Weighted Watson-Crick automata

Weighted Watson-Crick automata Mohd Izzuddin Mohd Tamrin, Sherzod Turaev, and Tengku Mohd Tengku Sembok Citation: AIP Conference Proceedings 1605, 302 (2014); doi: 10.1063/1.4887606 View online: http://dx.doi.org/10.1063/1.4887606 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1605?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Watson-Crick Petri net languages: The effect of labeling strategies AIP Conf. Proc. 1605, 690 (2014); 10.1063/1.4887673 Watson-Crick Petri net languages with finite sets of final markings AIP Conf. Proc. 1602, 876 (2014); 10.1063/1.4882587 Moving beyond Watson–Crick models of coarse grained DNA dynamics J. Chem. Phys. 135, 205102 (2011); 10.1063/1.3662137 Anharmonic Vibrational Signatures of DNA Bases and WatsonCrick Base Pairs Chin. J. Chem. Phys. 22, 563 (2009); 10.1088/1674-0068/22/06/563-570 Strong enhancement of vibrational relaxation by Watson-Crick base pairing J. Chem. Phys. 121, 5381 (2004); 10.1063/1.1785153

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Weighted Watson-Crick Automata Mohd Izzuddin Mohd Tamrina, Sherzod Turaevb and Tengku Mohd Tengku Sembokb a

Department of Information System and b Department of Computer Science, Kulliyyah of Information and Communication Technology, International Islamic University Malaysia, 50728 Gombak, Selangor, Malaysia Abstract. There are tremendous works in biotechnology especially in area of DNA molecules. The computer society is attempting to develop smaller computing devices through computational models which are based on the operations performed on the DNA molecules. A Watson-Crick automaton, a theoretical model for DNA based computation, has two reading heads, and works on double-stranded sequences of the input related by a complementarity relation similar with the Watson-Crick complementarity of DNA nucleotides. Over the time, several variants of Watson-Crick automata have been introduced and investigated. However, they cannot be used as suitable DNA based computational models for molecular stochastic processes and fuzzy processes that are related to important practical problems such as molecular parsing, gene disease detection, and food authentication. In this paper we define new variants of Watson-Crick automata, called weighted Watson-Crick automata, developing theoretical models for molecular stochastic and fuzzy processes. We define weighted Watson-Crick automata adapting weight restriction mechanisms associated with formal grammars and automata. We also study the generative capacities of weighted Watson-Crick automata, including probabilistic and fuzzy variants. We show that weighted variants of Watson-Crick automata increase their generative power. Keywords: DNA Computing, Weighted Watson Crick Automata, Generative Power PACS: 02.10.Ab, 02.20.Bb, 02.30.Gp, 87.14.gk

INTRODUCTION Information is encoded into the DNA molecules and is processed by building more complex DNA strands instead of merely crunching numbers. DNA molecules are double stranded helicoidally structure which comprised of four types of nucleotides: A (adenine), C (cytosine), G (guanine), and T (thymine). They are paired according to the Watson Crick complementary, adenine with thymine and guanine with cytosine. This is one of the fundamental features of DNA molecules in that the bonding between the nucleotides in the DNA molecules can give us the information about their counterpart with just the information about one nucleotide. The second feature of DNA computing is massive parallelism which could process a huge number of DNA strands at the same time with greater speed and using very small space compared to the supercomputer. With these abilities, DNA computing devices have the possibility to transform intractable problems due to the limitation of processing capacities to tractable problems with unlimited computing powers. The classical example for DNA computing was the biological experiment performed by Adleman [1] to solve the Hamiltonian Path problem. By building all possible DNA strands with different combination of the edges between seven vertices and at the end extracting only the strand visiting each vertices once but subject to specific requirement on the starting and ending vertices. This mark the birth of DNA computing and resulted in three prominent theoretical proposal for DNA based computing. A splicing system is one of the earliest formal model proposed in [2], using splicing operation for the cutting and recombining DNA molecules whereby the first part of the first strand is attached to the second part of the second strand and vice versa. The second theoretical proposal for DNA computing is a sticker system introduced in [3] and later extended in [4] by using sticker operation: strands of DNA are built by sticking smaller pieces of the DNA molecules at the end but subject to complimentary relations. The third theoretical proposal is Watson Crick (WK) finite automata that were introduced in [4] abstracting biological properties for verification purposes. In this paper, we are interested in the WK automata because we are focusing on application of automating the verification process on the strands of DNA encoded within its contextual information whereas the other two theoretical computing models are meant for generating strands of DNA. The WK automata has two head for reading double stranded DNA molecules and main feature is the complementary relations in that the character read on the Proceedings of the 21st National Symposium on Mathematical Sciences (SKSM21) AIP Conf. Proc. 1605, 302-306 (2014); doi: 10.1063/1.4887606 © 2014 AIP Publishing LLC 978-0-7354-1241-5/$30.00

