Conservation Laws in Cellular Automata

Conservation laws in physics are numerical invariants of the dynamics of a system. In cellular automata (CA), a similar concept has already been defined and studied. To each local pattern of cell states a real value is associated, interpreted as the “energy” (or “mass”, or . . . ) of that pattern.The overall “energy” of a configuration is simply the sum of the energy of the local patterns appearing on different positions in the configuration. We have a conservation law for that energy, if the total energy of each configuration remains constant during the evolution of the CA. For a given conservation law, it is desirable to find microscopic explanations for the dynamics of the conserved energy in terms of flows of energy from one region toward another. Often, it happens that the energy values are from non-negative integers, and are interpreted as the number of “particles” distributed on a configuration. In such cases, it is conjectured that one can always provide a microscopic explanation for the conservation laws by prescribing rules for the local movement of the particles. The onedimensional case has already been solved by Fuk´s and Pivato. We extend this to two-dimensional cellular automata with radius-0,5 neighborhood on the square lattice. We then consider conservation laws in which the energy values are chosen from a commutative group or semigroup. In this case, the class of all conservation laws for a CA form a partially ordered hierarchy. We study the structure of this hierarchy and prove some basic facts about it. Although the local properties of this hierarchy (at least in the group-valued case) are tractable, its global properties turn out to be algorithmically inaccessible. In particular, we prove that it is undecidable whether this hierarchy is trivial (i.e., if the CA has any non-trivial conservation law at all) or unbounded. We point out some interconnections between the structure of this hierarchy and the dynamical properties of the CA. We show that positively expansive CA do not have non-trivial conservation laws. We also investigate a curious relationship between conservation laws and invariant Gibbs measures in reversible and surjective CA. Gibbs measures are known to coincide with the equilibrium states of a lattice system defined in terms of a Hamiltonian. For reversible cellular automata, each conserved quantity may play the role of a Hamiltonian, and provides a Gibbs measure (or a set of Gibbs measures, in case of phase multiplicity) that is invariant. Conversely, every invariant Gibbs measure provides a conservation law for the CA. For surjective CA, the former statement also follows (in a slightly different form) from the variational characterization of the Gibbs measures. For one-dimensional surjective CA, we show that each invariant Gibbs measure provides a conservation law. We also prove that surjective CA almost surely preserve the average information content per cell with respect to any probability measure.

The European Union and Assembling Biofuel Development – topological investigations concerning the associations between law, policy and space

Biofuels for transport are a renewable source of energy that were once heralded as a solution to multiple problems associated with poor urban air quality, the overproduction of agricultural commodities, the energy security of the European Union (EU) and climate change. It was only after the Union had implemented an incentivizing framework of legal and political instruments for the production, trade and consumption of biofuels that the problems of weakening food security, environmental degradation and increasing greenhouse gases through land-use changes began to unfold. In other words, the difference between political aims for why biofuels are promoted and their consequences has grown – which is also recognized by the EU policy-makers. Therefore, the global networks of producing, trading and consuming biofuels may face a complete restructure if the European Commission accomplishes its pursuit to sideline crop-based biofuels after 2020. My aim with this dissertation is not only to trace the manifold evolutions of the instruments used by the Union to govern biofuels but also to reveal how this evolution has influenced the dynamics of biofuel development. Therefore, I study the ways the EU’s legal and political instruments of steering biofuels are coconstitutive with the globalized spaces of biofuel development. My analytical strategy can be outlined through three concepts. I use the term ‘assemblage’ to approach the operations of the loose entity of actors and non-human elements that are the constituents of multi-scalar and -sectorial biofuel development. ‘Topology’ refers to the spatiality of this European biofuel assemblage and its parts whose evolving relations are treated as the active constituents of space, instead of simply being located in space. I apply the concept of ‘nomosphere’ to characterize the framework of policies, laws and other instruments that the EU applies and construes while attempting to govern biofuels. Even though both the materials and methods vary in the independent articles, these three concepts characterize my analytical strategy that allows me to study law, policy and space associated with each other. The results of my examinations underscore the importance of the instruments of governance of the EU constituting and stabilizing the spaces of producing and, on the other hand, how topological ruptures in biofuel development have enforced the need to reform policies. This analysis maps the vast scope of actors that are influenced by the mechanism of EU biofuel governance and, what is more, shows how they are actively engaging in the Union’s institutional policy formulation. By examining the consequences of fast biofuel development that are spatially dislocated from the established spaces of producing, trading and consuming biofuels such as indirect land use changes, I unfold the processes not tackled by the instruments of the EU. Indeed, it is these spatially dislocated processes that have pushed the Commission construing a new type of governing biofuels: transferring the instruments of climate change mitigation to land-use policies. Although efficient in mitigating these dislocated consequences, these instruments have also created peculiar ontological scaffolding for governing biofuels. According to this mode of governance, the spatiality of biofuel development appears to be already determined and the agency that could dampen the negative consequences originating from land-use practices is treated as irrelevant.