page 14 Declaration page 15 Notes of copyright and the ownership of intellectual property rights page 15 The Author page 16 Acknowledgements page 16 1 Introduction page 17 1.1 Background page 17 1.2 The Style of Approach page 18 1.3 Motivation page 19 1.4 Style of Presentation page 20 1.5 Outline of the Thesis page 21 2 Models and Modelling page 23 2.1 Some Types of Models page 25 2.2 Combinations of Models page 28 2.3 Parts of the Modelling Apparatus page 33 2.4 Models in Machine Learning page 38 2.5 The Philosophical Background to the Rest of this Thesis page 41 Syntactic Measures of Complexity page 3 3 Problems and Properties page 44 3.1 Examples of Common Usage page 44 3.1.1 A case of nails page 44 3.1.2 Writing a thesis page 44 3.1.3 Mathematics page 44 3.1.4 A gas page 44 3.1.5 An ant hill page 45 3.1.6 A car engine page 45 3.1.7 A cell as part of an organism page 46 3.1.8 Computer programming page 46 3.2 Complexity as a Comparison page 46 3.2.1 The emergence of life page 47 3.3 What the Property of Complexity Could Usefully Refer to page 47 3.3.1 Natural systems page 47 3.3.2 The interaction of an observer with a system page 53 3.3.3 Patterns page 55 3.3.4 The modelling relation page 56 3.3.5 A model with respect to a specified framework page 56 3.4 Some Unsatisfactory Accounts of Complexity page 57 3.4.1 Size page 57 3.4.2 Size of rules page 58 3.4.3 Minimal size page 58 3.4.4 Processing time page 59 3.4.5 Ignorance page 60 3.4.6 Variety page 61 3.4.7 Midpoint between order and disorder page 62 3.4.8 Improbability page 63 3.4.9 Expressivity page 65 3.4.10 Dimension page 65 3.4.11 Ability to surprise page 66 3.4.12 Logical strength page 66 Syntactic Measures of Complexity page 4 3.4.13 Irreducibility page 67 3.5 Complexity is Relative to the Frame of Reference page 68 3.5.1 The level of application page 68 3.5.2 Goals type of difficulty page 69 3.5.3 Atomic parts page 69 3.5.4 The language of description page 69 4 A Definition of Complexity page 72 4.1 Aspects of the Definition page 75 4.1.1 Identity of a system page 75 4.1.2 Atomic components page 76 4.1.3 Difficulty page 77 4.1.4 Formulating overall behaviour page 78 4.1.5 Complexity vs. ignorance page 79 4.1.6 As a gap between the global and local page 80 4.1.7 The comparative nature of complexity page 80 4.1.8 The existence of complexity page 81 4.1.9 Relativisation to a language page 81 4.2 Examples page 81 4.2.1 The flight behaviour of a herd page 82 4.2.2 Cellular automata page 82 4.3 Relationship to Some Other Formulations page 83 4.3.1 Number of inequivalent descriptions page 83 4.3.2 Effective measure complexity page 84 4.3.3 Computational complexity page 84 4.3.4 Algorithmic information complexity page 84 4.3.5 Shannon entropy page 85 4.3.6 Crutchfield’s “topological complexity” page 85 5 Applications of Complexity to Formal Languages page 86 5.1 Types of Complexity Involving Formal Languages page 86 5.2 Expected Properties of “Analytic Complexity” page 87 Syntactic Measures of Complexity page 5 5.2.1 Independent of the particular symbols used page 87 5.2.2 The complexity of sub-expressions should be less than the whole page 87 5.2.3 Expressions with no repetitions are simple page 88 5.2.4 Small size should limit the possible complexity page 89 5.2.5 There should be no upper limit to complexity if the language is suitably generative page 89 5.2.6 The complexity of irrelevant substitutions page 90 5.2.7 The complexity of relevant relating of expressions page 91 5.2.8 Decomposability of expressions page 91 5.3 Measures of Analytic Complexity page 93 5.3.1 Notation page 93 5.3.2 Weak complexity measures page 95 5.3.3 Weak complexity measures where simple repetition does not increase complexity page 100 5.3.4 Weak complexity measures that respect the subformula relation and where simple repetition does not increase complexity page 102 5.3.5 Strong complexity measures page 104 5.4 The Cyclomatic Number as a Measure of Analytic Complexity page 106 5.5 Layers of Syntax and Complexity page 108 5.5.1 Example 1 a supply of variable names page 109 5.5.2 Example 2 WFFs of the implication fragment page 110 5.5.3 Example 3 The implicational fragment of E page 111 5.5.4 Discussion of syntactic structures page 112 5.6 Application to Axioms and Proof Systems page 113 5.6.1 Axiom complexity page 113 5.6.2 Proof complexity page 117 5.7 Application to Simplification page 120 5.7.1 Searching over equivalent expressions within a language page 120 5.7.2 Searching over equivalent derivations within a language page 121 5.7.3 Specialising the syntactic level page 121 Syntactic Measures of Complexity page 6 5.7.4 Searching over equivalent languages page 123 5.7.5 Simplification via trade-offs with specificity and accuracy page 124 6 Philosophical Applications page 126 6.1 Complexity and Relevance page 126 6.2 Complexity and Emergence page 126 6.3 Complexity and Language page 128 6.4 Complexity and Representation page 128 6.5 Complexity and “Simplicity” page 129 6.6 