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first head must correspond to the correct character on the second head. The two strands of input are separately scanned from left to right but controlled by the same state. Earlier study on the variant of WK automata investigated in [5, 6] such as initial stateless WK finite automata, reverse WK automata, WK two-way finite automata. Other studies on WK automata comprise of the WK Z-automata [7], 5'→3' sensing WK finite automata [8], 5′ →3′ WK automata with several runs [9] and deterministic WK automata [10]. However, they are not adequate to perform verification process with the element of uncertainty in their contextual properties. There are study done in [11, 12, 13] which add probabilities to production and transition for grammar and automata respectively which allow efficient parsing algorithm for the language processing. This weighted concept from formal language and automata theories can be adapted for a new variant of WK automata. In this paper, we introduce a new variant of WK automata, called weighted WK automata that associate weight to the transition function. The weights are calculated by applying one of the two types of operations, multiplication or addition, on every transition function fired from the starting state to the finishing state. We also define threshold languages that can be accepted weighted WK automata by using different weighting spaces and modes. The operation selected for calculating the weight will determine the type of algebraic structure used for the weighting space. The rest of this paper is organized as follows. In Section 2, the necessary definitions and results from the theories of formal languages and DNA computing are given. In Section 3, we introduce the concept of weighted WK automata, define the control on the accepted language through the use of threshold and show basic results on their generative power. In Section 4, we discuss on our main results and future direction of this research.

