Theoretical and Experimental DNA Computation (Natural ...

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2.4 The Random Access Machine 29 The example program given above implements unary addition. The input represents two numbers to be added, expressed in unary notation. To represent two integers {j, k} as an input string we start with a marker “0”, followed by j “1”s, followed by a separator “0”, and then k “1”s terminated by a final “0.” So, to add 2 and 3 we would specify an input string on 01101110. Running the program specified in Table 2.2 on the input string 01101110 involves the steps described in Table 2.3. Note that the position of the readwrite head on the tape is denoted by an arrow. Table 2.3. Run of TM program Tape State Action ˇ01101110 1 Write 0, go to state 2 0ˇ1101110 2 Scan over 1 01ˇ101110 2 Scan over 1 011ˇ01110 2 End of first number, go to next 0110ˇ1110 3 Change 1 to 0, go back 011ˇ10110 4 Copy 1, return to second number 01111ˇ010 2 End of first number, go to next 011110ˇ10 3 Change 1 to 0, go back 01111ˇ010 4 Copy 1, return to second number 0111110ˇ0 3 No second number, erase trailing 0 011111ˇ0 5 Halt It is clear that, after the TM halts, the tape represents the correct sum represented in unary notation. 2.4 The Random Access Machine Any function (problem) that can be computed by any other machine can also be computed by a Turing Machine. This thesis was established by Alonzo Church (see [145]). This means that Turing Machines are universal in the sense of computation. Note that this does not necessarily mean that any function can be computed by a Turing Machine – there exist uncomputable functions. The most famous uncomputable problem is the Halting Problem: given a Turing Machine with a given input, will it halt in a finite number number of steps or not? This can be re-phrased in the following fashion: is it possible to write a program to determine whether any arbitrary program will halt? The answer is no. So, we can see that the Turing Machine is useful for investigating fundamental issues of computability. However, the ease of its application in other domains is less clear. Writing a program for a Turing Machine is a very “lowlevel” activity that yields an awkward and non-intuitive result. TMs also differ
30 2 Theoretical Computer Science: A Primer from “traditional” computers in several important ways, the most important concerning memory access. Recall the run-through of the example TM program, given in Table 2.3. In order to reach the end of the first number on the tape, the TM was forced to scan linearly along the tape until it reached the middle marker. In more general terms, in order to reach a distant tape (memory) cell, the TM must first read all of the intermediate cells. This problem may be overcome by the introduction of the Random Access Machine (RAM) [5]. This machine is capable of reading an arbitrary memory cell in a single step. Like the TM, the RAM is also abstract in the sense that it has unbounded memory and can store arbitrarily large integers in each cell. The structure of the RAM is depicted in Fig. 2.5. The main components of the machine are the read-only input tape, a write-only output tape, a fixed program, and a memory. The input and output tapes are identical in structure to the TM tape. When a symbol is read from the input tape, the read head is moved one cell to the right. When a symbol is written to the output tape, the write head is moved one cell to the right. The RAM program is fixed, and is stored separately from the memory. The first memory location, M[0] is reserved for the accumulator, whereall computation takes place. PC Program Accumulator M[0] M[1] M[2] Fig. 2.5. Random Access Machine Input tape Memory Output tape In order to describe algorithms, we define an instruction set and a program. The instruction set defines the range of operations available to the RAM. The exact nature of the operations chosen is not important, and different authors specify different instruction sets. However, the set of instructions usually resemble those found in real computers. The program is simply a sequence of instructions, any of which may be labelled. Theprogram counter (PC) keeps track of the current execution point in the program. An example instruction set is given in Table 2.4. Instructions are of the form < operation >< operand >. An operand can be:
Page 2 and 3: Natural Computing Series Series Edi
Page 4: Martyn Amos Department of Computer
Page 8 and 9: Preface DNA computation has emerged
Page 10: Preface IX acknowledged. Finally, I
Page 13 and 14: XII Contents 4 Complexity Issues ..
Page 16 and 17: Introduction “Where a calculator
Page 18 and 19: Introduction 3 Even though their un
Page 20 and 21: 1 DNA: The Molecule of Life “All
Page 22 and 23: 1.3 DNA as the Carrier of Genetic I
Page 24 and 25: 1.3 DNA as the Carrier of Genetic I
Page 26 and 27: Synthesis 1.4 Operations on DNA 11
Page 28 and 29: Gel electrophoresis 1.4 Operations
Page 30 and 31: 5’ 3’ A T A G A G T T 3’ TCA
Page 32 and 33: 1.4 Operations on DNA 17 Others may
Page 34 and 35: Vector Strand to be inserted (targe
Page 36: 1.5 Summary 1.6 Bibliographical Not
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Page 93 and 94: 78 4 Complexity Issues Ω = {∧,
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80 4 Complexity Issues 0 1 0 1 x1 x
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82 4 Complexity Issues connecting a
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84 4 Complexity Issues Level simula
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86 4 Complexity Issues At level D(S
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88 4 Complexity Issues xANDy)) and
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90 4 Complexity Issues Input matrix
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92 4 Complexity Issues memory acces
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94 4 Complexity Issues denoted by P
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96 4 Complexity Issues The final st
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98 4 Complexity Issues 1)), ADDR[])
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100 4 Complexity Issues Size(STI)=O
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102 4 Complexity Issues best attain
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104 4 Complexity Issues 10. LDC 0 1
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106 4 Complexity Issues ACC[] := bi
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5 Physical Implementations “No am
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5.3 Initial Set Construction Within
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Vertex 1 Vertex 2 Vertex 3 ¡ ¡ ¡
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5.5 Evaluation of Adleman’s Imple
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5.6 Implementation of the Parallel
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5.8 Experimental Investigations 119
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Create initial strand library Confi
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Experiment 25. New full-length chai
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5.9 Other Laboratory Implementation
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5.11 Bibliographical Notes 145 whic
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148 6 Cellular Computing of truly i
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150 6 Cellular Computing 6.2 Succes
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152 6 Cellular Computing are “glu
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154 6 Cellular Computing x y z x (a
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156 6 Cellular Computing 6.7 Biblio
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158 References 14. Martyn Amos, Ste
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160 References 53. Dónall A. Mac D
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162 References 92. D. Knuth. The Ar
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164 References 135. Kensaku Sakamot
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Index ALGOL, 102 AND, 23-24, 42, 87
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iological, 74, 114, 150 Feynman, R.
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Petrinet,63 phage, 18-20 phosphate,
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Natural Computing Series W.M. Spear
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