Classical spin Hamiltonians are context-sensitive languages

TL;DR

This paper classifies classical spin Hamiltonians as formal languages within the Chomsky hierarchy, revealing their complexity levels and implications for recognizing configurations and energies, and compares these with computational complexity results.

Contribution

It introduces a novel classification of spin Hamiltonians as formal languages, linking physical models to formal language theory and complexity hierarchies.

Findings

Zero-dimensional Hamiltonians are regular languages.

One-dimensional Hamiltonians are deterministic context-free.

Higher-dimensional Hamiltonians are context-sensitive.

Abstract

Classical spin Hamiltonians are a powerful tool to model complex systems, characterised by a local structure given by the local Hamiltonians. One of the best understood local structures is the grammar of formal languages, which are central in computer science and linguistics, and have a natural complexity measure given by the Chomsky hierarchy. If we see classical spin Hamiltonians as languages, what grammar do the local Hamiltonians correspond to? Here we cast classical spin Hamiltonians as formal languages, and classify them in the Chomsky hierarchy. We prove that the language of (effectively) zero-dimensional spin Hamiltonians is regular, one-dimensional spin Hamiltonians is deterministic context-free, and higher-dimensional and all-to-all spin Hamiltonians is context-sensitive. This provides a new complexity measure for classical spin Hamiltonians, which captures the hardness of recognising spin configurations and their energies. We compare it to the computational complexity of the ground state energy problem, and find a different easy-to-hard threshold for the Ising model. We also investigate the dependence on the language of the spin Hamiltonian. Finally, we define the language of the time evolution of a spin Hamiltonian and classify it in the Chomsky hierarchy. Our work suggests that universal spin models are weaker than universal Turing machines.