## Preprint 2016-19

## Anahi Gajardo, Benjamin Hellouin, Diego Maldonado, Andres Moreira:

### Nontrivial turmites are Turing-universal

### 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 Langton ant, and they present very complex and diverse behaviours. 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 Langton ant; here we use a similar technique but prove a stronger notion of simulation, and for a more general family.

This preprint gave rise to the following definitive publication(s):

**Anahi GAJARDO, Benjamin HELLOUIN, Diego MALDONADO, Andres MOREIRA: ***Nontrivial turmites are Turing-universal*. Journal of Cellular Automata, vol. 13, 5-6, pp. 373-392, (2018).