PRELIMINARIES The basic concept from WK automata and formal language will be recalled to get the reader familiarize with the notations used throughout this paper. More information on these concepts are available from [14, 15], however we had replace the notation for set of states, from ‫ ܭ‬to ܳ when explaining the concept of WK automata. The following are the general notations we used in this paper: the symbol ‫ ه‬denotes strict inclusion whereas inclusion is denoted by the symbol‫ك‬. The member of an element in a set is denoted by ‫ א‬while the negation to the member in a set is denoted by‫ב‬. The empty set is denoted by‫׎‬. The sets of integers and positive rational numbers are denoted by Ժ and Էା respectively. The power set of ܺ is denoted by ȬሺܺሻǤ The notations used for WK automata are as follows: let ܸ be finite alphabet, ܸ ‫ כ‬is a set of any possible finite words over the alphabet inܸ. ߩ ‫ ܸ ك‬ൈ ܸdenotes the symmetric relation on ܸ which has direct reflection on the concept of Watson-Crick complementary relation on the nucleotides for double stranded DNA molecules. For example, if ܸ ൌ ሼ‫ܣ‬ǡ ‫ܥ‬ǡ ‫ܩ‬ǡ ܶሽ, then the ߩ ൌ ሼሺ‫ܣ‬ǡ ܶሻǡ ሺܶǡ ‫ܣ‬ሻǡ ሺ‫ܥ‬ǡ ‫ܩ‬ሻǡ ሺ‫ܩ‬ǡ ‫ܥ‬ሻሽ. A WK finite automaton is a 6-tuple ‫ ܯ‬ൌ ሺܸǡ ߩǡ ܳǡ ‫ݍ‬଴ ǡ ‫ܨ‬ǡ ߜሻ where ܳ denotes a finite set of states, with the initial ܸ (starting) state‫ݍ‬଴ ‫ܳ א‬, the set of accepting or final states ‫ܳ ك ܨ‬, and the transition functions ߜǣ ܳ ൈ ቀ ቁ ՜ Ȭሺܳሻ is ܸ ‫ݒ‬ଵ ‫כ‬ ‫כ‬ mapping such that ߜ ቆ‫ݍ‬ǡ ቀ‫ ݒ‬ቁቇ ് ‫ ׎‬and produced finite triplets ሺ‫ݍ‬ǡ ‫ݒ‬ଵ ǡ ‫ݒ‬ଶ ሻ ‫ ܳ א‬ൈ ܸ ൈ ܸ Ǥ The transition function ଶ ‫ݒ‬ଵ ‫ݒ‬ଵ ‫ݒ‬ଵ ᇱ ‫ ߜ א ݍ‬ቀ‫ݍ‬ǡ ቀ‫ ݒ‬ቁቁcan be expressed with re-writing rule as ‫ ݍ‬ቀ‫ ݒ‬ቁ ՜ ቀ‫ ݒ‬ቁ ‫ݍ‬Ԣ for getting better picture on the ଶ ଶ ଶ movement of the automata after firing individual transition function. The WK automata read the double stranded strings that have the same length and the pair of letters are complementary to each other such that for ‫ݓ‬ଵ ‫ݒ‬ଵ ‫ݔ‬ଵ ܸ ‫ݒ כ‬ଵ ‫ݓ‬ଵ ‫ݔ‬ଵ ቀ‫ ݒ‬ቁ ǡ ቀ‫ ݓ‬ቁ ǡ ቀ‫ ݔ‬ቁ ‫ א‬ቀ ቁ , ቂ‫ ݔ ݓ ݒ‬ቃ ‫ܭܹ א‬ఘ ሺܸሻ, and the transition is define as ܸ ଶ ଶ ଶ ଶ ଶ ଶ ‫ݔ ݓ‬ଵ ‫ݒ‬ଵ ‫ݒ‬ଵ ‫ݓ‬ ‫ݔ‬ଵ ቀ‫ ݒ‬ቁ ‫ ݍ‬ቀ‫ ݓ‬ቁ ቀ‫ ݔ‬ቁ ֜ ቀ‫ ݒ‬ቁ ቀ‫ ݓ‬ቁ ‫ ݍ‬ᇱ ቀ‫ ݔ‬ቁ ଶ

ଶ

ଶ

ଶ

ଶ

ଶ

‫ݒ‬ଵ where ‫ݍ‬ǡ ‫ݍ‬Ԣ ‫ܳ א‬, if and only if ‫ ݍ‬ᇱ ‫ א‬Ɂ ቆ‫ݍ‬ǡ ቀ‫ ݒ‬ቁቇ. ଶ

The transitive and reflexive closure of ֜ is denoted by ֜‫ כ‬and the language accepted by the WK automaton ‫ ܯ‬is defined by ‫ݓ‬ଵ ‫ݓ‬ଵ ‫ݓ‬ଵ ‫ܮ‬ሺ‫ܯ‬ሻ ൌ ቄ‫ݓ‬ଵ ‫ כ ܸ א‬ȁ‫ݍ‬଴ ቂ‫ ݓ‬ቃ ֜‫ כ‬ቂ‫ ݓ‬ቃ ‫ݍ‬௙ ‫ݍ‬௙ ‫ݓܨ א‬ଶ ‫ כ ܸ א‬ቂ‫ ݓ‬ቃ ‫ܭܹ א‬ఘ ሺܸሻቅǤ ଶ ଶ ଶ


The families of recursively enumerable, context-sensitive, context-free, linear, regular and finite languages are denoted by‫܀‬۳ǡ ۱‫܁‬ǡ ۱۴ǡ ‫ۺ‬۷‫ۼ‬ǡ ‫܀‬۳۵ǡ ۴۷‫ۼ‬, respectively. For these families of languages the following strict inclusions, called Chomsky hierarchy, hold: ۴۷‫܀ ؿ ۼ‬۳۵ ‫ۺ ؿ‬۷‫ ؿ ۼ‬۱۴ ‫ ؿ‬۱‫܀ ؿ ܁‬۳Ǥ

