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Computational Complexity: from Patience to Precision and Beyond

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Computational Complexity: from Patience to Precision and Beyond A Model-Independent Theory of Computational Complexity: From Patience to Precision and Beyond Ed Blakey The Queen's College, Oxford [email protected] Oxford University Computing Laboratory Wolfson Building, Parks Road, Oxford, OX1 3QD Submitted in Trinity Term, 2010 for the degree of Doctor of Philosophy in Computer Science Acknowledgements and Dedication We thank the author's supervisors, Dr Bob Coecke and Dr JoÄelOuaknine, for their continued support and suggestions during this DPhil project. We thank the author's Transfer/Con¯rmation of Status assessors, Prof. Samson Abramsky and Prof. Peter Jeavons, ¯rst for agreeing to act as assessors and secondly for their useful and for- mative comments about this project; we thank the author's Examiners, Prof. Peter Jeavons and Prof. John Tucker, for taking the time to act as such and for their valuable suggestions, of which some are incorporated here. We thank the organizers of Uncon- ventional Computing, the International Workshop on Natural Computing, Quantum Physics and Logic/Development of Computational Models, Science and Philosophy of Unconventional Computing and the International Conference on Systems Theory and Scienti¯c Computation for the opportunity (and, in the case of the last-mentioned con- ference, the kind invitation) to present work forming part of this project. We thank the participants of the above-mentioned conferences/workshops, as well as the British Colloquium for Theoretical Computer Science and Complexity Resources in Physical Computation, for their encouraging feedback and insightful discussion. We thank the reviewers of publications to which the author has contributed (including New Genera- tion Computing, the International Journal of Unconventional Computing and Natural Computing, as well as proceedings/publications associated with the conferences and workshops mentioned above) for their detailed comments and helpful suggestions; we thank also Prof. Jos¶eF¶elixCosta for his kind invitation to contribute to the last- mentioned journal. We thank collaborators and colleagues for showing an interest in this work, for useful discussion and corrections, for bringing to the author's attention many relevant references, and for helping to shape and direct this research (as have the conference participants and publication reviewers|and of course supervisors and assessors|mentioned above); notably, we thank Prof. Cristian Calude for fascinating discussions regarding accelerated Turing machines and the conjectured incomplete- ness of axiomatizations of resource, Dr Viv Kendon and colleagues (not least Rob Wagner) for the opportunity to explore the interface between the present project and the practical considerations of quantum computation/simulation, Dr Rebecca Palmer for noticing an omission relating to wave superposition, Andr¶asSalamon for inspir- ing parts of the present work with questions concerning gap theorems and combined resources, Prof. Susan Stepney for her championing the `natural' in `natural compu- tation', and Dr Damien Woods for his suggestions relating to restriction of precision. Academic assistance aside, we thank the administrative and support sta® both at the Oxford University Computing Laboratory and at the Queen's College for their help during the author's DPhil programme; we thank in particular Janet Sadler for her assistance with the organization of the workshop Complexity Resources in Physical Computation and with the administration of the below-mentioned EPSRC funding. We acknowledge the generous ¯nancial support of the EPSRC; this work forms part of, and as of 1.x.2008 is funded by, the EPSRC project Complexity and Decidability in Unconventional Computational Models (EP/G003017/1). Personal thanks are due from the author to his family; to his friends in Oxford, Warwick and elsewhere; and to his girlfriend, Gemma. The author dedicates this dissertation to the memory of his mother, Linda (1944 { 2009). Abstract A Model-Independent Theory of Computational Complexity: From Patience to Precision and Beyond Ed Blakey, The Queen's College Submitted in Trinity Term, 2010 for the degree of Doctor of Philosophy in Computer Science The ¯eld of computational complexity theory|which chiey aims to quan- tify the di±culty encountered when performing calculations|is, in the case of conventional computers, correctly practised and well understood (some impor- tant and fundamental open questions notwithstanding); however, such under- standing is, we argue, lacking when unconventional paradigms are considered. As an illustration, we present here an analogue computer that performs the task of natural-number factorization using only polynomial time and space; the sys- tem's true, exponential complexity, which arises from requirements