Complexity and Evolution page 130 6.7 Complexity and Holism page 131 6.8 Complexity and System Identity page 132 6.9 Complexity and Society page 133 7 Conclusion page 134 7.1 Further Work page 134 8 Appendix 1 A Brief Overview of Some Existing Formulations of Complexity page 136 8.1 Abstract Computational Complexity page 136 8.2 Algorithmic Information Complexity page 136 8.3 Arithmetic Complexity page 138 8.4 Bennett's ‘Logical Depth’ page 138 8.5 Cognitive Complexity page 139 8.6 Connectivity page 140 8.7 Cyclomatic Number page 140 8.8 Descriptive/Interpretative Complexity page 141 8.9 Dimension of Attractor page 141 8.10 Ease of Decomposition page 142 8.11 Economic Complexity page 142 Syntactic Measures of Complexity page 7 8.12 Entropy page 143 8.13 Goodman's Complexity page 143 8.14 Horn Complexity page 143 8.15 Information page 144 8.16 Information Gain in Hierarchically Approximation and Scaling page 145 8.17 Irreducibility page 145 8.18 Kemeny's Complexity page 146 8.19 Length of Proof page 146 8.20 Logical Complexity/Arithmetic Hierarchy page 146 8.21 Loop Complexity page 147 8.22 Low Probability page 148 8.23 Minimum Number of Sub Groups page 148 8.24 Minimum Size page 149 8.25 Mutual Information page 151 8.26 Network Complexity page 151 8.27 Number of Axioms page 152 8.28 Number of Dimensions page 152 8.29 Number of Inequivalent Descriptions page 152 8.30 Number of Internal Relations page 153 8.31 Number of Spanning Trees page 153 8.32 Number of States in a Finite Automata page 153 8.33 Number of Symbols page 154 8.34 Number of Variables page 155 8.35 Organised/Disorganised Complexity page 155 8.36 Shannon Information page 156 8.37 Simplicity page 156 8.38 Size page 157 Syntactic Measures of Complexity page 8 8.39 Size of Grammar page 157 8.40 Size of matrix page 158 8.41 Sober's Minimum Extra Information page 158 8.42 Sophistication page 159 8.43 Stochastic Complexity page 160 8.44 Syntactic Depth page 160 8.45 Tabular Complexity page 161 8.46 Thermodynamic Depth page 161 8.47 Time and Space Computational Complexity page 162 8.48 Variety page 163 9 Appendix 2 Longer Proofs page 164 9.1 (Non-existence of) Complexity Measures on Strings page 164 9.2 Cyclomatic Number as a Lower Bound for Minimal Damage Cut page 169 9.3 Decomposition of Formulas into Complexes page 169 9.4 Generating a Measure from a Function on the Complexes page 170 9.5 Three Conditions that are Equivalent on a Weak Complexity Measure page 175 10 Appendix 3 Formalisation of Syntactic Structure page 182 10.1 Formalisation page 182 10.1.1 The syntax of trees page 182 10.1.2 The syntax of rules page 182 10.1.3 The syntax of syntactic structures: page 182 10.1.4 Generation from syntactic structures page 183 10.1.5 Production from trees page 183 10.1.6 Complete production page 184 10.1.7 Complete productive generation from syntactic structures page 184 10.2 The Expressivity of Syntactic Structures page 184 10.3 Flattening Syntactic Structures page 187 Syntactic Measures of Complexity page 9 11 Appendix 4 A tool for exploring syntactic structures, complexity and simplification page 188 11.1 Overview page 188 11.2 Examples page 188 11.3 Programming page 190 11.4 Interface page 192 12 Appendix 5 A comparison of different rankings of logical formula page 193 13 Appendix 6 Complexity and Scientific Modelling page 199 Overview page 199 Complexity page 200 A Framework for Analysing Modelling page 201 Other Formulations of Complexity page 202 Order and Disorder page 203 Noise page 206 Complexity vs. Information page 207 Complexity and Induction page 207 Conclusion page 208 14 Appendix 7 Complexity and Economics page 210 What is Complexity? page 210 “Complexity” in economics page 210 The “Sciences of Complexity” page 210 Complexityper se page 211 The effects of complexity on modelling by agents page 213 Ideal rationality and perfect information page 213 Ideal rationality and noisy information page 213 Ideal rationality and inadequate information page 214 Bounded rationality and inadequate information page 214 The effects of modelling by agents on complexity page 215 Syntactic Measures of Complexity page 10 Ignoring the process of modelling by economic agents page 215 Including the process of modelling by economic agents page 215 Towards dealing with the complexity of modelling agents modelling modelling page 217 The Form meaning distinction page 217 The complexity, specificity, error trade-off page 218 The modelling language page 219 Processes of model development page 219 Some future directions for economic modelling page 220 Applying our model of modelling to ourselves page 220 Relatively new (non-numerical) techniques page 221 Conclusion complexity again page 222 15 References page 223 Syntactic Measures of Complexity page 11 List of Figures Figure 1: Entity, W, using A as a model of B page 24 Figure 2: An illustration of the syntactic view of models page 28 Figure 3: A semantic picture of modelling (from Giere in [176]) page 29 Figure 4: T

