LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

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124 Karel Culik II, Juhani Karhumäki, and Petri Salmela u3 = ab : a · ab · (bab) n /∈ Suf(X + ) implies aab(bab) n /∈ C(X) sothat aab(bab) n /∈ XC(X) and therefore ab(bab) n /∈ C(X) for all n ∈ N. Hence the number n3 does not exist. u4 = ba : ba · an · a/∈ XC(X) and therefore ba · an /∈ C(X) for all n ∈ N, and hence the number n4 does not exist. u5 = bb : bb ∈ X ⊆ C(X) sothatn5 =0. As a conclusion I = {0, 1, 5}, andG = � i∈I uiX ni =1· X + a + bb = X. This gives us the centralizer in other words we have established: C(X) =GX ∗ = XX ∗ = X + , Fact 1 C({a, bb, aba, bab, bbb})={a, bb, aba, bab, bbb} + . Next we prove that the fixed point approach applied to the language X leads to an infinite loop of iterations. We prove this by showing that there exist words in C(X) \ Xi for every Xi of the iteration (1). To do this we take a closer look on the language L =(bab) ∗ ab(bab) ∗ . Clearly L ⊆ X0 =Pref(X + ) ∩ Suf(X + )and L ∩ X + = ∅. By the definition of the fixed point approach, word w ∈ Xi is in Xi+1 if and only if Xw ⊆ XiX and wX ⊆ XXi. We will check this condition for an arbitrary word (bab) k ab(bab) n ∈ L with k, n ≥ 1. The first condition Xw ⊆ XiX leads to the cases: and a · (bab) k ab(bab) n =(aba)(bb · a) k−1 (bab) n+1 ∈ X + X ⊆ XiX, bb · (bab) k ab(bab) n =(bbb)a(bb · a) k−1 (bab) n+1 ∈ X + X ⊆ XiX, aba · (bab) k ab(bab) n = a(bab)a(bb · a) k−1 (bab) n+1 ∈ X + X ⊆ XiX, bbb · (bab) k ab(bab) n =(bb) 2 a(bb · a) k−1 (bab) n+1 ∈ X + X ⊆ XiX bab · (bab) k ab(bab) n =(bab) k+1 ab(bab) n−1 · bab ∈ XiX. However, the last one holds if and only if and ⎫ ⎪⎬ ⎪⎭ (3) (bab) k+1 ab(bab) n−1 ∈ Xi. (4) Similarly, the second condition wX ⊆ XXi yields us: (bab) k ab(bab) n · a = bab(bab) k−1 (a·bb) n aba ∈ XX + ⊆ XXi, (bab) k ab(bab) n · bb = bab(bab) k−1 (a·bb) n a · bbb ∈ XX + ⊆ XXi, (bab) k ab(bab) n · aba = bab(bab) k−1 (a·bb) n a·a·bab·a ∈ XX + ⊆ XXi, (bab) k ab(bab) n · bbb = bab(bab) k−1 (a·bb) n a·a·bb· bb ∈ XX + ⊆ XXi (bab) k ab(bab) n · bab = bab · (bab) k−1 ab(bab) n+1 ∈ XXi. ⎫ ⎪⎬ ⎪⎭ (5)
Here the last one holds if and only if Fixed Point Approach to Commutation of Languages 125 From (4) and (6) we obtain the equivalence (bab) k−1 ab(bab) n+1 ∈ Xi. (6) (bab) k ab(bab) n ∈ Xi+1 ⇐⇒ (bab) k+1 ab(bab) n−1 , (bab) k−1 ab(bab) n+1 ∈ Xi (7) Now, the result follows by induction, when we cover the cases k =0orn =0. In the case k =0andn ≥ 0wehavethatab(bab) n ∈ X0, but ab(bab) n /∈ X1, since a · ab(bab) n /∈ X0X. The same applies also for (bab) n ba by symmetry. �� Fig. 1. Language (bab) ∗ ab(bab) ∗ written as a pyramid. � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Fig. 2. Deleting words in (bab) ∗ ab(bab) ∗ from languages Xi during the iteration. In the case n =0andk ≥ 1wefirstnotethatX(bab) k ab ⊆ X0X, due to the equations in (3) and the fact that bab·(bab) k ab =(bab) k ba·bab ∈ X0X. Similarly, (bab) k abX ⊆ XX0, due to the equations in (5) and the fact (bab) k ab · bab = bab · (bab) k−1 ab(bab) ∈ XX0. These together imply that (bab) k ab ∈ X1. On the other hand, since (bab) k ba /∈ X1, thenbab · (bab) k ab /∈ X1X, and hence (bab) k ab /∈ X2.
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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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2 Definitions Definition 1. A strea
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5 Conclusions Eilenberg P Systems w
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Molecular Tiling and DNA Self-assem
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3 Molecular Self-assembly Processes
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Page 83 and 84: Molecular Tiling and DNA Self-assem
Page 85 and 86: 8 Hierarchical Tiling Molecular Til
Page 91 and 92: References Molecular Tiling and DNA
Page 95 and 96: On Some Classes of Splicing Languag
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Page 117 and 118: The Power of Networks of Watson-Cri
Page 129 and 130: Fixed Point Approach to Commutation
Page 133: Fixed Point Approach to Commutation
Page 139 and 140: � � � � � Fixed Point App
Page 143 and 144: Remarks on Relativisations and DNA
Page 149 and 150: Splicing Test Tube Systems and Thei
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Page 153 and 154: Splicing Test Tube Systems 143 to a
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Page 157 and 158: Splicing Test Tube Systems 147 (c)
Page 159 and 160: I0 Ai i Splicing Test Tube Systems
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Page 163 and 164: Digital Information Encoding on DNA
Page 177 and 178: DNA-based Cryptography Ashish Gehan
Page 179 and 180: DNA-based Cryptography 169 concern.
Page 181 and 182: DNA-based Cryptography 171 The one-
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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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