concerning precision, is overlooked by a traditional, `time-and-space' approach to complex- ity theory. Hence, we formulate the thesis that unconventional computers war- rant unconventional complexity analysis; the crucial omission from traditional analysis, we suggest, is consideration of relevant resources, these being not only time and space, but also precision, energy, etc. In the presence of this multitude of resources, however, the task of compar- ing computers' e±ciency (formerly a case merely of comparing time complexity) becomes di±cult. We resolve this by introducing a notion of overall complex- ity, though this transpires to be incompatible with an unrestricted formulation of resource; accordingly, we de¯ne normality of resource, and stipulate that considered resources be normal, so as to rectify certain undesirable complexity behaviour. Our concept of overall complexity induces corresponding complexity classes, and we prove theorems concerning, for example, the inclusions therebe- tween. Our notions of resource, overall complexity, normality, etc. form a model- independent framework of computational complexity theory, which allows: in- sightful complexity analysis of unconventional computers; comparison of large, model-heterogeneous sets of computers, and correspondingly improved bounds upon the complexity of problems; assessment of novel, unconventional systems against existing, Turing-machine benchmarks; increased con¯dence in the di±- culty of problems; etc. We apply notions of the framework to existing disputes in the literature, and consider in the context of the framework various fundamental questions concerning the nature of computation. Contents 1 Introduction 6 1.1 A Tale of Two Complexities . 6 1.2 Background . 7 1.2.1 Preliminaries . 7 1.2.2 Literature . 12 1.3 Overview . 12 2 Motivation 15 2.1 Introduction . 15 2.1.1 Factorization . 16 2.2 Original Analogue Factorization System . 27 2.2.1 Description . 27 2.2.2 Proof of Correctness . 52 2.2.3 Time/Space Complexity . 53 2.2.4 Precision Complexity . 55 2.2.5 Summary . 57 2.3 Improved Analogue Factorization System . 58 2.3.1 Relating the Original and Improved Systems . 58 2.3.2 Description . 59 2.3.3 Proof of Correctness . 66 2.3.4 Practical Considerations . 67 2.3.5 Time/Space Complexity . 69 2.3.6 RSA Factorization . 70 2.3.7 Precision Complexity . 72 2.3.8 Summary . 73 2.4 Conclusion . 74 2.4.1 Summary . 74 2.4.2 Discussion . 74 3 Resource 75 3.1 Unconventional Resources . 75 3.1.1 The Need Therefor. 75 3.1.2 . But the Neglect Thereof . 79 3.2 Interpreting `Resource' . 83 3.2.1 Possible Interpretations . 83 3.2.2 Recasting as Di®erent Resource Types . 88 3.2.3 Source of Complexity . 88 3.2.4 Commodity Resources . 91 1 CONTENTS 2 3.3 Precision|an Illustrative Unconventional Resource . 92 3.3.1 Motivation . 92 3.3.2 Examples . 93 3.3.3 De¯nitions . 105 3.3.4 Precision in the Literature . 114 3.4 Formalizing Resource . 116 3.4.1 First Steps . 116 3.4.2 Null Resource . 118 3.4.3 From Resource Comes Complexity . 118 3.5 Model-Independent Resources . 121 3.5.1 Speci¯c Examples . 122 3.5.2 Generic Criteria . 124 3.5.3 Two Approaches . 125 3.6 Model-Dependent Resources . 125 3.6.1 Resources for Non-Quantum Computers . 125 3.6.2 Resources for Quantum Computers . 132 3.6.3 Summary . 136 3.7 The Old versus the New . 136 3.7.1 Two Conventional Resources . 137 3.7.2 Moore's Law . 138 3.8 Underlying Resource . 139 3.8.1 Trade-O®s . 140 3.8.2 Combining Resources . 141 3.9 Summary . 142 4 Dominance 143 4.1 Comparison of Computers . 143 4.1.1 The Need to Compare Computers . 143 4.1.2 Problem-Homogeneity . 146 4.1.3 Comparing Turing Machines . 146 4.1.4 Comparing Unconventional Computers . 147 4.2 Dominance De¯ned . 149 4.2.1 Overall Complexity . 150 4.3 Dominance Classes . 152 4.3.1 De¯nitions . 152 4.3.2 Expected Results . 152 4.3.3 Other Internal Results . 153 4.3.4 Trade-O® Results . 160 4.3.5 Relationship to Traditional Hierarchy . 162 4.3.6 Transfer between Hierarchies of Results/Questions . 162 4.4 An Analogue of the Gap Theorem . 162 4.4.1 Introduction . 163 4.4.2 Dominance Gap Theorem . 165 4.4.3 Future Work . 168 4.4.4 Conclusion . 171 4.5 Normalization|Resources Revisited . 172 4.5.1 The Problem: Dominance versus Unrestricted Resource . 172 4.5.2 The Solution: Normalization . 173 4.5.3 Why Normalization? . 178 4.5.4 Summary of Resource . 178 CONTENTS 3 4.6 Discussion . 179 4.6.1 Summary . 179 4.6.2 Comments . 180 5 Case Studies 181 5.1 Introduction . 181 5.2 Ray-Tracing|Computing the Uncomputable? . 181 5.2.1 Background . 181 5.2.2 Resolution . 182 5.3 An Adiabatic Solution to Hilbert's Tenth Problem . 184 5.3.1 Background . 184 5.3.2 Resolution . 186 5.4 Cabello versus Meyer . 186 5.4.1 Background . 186 5.4.2 Resolution|First Steps . 188 5.5 Future Work . 189 5.6 Discussion . 190 5.6.1 General Features of Resolution . 190 5.6.2 Extent of Resolution . 191 6 Discussion 192 6.1 Other Applications . 192 6.1.1 Cryptography .
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