WEIGHTED WATSON-CRICK AUTOMATA AND THEIR GENERATIVE POWER In this section, we define weighted variants of Watson-Crick automata, where weights (from different weighting spaces) are assigned to each transition of an automaton and for each computation the weight is computed by applying a given operation (e.g., multiplication, addition, etc.) over weights of the transitions in the computation. In addition, we define thresholds with respect to a weighting space and an operation that select a subset of languages generated by the weighted Watson Crick automaton. Definition 1. A weighted Watson-Crick automaton is an 8-tuple ‫ ܭ‬ൌ ሺܸǡ ߩǡ ܳǡ ‫ݍ‬଴ ǡ ‫ܨ‬ǡ ߜ௪ ǡ ‫ܯ‬ǡْሻ where ܸǡ ߩǡ ܳǡ ‫ݍ‬଴ ܸ and‫ ܨ‬are the same for usual WK finite automaton, ߜ௪ ‫ ܳ ׷‬ൈ ቀ ቁ ՜ Ȭሺܳ ൈ ‫ܯ‬ሻis an extended transition function, ܸ M is a weighting space and ْ is an operation over weights from‫ܯ‬. Remark 1. In this paper we restrict our research into the weighting spaces and operations‫ ܯ‬ൌ ሺԺǡ ൅ሻand‫ ܯ‬ൌ ሺԷା ǡൈሻǤThen, if ሺ‫ܯ‬ǡْሻ ൌ ሺԺǡ ൅ሻǡa weighted WK automaton ‫ܭ‬is called a weighted WK automaton over Ժ (or additive for short) and, if ሺ‫ܯ‬ǡْሻ ൌ ሺԷା ǡൈሻǡa weighted WK automaton ‫ܭ‬is called a weighted WK automaton over Էା (or multiplicative for short). ‫ݓ‬ଵ ‫ݒ‬ଵ ‫ݔ‬ଵ ‫ݒ‬ଵ ‫ݓ‬ଵ ‫ݔ‬ଵ ܸ ‫כ‬ For ቀ‫ ݒ‬ቁ ǡ ቀ‫ ݓ‬ቁ ǡ ቀ‫ ݔ‬ቁ ‫ א‬ቀ ቁ such thatቂ‫ ݔ ݓ ݒ‬ቃ ‫ܭܹ א‬ఘ ሺܸሻ, we use the notation ܸ ଶ ଶ ଶ ଶ ଶ ଶ ‫ݔ ݓ‬ଵ ‫ݒ‬ଵ ‫ݒ‬ଵ ‫ݓ‬ ‫ݔ‬ଵ ቆቀ‫ ݒ‬ቁ ‫ ݍ‬ቀ‫ ݓ‬ቁ ቀ‫ ݔ‬ቁ ǡ ݉ଵ ቇ ֜ ቆቀ‫ ݒ‬ቁ ቀ‫ ݓ‬ቁ ‫ ݍ‬ᇱ ቀ‫ ݔ‬ቁ ǡ ݉ଵ ْ ݉ଶ ቇ ଶ ଶ ଶ ଶ ଶ ଶ ‫ݒ‬ଵ where‫ݍ‬ǡ ‫ݍ‬Ԣ ‫ܳ א‬, if and only if ሺ‫ ݍ‬ᇱ ǡ ݉ଶ ሻ ‫ א‬Ɂ ቆ‫ݍ‬ǡ ቀ‫ ݒ‬ቁቇ. The transitive and reflexive closure of ֜ is denoted by ֜‫ כ‬Ǥ ଶ

