Hypergraph Rewriting and the Wolfram Model

Abstract

This talk will summarize some recent results obtained in connection with the Wolfram Physics Project: a systematic attempt to reduce fundamental physics to the problem of hypergraph rewriting. Specifically, we will discuss the construction of an efficient hypergraph rewriting scheme using the formalism of Wolfram model multiway systems, with a categorical semantics defined via double-pushout rewrites over selective adhesive categories, and equipped with a dagger-symmetric monoidal structure that exhibits surprising formal connections to categorical quantum mechanics. We will also demonstrate that, by equipping such a multiway system with an additional notion of causal structure (defined via the semantics of precausal categories), we are able to introduce new inference rules of selective resolution, paramodulation and factoring that generalize the standard Knuth-Bendix completion rules for term rewriting systems, and significantly improve the efficiency of automated diagrammatic reasoning algorithms over hypergraphs. Time permitting, we will discuss some near-term practical applications of these methods for quantum information theory (allowing for more efficient diagrammatic simplification of quantum circuits), as well as numerical general relativity (allowing for more explicit numerical simulation of graph-theoretic models of space and time).