# Nontrivial Turmites are Turing-universal

## TL;DR

This paper proves that most Turmites, a type of two-dimensional Turing machine, are capable of universal computation, and establishes the computational complexity of predicting their behavior.

## Contribution

It demonstrates that all Turmites with symbol-dependent rules are Turing-universal and proves the P-completeness of predicting their future states.

## Key findings

- Most Turmites are Turing-universal.
- Prediction of Turmite behavior is P-complete.
- The results generalize previous findings on Langtons ant.

## Abstract

A Turmit is a Turing machine that works over a two-dimensional grid, that is, an agent that moves, reads and writes symbols over the cells of the grid. Its state is an arrow and, depending on the symbol that it reads, it turns to the left or to the right, switching the symbol at the same time. Several symbols are admitted, and the rule is specified by the turning sense that the machine has over each symbol. Turmites are a generalization of Langtons ant, and they present very complex and diverse behaviors. We prove that any Turmite, except for those whose rule does not depend on the symbol, can simulate any Turing Machine. We also prove the P-completeness of prediction their future behavior by explicitly giving a log-space reduction from the Topological Circuit Value Problem. A similar result was already established for Langtons ant; here we use a similar technique but prove a stronger notion of simulation, and for a more general family.