Wolfram Computation Meets Knowledge

Wolfram Summer School


Joscha Bach

Science and Technology

Class of 2016


Joscha Bach, PhD, is a research scientist at the MIT Media Lab and an affiliate at the Harvard Program for Evolutionary Dynamics. He has mostly been working in cognitive science, trying to understand the functional structure of our minds, and is the author of the cognitive architecture MicroPsi. He currently lives in Cambridge, Massachusetts.

Project: Hypergraph Visualization for Understanding Entanglement

Using hypergraphs with rewriting rules seems to be a promising and general paradigm to think about a possible lowest level for digital physics. Hypergraphs are a generalization of graphs where links are not properties of pairs of nodes, but of 1 to n nodes. Think of links as states that are shared between locations. While locations are stateless, links may possibly hold (discrete) values/have types.

Inhabitants of the graph cannot directly observe the values of the links; they can only observe differentials. To perceive differentials, there needs to be a computational process that makes these differentials functionally relevant, i.e. gives rise to state change based on an existing differential. This state change can possibly be a change of the values of a link (for instance, whenever two links of the same type meet in a location but have different values, the transition function of the universe tries to equalize them), or it can change the linkage of the graph, or both. As we will see, the latter seems mandatory for our universe and sufficient to model the former effect.

The equalization of differentials will usually not make the differential disappear. Instead, it will travel through the graph. A particle may be seen as a set of traveling differentials in the graph.

Locations do not have coordinates. Through the links, they have distances from each other (the number of links one has to traverse to get from one location to another), and as a result, an inhabitant of the graph can project the graph into a (virtual) space. Let us assume that all links have the same “length”, i.e. it takes the same number of steps (1) to change their state or change the state of other links based on their state.

Spacetime is the set of possible causal evolutions from the point of an observer, i.e. the set of all trajectories in which information can travel through the graph. In the case of our universe, these trajectories can be assembled into a (curved) 3-space with a temporal dimension. Why is space three-dimensional?

A possible solution (similar to loop quantum gravity) may assume that the links form a regular lattice with only local and sparse connectivity—for instance, an orthogonal or tetrahedron packing that forms a 3-space. However, this does not seem to be compatible with Lorentz invariance (information can travel in all directions at roughly the same speed) or with spacetime curvature. Nonlocality suggests that it is possible that links can be nonlocal, but subsequent operations on these links tend to obliterate the nonlocality. Thus, the locality of the connectivity possibly results from the transition function itself. Lorentz invariance, however, suggests a dense local connectivity, or stochastic linking that averages out over large distances. It is also conceivable that particles have properties similar to gliders in the Game of Life, and tend to break down in higher dimensions, i.e. only use a three-dimensional subset of the graph structure. High-dimensional particles might be unstable and break down after traversing a few Planck lengths, not unlike the strong force and the weak force.

Spacetime curvature suggests that particles change the effective topology around them, i.e. distort the way information travels around them. Also, gravity has no limit of action, i.e. information traveling from a massive particle creates a permanent gradient.

Favorite 3-Color 2D Totalistic Cellular Automaton

Rule 599071137