The Lattice of Chemical Organisations Pietro Speroni di Fenizio 1 1 Bio Systems Analysis Group, Jena Centre for Bioinformatics, Friedrich Schiller-University Jena, D-07737 Jena ecal2015@piespe.net Abstract The paper describes how, in an Artificial Chemistry under flow conditions, the set of organisations form a lattice. The consequences of this are described, in particular how a series of theorems, valid for lattices, can be applied to more easily discover the complete set of organisations. An algorithm is then developed that uses such theorems to explore such lattice. The algorithm is applied first to the NTop Artificial Chemistry and then to an extension of it. Due to its complexity this system is also suggested as a benchmark case to test new Artificial Chemistries’ algorithms. Introduction In 1994 Walter Fontana and Leo Buss introduced the concept of Organisation to represent the fixed points in Constructive Dynamical Systems (Fontana and Buss, 1994a, 1994b, 1996). Looking at Dynamical Systems they observed how they could predict only quantitative dynamics, and no qualitative development. True novelty could never appear through Ordinary Differential Equations. But novelty is inherent in this world. And it is unpredictability is what makes the world so interesting. So, to describe how a system could transform the qualitative space of possibilities, they presented the concept of Constructive Dynamical Systems. Those were a molecular based system, where new molecules could be generated through the interaction of existing ones. Such system was thought as a model for the macro-molecules inside a cell, but this was just one of the possible examples, of such a general theory. Using standard dynamical systems as a metaphor, they then went on describing what would be the equivalent fixed points in their system; points where no novelty would appear, and called such cases Organisations. A set of molecules would be an organisation if, and only if, for each molecule inside the set, there was a reaction among the molecules in the set that would produce it; and, given any reaction among the molecules of the set, the result would always be a molecule of the set. The first property was called Self-Maintenance and the second Closure. A set satisfying both of those properties would be called an Organisation. Later (Dittrich, Speroni di Fenizio, 2007) Organisations were then renamed Semi-Organisations when it became clear that those properties were not enough to permit to those sets to be dynamically stability. Organisations were so re-defined as special Semi-Organisations where it is possible for all the reactions among molecules inside them to be active, and have no molecule type diminish. Studying Organisations and Semi-Organisations, it was soon found that under quite common assumptions, those structures would form a Lattice. It should be noted that this it is not always true for every reaction system, but it is true if we assume that every molecule has a certain probability to disappear (i.e. every molecule would have an out-flux greater than zero). Those systems were every molecule has an out flux were called Flow Systems to distinguish them from the more general General Reaction Systems (Dittrich, Speroni, 2007). It should be noted how all of Fontana and Buss’ models had an out-flux applied to each molecule that would destroy it at the same speed at which others were produced. So those models were Flow Systems, and the set of organisations would form a Lattice. Other Flow Systems are also possible; for example if we consider cells, and thus cell growth, although it is not true that each molecule has a certain probability higher than zero of being excreted, it is true that as the cell grows the relative concentration of each molecule that is not generates diminishes, having the same effective result as if each molecule was subject to an out-flux. Then when the cell reproduces the average amount of that molecule halves, eventually reaching zero after enough reproductions. Thus the set of molecules inside a cell form a Flow System, and the set of possible Organisations in a living system form a lattice. Vice-versa, the reaction system in the atmosphere of a planet is not a Flow System, but a General Reaction System, and the Organisations do not form a lattice but just a Partially Ordered Set. Flow Systems and General Reaction Systems are not the only possible type of Artificial Chemistries: a more detailed analysis can distinguish also Catalytic Flow Systems as a specific type of Flow System where each molecule reacts in a catalytic way. In other words they are not used up in the Catalytic Flow Systems Flow Systems General Reaction Systems All molecules in each reaction are catalytic; all molecules have an out flux All molecules have an out-flux No requirements on the out-flux nor on which molecules are catalytic each semi- organisation is an organisations some semi- organisations are organisations some semi- organisations are organisations both organisations and semi-organisations form a lattice both organisations and semi-organisations form a lattice neither organisations nor semi-organisations form a lattice Table 1: Different type of Artificial Chemistries produce sets of organisations with different properties. Pietro Speroni di Fenizio (2015) The Lattice of Chemical Organisations. Proceedings of the European Conference on Artificial Life 2015, pp. 242-248 DOI: http://dx.doi.org/10.7551/978-0-262-33027-5-ch048