LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

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330 Gheorghe Păun targets for the same rule, we can either proceed as in the definition given above, or we can restrict the use of metarules so that only one target is used. On the other hand, we can complicate the matter by adding further features, such as the possibility to dissolve a membrane (then both the objects and the rules should remain free in the immediately upper membrane), or the possibility to also move objects through membranes. Furthermore, we can take as the result of a computation the trace of a designated rule in its passage through membranes, in the same way as the trace of a traveller-object was considered in “standard” (symport/antiport) P systems. Then, we can pass to systems with symport/antiport rules; in such a case, out will mean that the rule moves up and gets associated with the membrane immediately above the membrane with which the rule was associated (and applied), while in will mean going one step down in the membrane structure. Otherwise stated, the targets are, in fact, here, up, down. Plenty of problems, but we switch to another “exotic” idea. 3 Counter-Dividing Membranes The problem to consider reversible P systems was formulated several times – with “reversible” meaning both the possibility to reverse computations, like in dynamic systems, and local/strong reversibility, at the level of rules: considering v → u at the same time with u → v. For multiset rewriting-like rules this does not seems very spectacular (although most proofs from the membrane computing literature, if not all proofs, will get ruined by imposing this restriction to the existing sets of rules, at least because the synchronization is lost). The case of membrane division rules looks different/strange. A rule [ i a] i → [ i b] i [ i c] i produces two identical copies of the membrane i, replicating the contents of the divided membrane, the only difference between the two copies being that objects b and c replace the former object a. Moreover, generalizations can be considered, with division into more than two new membranes, with new membranes having (the same contents but) different labels. Also, we can divide both elementary and non-elementary membranes. Let us stay at the simplest level, of rules [ i a] i → [ i b] i [ i c] i which deal with elementary membranes only, 2 division, and the same label for the new membranes. Reversing such a rule, hence considering a rule [ i b] i [ i c] i → [ i a] i , will mean to pass from two membranes with the label i and identical contents up to the two objects b and c present in them to a single membrane, with the same contents and with the objects b, c replaced by a new object, a. Thisoperation looks rather powerful, as in only one step it compares the contents of two membranes, irrespective how large they are, and removes one of them, also changing one object. How to use this presumed power, for instance, for obtaining solutions to computationally hard problems, remains to be investigated. Here we are only mentioning the fact that this counter-division operation can be used in order to build a system which computes the intersection of the sets of numbers/vectors computed by two given systems. The idea is suggested in Figure 2. Consider two
Membrane Computing: Some Non-standard Ideas 331 Psystems,Π1 and Π2. Associate with each of them one membrane with label i. Embed one of the systems, say Π2, together with the associated membrane i, in a further membrane, with label 2, and then all these membranes into a skin membrane (1, in Figure 2). Let the systems Π1,Π2 work and halt; the objects defining the results are moved out of the two systems and then into the corresponding membranes with label i. At some (suitable) moment, dissolve membrane 2. Now, by a rule [ i b] i [ i c] i → [ i a] i we check whether or not the two systems have produced the same number (or vector of numbers), and, in the affirmative case, we stop. (For instance, each membrane i can evolve forever by means of suitable rules applied to b and c, respectively; after removing one of the membranes, the object a newly introduced will not evolve, hence the computation can stop.) Of course, several details are to be completed, depending onthetypeofthesystemsweworkwith. 1✬ ✬ Π1 ✩✬ 2 ✤ Figure 2: Computing the intersection. ✩ ✩ ✜ ✫ ❄ ❄ ❄ ✪✣ ❄ ❄ ❄ ✢ i ✓ ❄ ❄ ❄ ✏ i✓ ❄ ❄ ❄ ✏ ✒b ✑ ✒c ✑ ✫ ✪ ✫ ✪ Now, again the question arises: is this way to compute the intersection of any usefulness? We avoid addressing this issue, and pass to the third new type of P systems. Π2 4 Rules (and Membranes) Acceleration A new type of P systems, but not a new idea: accelerated Turing machines were considered since many years, see [1], [2], [3] and their references. Assume that our machine is so clever that it can learn from its own functioning, and this happens at the spectacular level of always halving the time needed to perform a step; the first step takes, as usual, one time unit. Then, in 1 + 1 1 1 2 + 4 + ...+ 2n + ...= 2 time units the machine performs an infinity of steps, hence finishing the computation. . . Beautiful, but exotic. . . Let us consider P systems using similarly clever rules, each one learning from its own previous applications and halving the time needed at the next
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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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8 Hierarchical Tiling Molecular Til
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References Molecular Tiling and DNA
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On Some Classes of Splicing Languag
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ˆxa ˆby ✬ ✩✬ ✩ a b x y
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The Power of Networks of Watson-Cri
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Fixed Point Approach to Commutation
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Here the last one holds if and only
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� � � � � Fixed Point App
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Remarks on Relativisations and DNA
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Splicing Test Tube Systems and Thei
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Splicing Test Tube Systems 141 the
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Splicing Test Tube Systems 143 to a
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Splicing Test Tube Systems 145 6. R
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Splicing Test Tube Systems 147 (c)
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I0 Ai i Splicing Test Tube Systems
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Splicing Test Tube Systems 151 4. J
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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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Page 289 and 290: Definition 1. Define the relation
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Communicating Distributed H Systems
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Communicating Distributed H Systems
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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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