Syntactic Measures of Complexity

Two-level logic minimization: an overview

Logical inference is of central importance in the information and decision sciences but presents a very hard computational problem. Since the traditional symbolic inference methods have had limited success on large knowledge bases, this papers investigates a quantitative approach. It surveys the application of integer programming methods to inference problems in propositional logic. It displays a number of remarkable parallels between logic and mathematics and shows that these can lead to fast inference methods, both quantitative and symbolic. In particular it explains why the logical concepts of resolution, extended resolution, input and unit refutation, the Davis-Putnam procedure, and drawing of inferences pertinent to a given topic are closely related to the mathematical concepts of cutting planes, Chvatal's method, elementary closure, branch and bound, and projection of a polytope, respectively. Much of the paper should be intelligible to persons with limited background in logic and mathematical programming, but recent mathematical results are stated precisely.

A quantitative approach to logical inference

Identifying control strategies for biological networks is paramount for practical applications that involve reprogramming a cell’s fate, such as disease therapeutics and stem cell reprogramming. Here we develop a novel network control framework that integrates the structural and functional information available for intracellular networks to predict control targets. Formulated in a logical dynamic scheme, our approach drives any initial state to the target state with 100% effectiveness and needs to be applied only transiently for the network to reach and stay in the desired state. We illustrate our method’s potential to find intervention targets for cancer treatment and cell differentiation by applying it to a leukemia signaling network and to the network controlling the differentiation of helper T cells. We find that the predicted control targets are effective in a broad dynamic framework. Moreover, several of the predicted interventions are supported by experiments.

/pdf/cell-fate-reprogramming-by-control-of-intracellular-network-lla47my49a.pdf

Cell fate reprogramming by control of intracellular network dynamics

/pdf/cell-fate-reprogramming-by-control-of-intracellular-network-xn1eetyp2d.pdf

Cell Fate Reprogramming by Control of Intracellular Network Dynamics

We show that any Boolean function f which can be expressed in a sum-of-products form using m product terms can contain as many as 2m− 1 implicants but no more.

McMullen

Papers

Prime Implicants, Minimum Covers, and the Complexity of Logic Simplification