Computational Irreducibility and Computational Analogy

In a previous paper [1], we provided a formal definition for the concept of computational irreducibility (CIR), that is, the fact that for a function f from N to N it is impossible to compute f (n) without following approximately the same path as computing successively all the values f (i) from i = 1 to n. Our definition is based on the concept of enumerating Turing machines (E-Turing machines) and on the concept of approximation of E-Turing machines, for which we also gave a formal definition. Here, we make these definitions more precise through some modifications intended to improve the robustness of the concept. We then introduce a new concept: the computational analogy, and prove some properties of the functions used. Computational analogy is an equivalence relation that allows partitioning the set of computable functions in classes whose members have the same properties regarding their CIR and their computational complexity. Introduction 1

Generalized Toric Codes Coupled to Thermal Baths

We have studied the dynamics of a generalized toric code based on qudits at finite temperature by finding the master equation coupling the code's degrees of freedom to a thermal bath. As a consequence, we find that for qutrits new types of anyons and thermal processes appear that are forbidden for qubits. These include creation, annihilation and diffusion throughout the system code. It is possible to solve the master equation in a short-time regime and find expressions for the decay rates as a function of the dimension $d$ of the qudits. Although we provide an explicit proof that the system relax to the Gibbs state for arbitrary qudits, we also prove that above a certain crossing temperature, qutrits initial decay rate is smaller than the original case for qubits. Surprisingly this behavior only happens with qutrits and not with other qudits with $d>3$.Comment: Revtex4 file, color figures. New Journal of Physics' versio

The limits to prediction in ecological systems

Predicting the future trajectories of ecological systems is increasingly important as the magnitude of anthropogenic perturbation of the earth systems grows.We distinguish between two types of predictability: the intrinsic or theoretical predictability of a system and the realized predictability that is achieved using available models and parameterizations. We contend that there are strong limits on the intrinsic predictability of ecological systems that arise from inherent characteristics of biological systems. While the realized predictability of ecological systems can be limited by process and parameter misspecification or uncertainty, we argue that the intrinsic predictability of ecological systems is widely and strongly limited by computational irreducibility. When realized predictability is low relative to intrinsic predictability, prediction can be improved through improved model structure or specification of parameters. Computational irreducibility, however, asserts that future states of the system cannot be derived except through computation of all of the intervening states, imposing a strong limit on the intrinsic or theoretical predictability. We argue that ecological systems are likely to be computationally irreducible because of the difficulty of pre-stating the relevant features of ecological niches, the complexity of ecological systems and because the biosphere can enable its own novel system states or adjacent possible. We argue that computational irreducibility is likely to be pervasive and to impose strong limits on the potential for prediction in ecology. Copyright