LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

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102 Rodica Ceterchi, Carlos Martín-Vide, and K.G. Subramanian 7 Semi-simple Circular Splicing Systems For fixed letters a, b ∈ V , we will consider the following four types of semi-simple circular splicing rules: (a, 1; b, 1), (1,a;1,b), (1,a; b, 1), (a, 1; 1,b). The action of the four rules on circular strings is depicted in Figures 7, 8, 9 and 10. ˆxa ˆyb ✬ ✩✬ ✩ x a y b ✫ ⊢ (a,1;b,1) ✪✫ ✪ ˆxayb ✬ �� a x y ✫ �� b Fig. 7. Circular splicing using the semi-simple rule (a,1;b,1) âx ˆby ✬ ✩✬ ✩ b x y a ✫ ⊢ (1,a;1,b) ✪✫ ✪ âxby ✬ ❅ b x y a ✫❅ Fig. 8. Circular splicing using the semi-simple rule (1,a;1,b) âx ˆyb ✬ ✩✬ ✩ x y a b ✫ ⊢ (1,a;b,1) ✪✫ ✪ âxyb ✬ x y ✫❅ �� a b Fig. 9. Circular splicing using the semi-simple rule (1,a;b,1) ✩ ✪ ✩ ✪ ✩ ✪ We will consider the semi-simple splicing rules as functions from V ◦ × V ◦ to ◦ V 2 . Using the mirror image function we can express some relationships between the four types of rules seen as functions.
ˆxa ˆby ✬ ✩✬ ✩ a b x y ✫ ✪✫ ✪ On Some Classes of Splicing Languages 103 ⊢ (a,1;1,b) ˆxaby ✬ �� ❅ a b x y ✫ Fig. 10. Circular splicing using the semi-simple rule (a,1;1,b) Lemma 13. The following equalities of functions from V ◦ a (a, 1; b, 1)mi =(mi × mi)(1,a;1,b), (a, 1; 1,b)mi =(mi × mi)(1,a; b, 1). × V ◦ b to 2V ◦ Theorem 8. The classes SSH ◦ (1, 3) and SSH ◦ (2, 4) are incomparable. ✩ ✪ hold: Proof. Note first that simple circular splicing languages (in SH◦ (1, 3) = SH◦ (2, 4)) are in the intersection SSH◦ (1, 3) ∩ SSH◦ (2, 4). The semi-simple H system S1 =({a, b, c}, {âac,ˆb}, {(c, 1; b, 1)}) generates a language L1 = L(S1) ∈ SSH◦ (1, 3) \ SSH◦ (2, 4). Note that the words of L1 have the following two properties: (I) All words in L1, with the exception of the two axioms, have occurrences of all three letters a, b, andc. (II) All occurrences of a in the words of L1 are in subwords of the form aac. Now, suppose L1 is (2, 4) generated. Then it would be respected by one or several of the rules of the forms: (i) (1,c;1,b), (ii) (1,b;1,c), (iii) (1,a;1,b), (iv) (1,b;1,a), (v) (1,c;1,a), (vi) (1,a;1,c), (vii) (1,a;1,a), (viii) (1,b;1,b), (ix) (1,c;1,c). But we have: (v) (âac,âac) ⊢ (1,c;1,a) âacaca /∈ L1 (no b’s), (vi) (âac,âac) ⊢ (1,a;1,c) âacaca /∈ L1 (no b’s), (vii) (âac,âac) ⊢ (1,a;1,a) ˆcaacaa /∈ L1 (no b’s), (ix) (âac,âac) ⊢ (1,c;1,c) ˆcaacaa /∈ L1 (no b’s), (viii) (ˆb,ˆb) ⊢ (1,b;1,b) ˆbb /∈ L1 (no a’s, no c’s), (i) (âac,ˆb) ⊢ (1,c;1,b) ˆcaab /∈ L1 (contradicts (II)), (ii) (ˆb,âac) ⊢ (1,b;1,c) ˆbcaa /∈ L1 (contradicts (II)), (iii) (âac,ˆb) ⊢ (1,a;1,b) âcab /∈ L1 (contradicts (II)), (iv) (ˆb,âac) ⊢ (1,b;1,a) ˆbaca /∈ L1 (contradicts (II)). The semi-simple H system S2 =({a, b, c}, {âac,ˆb}, {(1,c;1,b)}) generates a language L2 = L(S2) ∈ SSH◦ (2, 4) \ SSH◦ (1, 3). The proof that L2 is not in SSH◦ (1, 3) is similar to the above one. We note that the words of L2 have the following properties:
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Lecture Notes in Computer Science 2
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Nata˘sa Jonoska Gheorghe Păun Grz
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Thomas J. Head
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VIII Preface portant to keep in min
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X Table of Contents Formal Properti
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Solving Graph Problems by P Systems
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where Solving Graph Problems by P S
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Writing Information into DNA Masano
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Writing Information into DNA 25 Ham
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Writing Information into DNA 27 Fig
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Writing Information into DNA 29 Dea
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5 Results 5.1 DNA Code for the Engl
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Writing Information into DNA 33 3.
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Writing Information into DNA 35 101
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Balance Machines: Computing = Balan
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+ .. . + Balance Machines: Computin
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x Balance Machines: Computing = Bal
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1 Balance Machines: Computing = Bal
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Eilenberg P Systems with Symbol-Obj
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Digital Information Encoding on DNA
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DNA-based Cryptography Ashish Gehan
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DNA-based Cryptography 169 concern.
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DNA-based Cryptography 171 The one-
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DNA-based Cryptography 173 of DNA t
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DNA-based Cryptography 175 Fig. 3.
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DNA-based Cryptography 177 the comp
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DNA-based Cryptography 179 can be c
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DNA-based Cryptography 181 4.4 DNA-
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DNA-based Cryptography 183 Fig. 7.
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DNA-based Cryptography 185 offer li
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DNA-based Cryptography 187 30. C. M
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Splicing to the Limit Elizabeth Goo
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Splicing to the Limit 191 molecular
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Splicing to the Limit 193 Discussio
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Example 9. The splicing rules are S
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−2α � kNNk = −2αMN, k α
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Finally we define the limit languag
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6 Conclusion Splicing to the Limit
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Formal Properties of Gene Assembly
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(a) (b) Formal Properties of Gene A
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n-Insertion on Languages Masami Ito
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n-Insertion on Languages 215 3. ∀
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n-Insertion on Languages 217 Theore
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Transducers with Programmable Input
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1 01 s 0 Transducers with Programma
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order β l Transducers with Program
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start tile Transducers with Program
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Methods for Constructing Coded DNA
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u u Methods for Constructing Coded
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On the Universality of P Systems wi
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An Algorithm for Testing Structure
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Definition 1. Define the relation
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280 Manfred Kudlek Definition 5. Co
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On Languages of Cyclic Words 283 Th
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On Languages of Cyclic Words 285 No
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On Languages of Cyclic Words 287 Th
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A DNA Algorithm for the Hamiltonian
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Formal Languages Arising from Gene
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A Proof of Regularity for Finite Sp
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--- ◦ ❄ ⊗ γ ✲ A Proof of R
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The Duality of Patterning in Molecu
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The Duality of Patterning in Molecu
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Membrane Computing: Some Non-standa
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where: Membrane Computing: Some Non
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where: Membrane Computing: Some Non
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The P Versus NP Problem Through Cel
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Realizing Switching Functions Using
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Realizing Switching Functions Using
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AND Gate Realizing Switching Functi
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NOR Gate Realizing Switching Functi
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Plasmids to Solve #3SAT Rani Siromo
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Plasmids to Solve #3SAT 363 A comme
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Plasmids to Solve #3SAT 365 from th
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Communicating Distributed H Systems
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Proof Communicating Distributed H S
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Books: Publications by Thomas J. He
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Publications by Thomas J. Head 387
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LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

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