Related papers: Causal Graph Dynamics and Kan Extensions
Causal graph dynamics are transformations over graphs that capture two important symmetries of physics, namely causality and homogeneity. They can be equivalently defined as continuous and translation invariant transformations or functions…
Consider a graph having quantum systems lying at each node. Suppose that the whole thing evolves in discrete time steps, according to a global, unitary causal operator. By causal we mean that information can only propagate at a bounded…
Causal Graph Dynamics generalize Cellular Automata, extending them to bounded degree, time varying graphs. The dynamics rewrite the graph at each time step with respect to two physics-like symmetries: causality (bounded speed of…
We extend the theory of Cellular Automata to arbitrary, time-varying graphs. In other words we formalize, and prove theorems about, the intuitive idea of a labelled graph which evolves in time - but under the natural constraint that…
Causal Graph Dynamics extend Cellular Automata to arbitrary, bounded-degree, time-varying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physics-like symmetries:…
A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal…
Based on a recent development in the area of error control coding, we introduce the notion of convolutional factor graphs (CFGs) as a new class of probabilistic graphical models. In this context, the conventional factor graphs are referred…
We propose Universal Causality, an overarching framework based on category theory that defines the universal property that underlies causal inference independent of the underlying representational formalism used. More formally, universal…
Based on methods of structural convergence we provide a unifying view of local-global convergence, fitting to model theory and analysis. The general approach outlined here provides a possibility to extend the theory of local-global…
Given a causal graph representing the data-generating process shared across different domains/distributions, enforcing sufficient graph-implied conditional independencies can identify domain-general (non-spurious) feature representations.…
It was recently suggested that causal structures are both dynamical, because of general relativity, and indefinite, due to quantum theory. The process matrix formalism furnishes a framework for quantum mechanics on indefinite causal…
Evolution in finite populations is often modelled using the classical Moran process. Over the last ten years this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such…
Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to a point; and the fact that they have a well-defined notion of…
In this paper, we formalize precisely the sense in which the application of cellular automaton to partial configuration is a natural extension of its local transition function through the categorical notion of Kan extension. In fact, the…
We construct a generalised formalism for group field theories, in which the domain of the field is extended to include additional proper time variables, as well as their conjugate mass variables. This formalism allows for different types of…
Recently there has been significant interest in using causal modelling techniques to understand the structure of physical theories. However, the notion of `causation' is limiting - insisting that a physical theory must involve causal…
Artificial Neural Networks (ANNs), including fully-connected networks and transformers, are highly flexible and powerful function approximators, widely applied in fields like computer vision and natural language processing. However, their…
The proposed theory of causally structured discrete fields studies integer values on directed edges of a self-similar graph with a propagation rule, which we define as a set of valid combinations of integer values and edge directions around…
In this dissertation we develop a new formal graphical framework for causal reasoning. Starting with a review of monoidal categories and their associated graphical languages, we then revisit probability theory from a categorical perspective…
We develop a framework based on the covariant phase space formalism that identifies gravitational edge modes as dynamical reference frames. They enable the identification of the associated spacetime region and the imposition of boundary…