The Computable Universe Hypothesis

When can a model of a physical system be regarded as computable? We provide the definition of a computable physical model to answer this question. The connection between our definition and Kreisel's notion of a mechanistic theory is discussed, and several examples of computable physical models are given, including models which feature discrete motion, a model which features non-discrete continuous motion, and probabilistic models such as radioactive decay. We show how computable physical models on effective topological spaces can be formulated using the theory of type-two effectivity (TTE). Various common operations on computable physical models are described, such as the operation of coarse-graining and the formation of statistical ensembles. The definition of a computable physical model also allows for a precise formalization of the computable universe hypothesis--the claim that all the laws of physics are computable.Comment: 33 pages, 0 figures; minor change

Facticity as the amount of self-descriptive information in a data set

Using the theory of Kolmogorov complexity the notion of facticity {\phi}(x) of a string is defined as the amount of self-descriptive information it contains. It is proved that (under reasonable assumptions: the existence of an empty machine and the availability of a faithful index) facticity is definite, i.e. random strings have facticity 0 and for compressible strings 0 < {\phi}(x) < 1/2 |x| + O(1). Consequently facticity measures the tension in a data set between structural and ad-hoc information objectively. For binary strings there is a so-called facticity threshold that is dependent on their entropy. Strings with facticty above this threshold have no optimal stochastic model and are essentially computational. The shape of the facticty versus entropy plot coincides with the well-known sawtooth curves observed in complex systems. The notion of factic processes is discussed. This approach overcomes problems with earlier proposals to use two-part code to define the meaningfulness or usefulness of a data set.Comment: 10 pages, 2 figure

Embeddings and immersions of tropical curves

We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio