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. | DNA Computing | Sticker system |Probabilistic | Regular languages | Computationalpower | ® 2013IbnuSina Institute. All rights reserved. http://dx.doi.org/10.11113/mjfas.v9n3.101


INTRODUCTION
The first formal tool for the generation of languages from DNA recombination was introduced by Head in 1987, known as splicing systems.Another language generating model which uses the sequential recombinant behaviour ("sticker operation") of DNA molecules is a sticker system.The sticker operation is used by Adleman in his experiment of computing a Hamiltonian path in a graph by using DNA molecules [1].DNA or deoxyribonucleic acid is a genetic material in humans and all organisms.The structure of DNA is a double helix (helicoidal) which is composed of four nucleotides: A (adenine), C (cytosine), G (guanine), and T (thymine), which is paired as A-T, C-G according to Watson-Crick complementary [2].
The concept of sticker system as a language generating model based on sticker operations was first proposed by Kari [2].The axioms and strings generated bya sticker system are considered as encoded modelsof single and double stranded DNA molecules.Moreover, the sticker operations have the advantages over splicing operations used in splicing systemsbecause the sticker operations require no strands extension and use no enzymes [3].The initial sequences of DNA are prolonged to the left and right, producing computations of possible arbitrary length and the process stop when a complete double stranded sequence is obtained and no sticky ends exist [2].

Introduction
The sticker systems are language generating devices based on the sticker operation introduced by Adleman which has been turned into a model of the techniques used in his successful experiment of computing a Hamiltonian path in a graph by using DNA [1].
The conceptof regular sticker systems (one-sided sticker systems) was first considered in [2] and extended to bidirectional sticker systems in [4].When forming new complete double stranded sequences, the initial strands called axioms and a well started sequence are utilized and prolonged either to the left or to the right direction by the process of the sticker operation  [5].Starting from the axiom and iteratively using the operation of sticking, strands are prolonged in order to obtain a complete double stranded sequence.
A sticker system is a construct of 4-tuple  = (, , , ), where is an alphabet,  ⊆  ×  is the symmetric relation in ,  is a finite subset of axioms strand to the left or to the right according to the sticker operation  in order [2,3,4].For a given sticker system  = (, , ,   ,   ) and two sequences ,  ∈ (  ()) , we write,  ⇒ *  if and only if  = (  , (,   )).Note that, sequence �  , (,   )� = ((  , ),   ) because the prolongation to the left is independent with the prolongation to the right [2].By ⇒ * , we denote the reflexive and transitive closure of the relation ⇒.A sequence  1 ⇒  2 ⇒ ⋯ ⇒   , where  1 ∈ , is called a computation in  with length  − 1 (notice that, we start from  1 ∈ ).If   ∈   (), where no sticky end and hence blank symbol is present in the last string of composite symbol, the above computation is considered as complete [2].
A complete computation of a sticker system will produce a complete string, denoted by  ∈   ().The sticker system will generate the set of all possible languages of the completed strings.The language of all such strings in the sticker operation is the language generated by  and is called a sticker language [5,6,7].Sticker operations and languages will be explained in the next section.

Sticker Operations
Let V be an alphabet (a finite set of abstract symbols) endowed with a symmetric relation  (of complementarity),  ⊆  × .Let # be a special symbol not in V, denoting an empty space (the blank symbol).
Using the elements of  ∪ {#}, we construct the composite symbols of the following sets: We denote where ()is the set of well-started strings and () is the set ofthe combinations of upper and lower single strings.We call the elements of these set well-started sequences.In general,  * is the set of all strings, including the empty string denotedby , composed of elements of V, and  + is the set  * − {}.

Sticker Languages
A sticker language (SL) is the language generated by a sticker system which consists of all strings formed by the set of upper strands of all complete strings derived from the axioms for which an exactly matching sequence of lower stickers can be found [5,6].Sticker system will normally generate only regular languages without restrictions [8].

Definition 1[9]
A deterministic finite accepter or dfa is defined by the quintuple  = (, Σ, ,  0 , ), where •  is a finite set of internal states, • Σ is a finite set of symbols called the input alphabet, •  ∶  × Σ →  is a total function called the transition function, •  0 ∈  is the initial state, •  ⊆ is a set of final states.

Definition 2[9]
A language L is called regular if and only if there exists some deterministic finite accepterM such that L = L(M).
Some restrictions have been introduced in [2, 4, 6, 7, 9] on matching pairs of strands to obtain more powerful languages of sticker systems such as simple sticker languages, primitive sticker languages, and balance sticker languages.
Let  = (, , , ) be a sticker system, the unrestricted molecular language generated by is defined in [9] as follows: () = {(, ) ∈ |(, )is a computation of }.Furthermore, the unrestricted sticker language generated by  is the projection onto the first (upper) component of the molecular language, because of the complementarity relation of .The language denoted as () is defined by: The similarity of this process to Adleman's experiment can be found in [2].
There are few types of languages generated by sticker systems which are the one-sided sticker languages, regular sticker languages, simple sticker languages, simple one-sided sticker languages and simple regular sticker languages denoted by OSSL, RSL, SSL, SOSSL and SRSL,respectively.The definitions of these families of languages are listed in the following.

Definition 3 [4]
A language resulting from a sticker system is called a onesided sticker language (OSSL) if for each pair (, ) ∈ , we have either  = or  = .

Definition 4 [4]
A language resulting from a sticker system is called a regular sticker language (RSL) if for each pair (, ) ∈ , we have  = .

Probabilistic Sticker Systems
In this section, some notations of probabilistic sticker systems, operations and languages over probability are introduced.

Definition 8
A probabilistic sticker system (pSS) is a 5-tuple

Lemma 1
The relation ⊆  holds whereindicates the set of all languages generated by probabilistic sticker systems.
Let  1 = �, ,   ,   , �be a probabilistic sticker system where . The language generated by the probabilistic sticker system  1 : We define the threshold language generated by  1 as ( 1 , > 0), then it is not difficult to see that () = ( 1 , > 0).

Example 1
Given a probabilistic sticker system { } , , , , For the given sticker system, we start the computation with the axiom in A being attached to its complementary axiom from D. The first step of computation starts with prolongation to the right side and it is shown below: . 28 28 The computation is complete when a complete double strand sequence is obtained, that is when no sticky end exists in the string.Here, sticky end still exists from the first step.Therefore, the computation has to be continued until the double strand sequence is obtained.
For this example, the second step of the computation is shown in the following: To obtain a complete computation on the right side, the sticking operations have to follow the above steps.Now, to obtain a complete computation on the left side, the computation follows the steps below: , By joining both the stickers, the complete double strand string with the probability 2 3 7 5 11 28 28 . The computation is shown below: Hence, a complete computation is obtained.We can iteratively prolonged a complete double strand DNA sequences to generate the general form of the language produced using the same sequences as done earlier.
We can produce a language of sticker system as Using the threshold properties, we can conclude the following: i : For any probabilistic sticker system (G), the threshold language (, = 0)is the empty set, i.e. (, = 0) = ∅.

RESTRICTED VARIANTS OF PROBABILIS-TIC STICKER SYSTEMS
In this section, we define probabilistic variants of some restrictions of sticker systems.

CONCLUSION
In this paper, the definition of a new restriction of sticker system namely the probabilistic sticker system has been introduced.In addition, some results show that the generative power of sticker systems can be increased with the presence of probability.In addition, the definitions of some new restrictions of variants for sticker system, namely, probabilisticone-sided sticker systems, probabilisticregular sticker systems, probabilisticsimple sticker systems, probabilistic simple one-sided sticker systems and probabilistic simple regular sticker systemsare also introduced.