Definition 2. The language accepted by the weighted WK automaton ‫ܭ‬over Ժ is defined by ‫ݓ‬ଵ ‫ݓ‬ଵ ‫ݓ‬ଵ ‫ܮ‬௔ ሺ‫ܭ‬ሻ ൌ ቄሺ‫ݓ‬ଵ ǡ ‫݌‬ሻ ‫ א‬ሺܸ ‫ כ‬ǡ Ժሻȁቀ‫ݍ‬଴ ቂ‫ ݓ‬ቃ ǡ Ͳቁ ֜‫ כ‬ቀቂ‫ ݓ‬ቃ ‫ݍ‬௙ ǡ ‫݌‬ቁ ‫ݍ‬௙ ‫ݓܨ א‬ଶ ‫ כ ܸ א‬ቂ‫ ݓ‬ቃ ଶ ଶ ଶ ‫ܭܹ א‬ఘ ሺܸሻቅǤ Definition 3. The language accepted by the weighted WK automaton ‫ܭ‬over Է is defined by ‫ݓ‬ଵ ‫ݓ‬ଵ ‫ݓ‬ଵ ‫ܮ‬௠ ሺ‫ܭ‬ሻ ൌ ቄሺ‫ݓ‬ଵ ǡ ‫݌‬ሻ ‫ א‬ሺܸ ‫ כ‬ǡ Էା ሻȁቀ‫ݍ‬଴ ቂ‫ ݓ‬ቃ ǡ ͳቁ ֜‫ כ‬ቀቂ‫ ݓ‬ቃ ‫ݍ‬௙ ǡ ‫݌‬ቁ ‫ݍ‬௙ ‫ݓܨ א‬ଶ ‫ כ ܸ א‬ቂ‫ ݓ‬ቃ ଶ ଶ ଶ ‫ܭܹ א‬ఘ ሺܸሻቅǤ

Definition 4. Let ‫ܮ‬௫ ሺ‫ܭ‬ሻǡwhere ‫ א ݔ‬ሼܽǡ ݉ሽǡbe the language accepted by a weighted Watson-Crick automaton ‫ ܭ‬ൌ ሺܸǡ ߩǡ ܳǡ ‫ݍ‬଴ ǡ ‫ܨ‬ǡ ߜ௪ ǡ ‫ܯ‬ǡْሻ. A threshold language ‫ܮ‬௫ ሺ‫ܭ‬ǡ‫߬ څ‬ሻwith respect to a threshold (cut-point)߬ ‫ ܯ א‬is a subset of ‫ܮ‬௫ ሺ‫ܭ‬ሻ defined by ‫ܮ‬௫ ሺ‫ܭ‬ǡ‫߬ څ‬ሻ ൌ ሼ‫ݓ‬ଵ ‫ כ ܸ א‬ȁሺ‫ݓ‬ଵ ǡ ‫݌‬ሻ ‫ܮ א‬௫ ሺ‫ܭ‬ሻܽ݊݀‫߬ څ ݌‬ሽ where‫ אڅ‬ሼൌǡ ൐ǡ ൏ሽis called the mode of ‫ܮ‬௫ ሺ‫ܭ‬ǡ‫߬ څ‬ሻ. The family of languages accepted by Watson-Crick automata and weighted (additive, multiplicative) WatsonCrick automata are denoted by ‫܅‬۹ and ‫܅ݓ‬۹ (ܽ‫܅‬۹ǡ ݉‫܅‬۹), respectively. From the definitions above the next lemma follows immediately.


Lemma 1. ‫ۺ‬۷‫܅ ؿ ۼ‬۹ ‫܅ݔ ك‬۹ǡ where ‫ א ݔ‬ሼܽǡ ݉ሽǤ Proof. For an additive WK automaton, we choose Ͳ as the weight for each transition and for a multiplicative WK automaton, we chose ͳ as the weight for each transition. Theorem 1. ܽ‫܅‬۹ ‫܅݉ ك‬۹. Proof. Let L‫܅ܽ א‬۹. Then there is an additive WK automaton K = (ܸǡ ߩǡ ܳǡ ‫ݍ‬଴ ǡ ‫ܨ‬ǡ ߜ௪ ǡ ‫ܯ‬ǡ ൅), ‫ ك ܯ‬Ժ, such that L = ‫ ܮ‬ൌ ‫ܮ‬௔ ሺ‫ܭ‬ǡ‫߬ څ‬ሻ. We construct the multiplicative WK automaton ᇱ ൌ ሺܸǡ ߩǡ ܳǡ ‫ݍ‬଴ ǡ ‫ܨ‬ǡ ߜ௪ ǡ ‫ܯ‬ᇱ ǡൈሻ, ‫ܯ‬Ԣ ‫ ك‬Էା Ǥ For each ‫ݔ ݓ‬ଵ ‫ݒ‬ଵ ‫ݒ‬ଵ ‫ݓ‬ ‫ݔ‬ଵ ቆቀ‫ ݒ‬ቁ ‫ ݍ‬ቀ‫ ݓ‬ቁ ቀ‫ ݔ‬ቁ ǡ ݉ଵ ቇ ֜ ቆቀ‫ ݒ‬ቁ ቀ‫ ݓ‬ቁ ‫ ݍ‬ᇱ ቀ‫ ݔ‬ቁ ǡ ݉ଵ ൅ ݉ଶ ቇ ଶ

