Graph Rewrite Systems for Classical Structures in †-Symmetric Monoidal Categories

This paper introduces several related graph rewrite systems derived from known identities on classical structures within a †-symmetric monoidal category. First, we develop a rewrite system based on a single classical structure, and use it to develop a proof of the so-called “spider-theorem,” where a connected graph containing a single classical structure can be uniquely determined by the number of inputs and outputs (i.e. it can be rewritten as a graph resembling an n-legged spider). These spiders are shown to be the normal forms of graphs containing a single classical structure. Next, complementary classical structures are introduced, as well as a new rewrite system on graphs of red and green spiders. A proof of convergence is given for a limited two-colour rewrite system, as well as insights into ways to approach normalisation in a more powerful rewrite system.