Algebraic approaches to artificial chemistries Penelope Selina Margaret Faulkner Rainford PhD University of York Chemistry December 2018 Abstract We have developed a new systematic framework, MetaChemisty for the description of ar- tificial chemistries (AChems). It encompasses existing systems. It has the flexibility and complexity to allow for new features and new systems. A joint description language will allow comparisons to be drawn between systems. This will allow us to write metrics and benchmarks for artificial chemistries. It also enables us to combine existing systems in dif- ferent ways to give a wealth of more complex and varied systems. We will be able to build novel chemistries quicker through reuse of code and features between chemistries allowing new chemistries to start from a more complex base line. We have also developed an algebraic artificial chemistry, Jordan Algebra Artificial Chem- istry (JA AChem). This chemistry is based on existing algebra which is leverage to ensure features such as isomers and isotopes are possible in our system. The existence of isotopes leads naturally to the existence of elements for this chemistry. It is a chemistry with both constructive and destructive reactions making it a good candidate for further study as an open-ended system. We analyze the effect of changing probabilistic processes in JA AChem by modifying the probability spawning functions that control them. We also look at the algebraic properties of these probability spawning functions. We have described Swarm Chemistry, Sayama (2009), in the MetaChem showing it is at least more expressive than the previous framework for artificial chemistries, Dittrich et al. (2001). We use the framework to combine two artificial chemistries using a simple environment link structure to produce eight new modular AChems with a modular approach. This link structure requires minimal addition to existing code for artificial chemistry systems and no modification to most modules. 2 Contents Abstract 2 List of Tables 8 List of Figures 9 Acknowledgments 12 Declaration 13 1 Introduction 14 1.1 Artificial life and transitions . 14 1.1.1 Different scale perspectives on artificial life . 15 1.2 Artificial Chemistry . 15 1.2.1 Different artificial chemistries . 15 1.2.2 Issues with (S; R; A) description . 16 1.3 Rigorous Design . 17 1.3.1 Challenges of designing a framework . 17 1.3.2 Why algebraic structures? . 17 1.3.3 Using but not being constrained by algebra . 17 1.4 Structure of thesis . 18 2 Literature Review 19 2.1 Artificial Life . 19 2.1.1 Origins of Life . 19 2.1.2 Open ended evolution . 20 2.2 Artificial Chemistries . 21 2.2.1 Subsymbolic artificial chemistries . 21 2.2.2 AChems for behaviours . 22 2.2.3 AChems for modelling . 22 2.2.4 AChems for computation . 23 2.2.5 AChems for Open Ended Evolution . 23 3 CONTENTS 4 2.2.6 Problem with (S,R,A) format for comparison of systems . 23 2.3 Applications of Algebraic structures to real world and computational problems 24 3 Research Hypothesis 25 3.1 Expressing Artificial Chemistries in a single logical framework . 25 3.2 Rational Design . 25 3.2.1 Framework . 25 3.2.2 Artificial Chemistry . 25 3.3 Combining existing artificial chemistries to increase complexity . 26 4 MetaChemisty 27 4.1 Modularisation: Components of an Artificial Chemistry System . 28 4.1.1 Particles . 29 4.1.2 Container Nodes . 30 4.1.3 Action Nodes . 31 4.1.4 Admin Nodes . 31 4.1.5 Edges . 32 4.1.6 Graph as an Executable Algorithm . 32 4.2 Overview of descriptive levels . 33 4.2.1 Macro Level . 33 4.2.2 Micro Level . 34 4.2.3 Physics Level . 35 4.2.4 Abstraction levels . 36 4.3 Formalisation of MetaChem . 36 4.3.1 Static Graph MetaChem . 36 4.3.2 Dynamic System State . 37 4.3.3 Transition Functions . 38 4.3.4 Examples from StringCatChem . 42 4.4 Static and Grown Graphs/Graph Language . 44 4.5 Next Steps . 45 5 Jordan Algebra Artificial Chemistry 46 5.1 Introduction . 46 5.2 Open-Endedness through rigorous design . 46 5.3 Assessment of Desirable Mathematical Properties of Link for correct structure of Composite Particles . 47 5.3.1 Desirable Properties for Particles . 47 5.3.2 Algebraic Axioms . 48 5.3.3 Expanding to n-ary linking operator . 50 5.3.4 Jordan Algebras . 50 5.4 Particles and their subsymbolic structure . 53 CONTENTS 5 5.4.1 Hermitian Matrices as Subsymbolic Particles . 53 5.4.2 Atoms . 54 5.5 Macro Level Description of Jordan Algebra AChem . 54 5.6 Linking Process . 55 5.6.1 read() . 55 5.6.2 check() . 56 5.6.3 pull() . 57 5.6.4 process() . 57 5.6.5 push() . 57 5.6.6 Subsymbolic Link . 57 5.6.7 Elemental Table . 58 5.6.8 Structure . 59 5.7 Decomposition Process . 61 5.7.1 read() . 61 5.7.2 check() . 61 5.7.3 pull() . 63 5.7.4 process() . 63 5.7.5 push() . 63 5.8 Resolving \hanging" link . 63 5.8.1 Example . 68 5.9 Conclusion . 69 6 Modifying AChems with PSFs 70 6.1 Probabilistic AChems . 70 6.2 Jordan Algebra AChems . 71 6.2.1 Probability Spawning Functions Options . 71 6.2.2 Psfs in linking in Jordan Algebra AChem . 73 6.2.3 Psfs in decomposition in Jordan Algebra AChem . 74 6.3 Effects of psfs on JA AChem . 75 6.3.1 Known effect: Speed and Area . 75 6.3.2 Looking for unknown effects . 75 6.4 Variations of functions for tuning and testing effects . 78 6.4.1 Largest Link . 79 6.4.2 Strength . 79 6.4.3 Self-synthesis . 80 6.5 Combinations of psfs . 81 6.5.1 Semi-rings . 81 6.5.2 Semi-ring of probability spawning functions . 82 6.5.3 Example . 83 6.5.4 Combinations in JA AChem . 85 CONTENTS 6 6.6 Summary . 88 6.7 Further Work . 89 7 Swarm Chemistry 90 7.1 Swarm Chemistry . 92 7.2 Description . 93 7.3 Flocking . 94 7.3.1 read() . 95 7.3.2 check() . 95 7.3.3 pull() . ..