Hardness of Approximation for Langton’s Ant on a Twisted Torus

Takeo Hagiwara,Tatsuie Tsukiji

doi:10.3390/a13120344

Abstract

Langton’s ant is a deterministic cellular automaton studied in many fields, artificial life, computational complexity, cryptography, emergent dynamics, Lorents lattice gas, and so forth, motivated by the hardness of predicting the ant’s macroscopic behavior from an initial microscopic configuration. Gajardo, Moreira, and Goles (2002) proved that Langton’s ant is PTIME -hard for reachability. On a twisted torus, we demonstrate that it is PSPACE hard to determine whether the ant will ever visit almost all vertices or nearly none of them.

Highlights

In 1986, Chris Langton proposed an artificial life on square lattice Z × Z [1,2,3,4,5] where each vertex is either to-right or to-left
This paper aims to strengthen GMG in both computational complexity and highway conjecture
An h-twist (h/N-pitch) of the torus is φ( x, y) = φ( x + N, y) = φ( x + h, y + N ). It appears in related models, for example, square-lattice Ising models (Langton’s ant is a typical one) [26,27]

Summary

Introduction

In 1986, Chris Langton proposed an artificial life on square lattice Z × Z [1,2,3,4,5] where each vertex is either to-right (white) or to-left (black). Suppose that Langton’s ant is a particle system where a finite initial configuration and its consequent macroscopic quantities are the only observables. Macroscopic observables are fixed constants regardless of the initial state In any case, they must have some calculation from the initial finite configuration since the system is deterministic. An h-twist (h/N-pitch) of the torus is φ( x, y) = φ( x + N, y) = φ( x + h, y + N ) It appears in related models, for example, square-lattice Ising models (Langton’s ant is a typical one) [26,27]. No computer-aided analysis can calculate Langton’s ant’s macroscopic quantities from an initial finite configuration in polynomial time, unless P = PSPACE. Theorem 2 indicates that these currently known methods cannot measure them for square lattice

A Proof Outline

A Hamiltonian-tour gate to visit almost all vertices is easy to realize

Lattice Embedding

Hardness of Approximation for Langton’s Ant

Conclusions