ଶ

ଶ

ଶ

ଶ

ଶ

‫ݒ‬ଵ where ‫ݍ‬ǡ ‫ݍ‬Ԣ ‫ܳ א‬, if and only if ሺ‫ ݍ‬ᇱ ǡ ݉ଵ ሻ ‫ א‬Ɂ ቆ‫ݍ‬ǡ ቀ‫ ݒ‬ቁቇ in ‫ܭ‬, we define ଶ

‫ݔ ݓ‬ଵ ‫ݒ‬ଵ ‫ݓ‬ ‫ݔ‬ଵ ‫ݒ‬ଵ ቆቀ‫ ݒ‬ቁ ‫ ݍ‬ቀ‫ ݓ‬ቁ ቀ‫ ݔ‬ቁ ǡ ʹ௠భ ቇ ֜ ቆቀ‫ ݒ‬ቁ ቀ‫ ݓ‬ቁ ‫ ݍ‬ᇱ ቀ‫ ݔ‬ቁ ǡ ʹ௠భ ൈ ʹ௠మ ቇ ଶ ଶ ଶ ଶ ଶ ଶ ‫ݒ‬ଵ if and only if ሺ‫ ݍ‬ᇱ ǡ ʹ௠మ ሻ ‫ א‬Ɂ ቆ‫ݍ‬ǡ ቀ‫ ݒ‬ቁቇ in ‫ ܭ‬ᇱ Ǥ ଶ

Then it is clear that ௔ ሺ‫ܭ‬ǡ‫߬ څ‬ሻ = ௠ ሺ‫ ܭ‬ᇱ ǡ‫ʹ څ‬ఛ ሻ. Since we can use ߣ transitions in WK automata, we can “represent” any weight ‫ ݓ‬൐ ͳ or ‫ ݓ‬൏ ͳ as the sum of ones or negative ones, respectively, thus, the following theorem follows Theorem 2. For each additive WK automaton ‫ܭ‬, there is an equivalent additive WK automaton ‫ ܭ‬ᇱ ൌ ሺܸǡ ߩǡ ܳǡ ‫ݍ‬଴ ǡ ‫ܨ‬ǡ ߜ௪ ǡ ‫ܯ‬ǡْሻ, such that ‫ ܯ‬ൌ ሼെͳǡ Ͳǡ ͳሽ. Similarly, for any multiplicative WK automaton, we can represent any weight ‫ ݓ‬as the product of prime numbers, and construct an equivalent multiplicative WK automaton the weights of whose transitions are only prime numbers or one over prime numbers. Theorem 3. For each multiplicative WK automaton ‫ܭ‬, there is an equivalent multiplicative WK automaton ‫ ܭ‬ᇱ ൌ ሺܸǡ ߩǡ ܳǡ ‫ݍ‬଴ ǡ ‫ܨ‬ǡ ߜ௪ ǡ ‫ܯ‬ǡْሻ, such that ‫ ܯ‬ൌ ሼ‫ݔ‬ȁ‫ ݔ‬ൌ ‫ ݔݎ݋݌‬ൌ ͳȀ‫݌݌‬ሽ.

CONCLUSION We have introduced weighted variants of WK automata by associating weights (integer or rational numbers) with transitions of a WK automaton and computing the weight of the acceptance using weight operations over the weights of the transitions participated in the recognition of a string. The results show that the generative power of the WK automata is increased by imposing weights. There are many interesting topics that can be explored in the future such as: x

study the generative power of the weighted variants of WK automata such as stateless, all-final, simple and 1limited;

x

study the closure properties and complexity problems of the weighted WK automata;

x

construction of tools for testing the correctness of simulated weighted WK automata.

ACKNOWLEDGMENTS This work has been supported by IIUM Research Endowment Funds EDW B13-053-0938, EDW B14-136-1021 and Ministry of Education Malaysia FRGS13-066-0307.


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