Formal languages, spin systems, and quasicrystals

TL;DR

This paper develops a categorical framework linking formal languages to spin systems and quasicrystals, generalizing models to higher dimensions and exploring their physical properties and solvability.

Contribution

It introduces a functorial mapping from context-free and mildly context-sensitive grammars to spin systems, extending models to higher dimensions and connecting them with quasicrystals and integrable systems.

Findings

D0L-systems as special cases of the formalism

Construction of spin systems for quasicrystals and higher dimensions

Analysis of solvability, free energy, and criticality of these models

Abstract

We present a categorical formalism for context-free languages with morphisms given by correspondences obtained from rational transductions. We show that D0L-systems are a special case of the correspondences that define morphisms in this category. We construct a functorial mapping to aperiodic spin chains. We then generalize this construction to a class of mildly context sensitive grammars, the multiple-context-free grammars (MCFG), with a similar functorial mapping to spin systems in higher dimensions, with Boltzmann weights describing interacting spins on vertices of hypercubes. We show that a particular motivating example for this general construction is provided by the Korepin completely integrable model on the icosahedral quasicrystal, which we construct as the spin system associated to a multiple-context-free grammar describing the geometry of the Ammann planes quasilattice. We review the main properties of this spin system, including solvability, bulk free energy, and criticality, based on results of Baxter and the known relation to the Zamolodchikov tetrahedron equation. We show that the latter has a generalization for the Boltzmannweights on hypercubes of the spin systems associated to more general MCFGs in terms of two dual cubulations of the n-simplex. We formulate analogous questions about bulk free energy and criticality for our construction of spin systems.