids is defined in this study.

Probabilistic sticker systems

A model for DNA computing using the recombination behaviour of DNA molecules known as a sticker system has been introduced by Adleman in 1994. A sticker model is an abstract computational model which uses the Watson-Crick complementary principle of DNA molecules. Starting from the axioms – incomplete double stranded sequences, and iteratively using sticking operations, complete double stranded sequences are obtained. It is known that sticker systems with finite sets of axioms and sticker rules generate only regular languages. Hence, different types of restrictions have been considered to increase the computational power of sticker systems. In this paper, we introduce probabilistic sticker systems in which probabilities are initially associated with the axioms, and the probability of the generated string is computed by multiplying the probabilities of all occurrences of the initial strings used in the computation of the string.

Sticker P System

A computability model based on DNA sequences called sticker systems was introduced by Kari et al. (1998). Sticker systems are language generating devices based on the sticker operation. In this work, a theoretical study about a new class of P system based on the sticker operation has been proposed. In this system, objects are double stranded sequences with sticky ends and evolution rules are sticking rules based on sticker operation. The authors compare the language generated by the proposed system with regular languages. The authors suggest another P system based on the bidirectional sticker system.