Cover Page
The handle http://hdl.handle.net/1887/37052 holds various files of this Leiden University dissertation.
Author: Vliet, Rudy van
Title: DNA expressions : a formal notation for DNA Issue Date: 2015-12-10
DNA Expressions
A Formal Notation for DNA
Rudy van Vliet
The work described in this thesis has been carried out under the auspices of the research school IPA (Institute for Programming research and Algorithmics).
© 2015 Rudy van Vliet, except for the Calvin and Hobbes comic strip Typeset using LATEX
Printing: Ridderprint BV
Printed on BalancePure® recycled paper ISBN: 978-94-6299-254-2
IPA Dissertation Series 2015-23
Despite the effort put into the careful writing of this thesis, it is inevitable that it contains errors. Errors detected can be reported to the author at rvvliet@liacs.nl . He will maintain a list of errata at his website on DNA expressions, which is currently to be found at
http://www.liacs.leidenuniv.nl/~vlietrvan1/dnaexpressions/
The first report of any indisputable error will be rewarded for with e0.10 and an honour- able mention in the list of errata.
DNA Expressions
A Formal Notation for DNA
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden, op gezag van Rector Magnificus prof. mr C.J.J.M. Stolker,
volgens besluit van het College voor Promoties te verdedigen op donderdag 10 december 2015
klokke 12.30 uur
door
Rudy van Vliet
geboren te Alphen aan den Rijn in 1969
Promotiecommissie
Promotor: prof. dr J.N. Kok Copromotor: dr H.J. Hoogeboom
Overige leden: prof. dr N. Jonoska (University of South Florida) dr R. Brijder (Universiteit Hasselt)
prof. dr H.P. Spaink prof. dr T.H.W. B¨ack
© 1992 Watterson.
Dist. by UNIVERSAL UCLICK. All rights reserved.
Contents
1 Introduction 1
1.1 Background of the thesis . . . 1
1.2 Contribution of the thesis . . . 2
1.3 Set-up of the thesis . . . 4
1.4 Resulting publications . . . 5
2 Preliminaries 9 2.1 Strings, trees, grammars, relations and complexity . . . 9
2.2 DNA molecules . . . 19
2.3 DNA computing . . . 25
2.3.1 Splicing systems . . . 25
2.3.2 Adleman’s experiment . . . 26
I DNA Expressions in General 31
3 Formal DNA Molecules 33 3.1 N -words . . . 333.2 Definition of formal DNA molecules . . . 33
3.3 Components of a formal DNA molecule . . . 36
3.4 Properties, relations and functions of formal DNA molecules . . . 39
4 DNA Expressions 43 4.1 Operators and DNA expressions . . . 43
4.2 Brackets, arguments and DNA subexpressions . . . 51
4.3 Recognition of DNA expressions . . . 54
4.4 Computing the semantics of a DNA expression . . . 58
4.5 A context-free grammar for D . . . 67
4.6 The structure tree of a DNA expression . . . 73
4.7 Equivalent DNA expressions . . . 75
5 Basic Results on DNA Expressions 79 5.1 Expressible formal DNA molecules . . . 79
5.2 Nick free DNA expressions . . . 82
5.3 Some equivalences . . . 83
II Minimal DNA Expressions 99
6 The Length of a DNA Expression 101
I
6.1 The operators in a DNA expression . . . 101
6.2 Blocks of components of a formal DNA molecule . . . 103
6.3 Lower bounds for the length of a DNA expression . . . 124
7 The Construction of Minimal DNA Expressions 137 7.1 Minimal DNA expressions for a nick free formal DNA molecule . . . 138
7.2 Minimal DNA expressions for a formal DNA molecule with nick letters . . 171
8 All Minimal DNA Expressions 183 8.1 Reverse construction of a minimal DNA expression . . . 183
8.2 Operator-minimal l-expressions . . . 200
8.3 Characterization of minimal DNA expressions . . . 204
8.4 The structure tree of a minimal DNA expression . . . 215
8.5 The number of (operator-)minimal DNA expressions . . . 217
9 An Algorithm for Minimality 237 9.1 The algorithm and its correctness . . . 237
9.1.1 The procedure MakelExprMinimal . . . 255
9.1.2 The procedure Denickify . . . 262
9.1.3 The procedure RotateToMinimal . . . 271
9.2 The algorithm for an example . . . 274
9.3 Detailed implementation and complexity of the algorithm . . . 284
9.4 Decrease of length by the algorithm . . . 302
III Minimal Normal Form 311
10 A Minimal Normal Form for DNA Expressions 313 10.1 Definition of the minimal normal form . . . 31410.2 Characterization of the minimal normal form . . . 317
10.3 The structure tree of a DNA expression in minimal normal form . . . 324
10.4 Regularity of DMinNF . . . 325
11 Algorithms for the Minimal Normal Form 341 11.1 Recursive algorithm for the minimal normal form . . . 341
11.2 Two-step algorithm for the minimal normal form . . . 348
11.3 Implementation and complexity of the algorithm . . . 354
12 Conclusions and Directions for Future Research 367
Samenvatting 369
Over de Auteur 375
Dankwoord 377
Bibliography 379
II
List of Symbols 383
Index 385
Titles in the IPA Dissertation Series since 2009 393
III
