朗顿蚂蚁

Langton's ant after 11,000 steps. A red pixel shows the ant's location.

Langton's ant after 11,000 steps. A red pixel shows the ant's location.

11000步后的兰顿蚂蚁。一个红色的像素显示蚂蚁的位置。

Langton's ant is a two-dimensional universal Turing machine with a very simple set of rules but complex emergent behavior. It was invented by Chris Langton in 1986 and runs on a square lattice of black and white cells.[1] The universality of Langton's ant was proven in 2000.[2] The idea has been generalized in several different ways, such as turmites which add more colors and more states.

Langton's ant is a two-dimensional universal Turing machine with a very simple set of rules but complex emergent behavior. It was invented by Chris Langton in 1986 and runs on a square lattice of black and white cells. The universality of Langton's ant was proven in 2000. when starting on a completely white grid.

Rules

Simplicity. During the first few hundred moves it creates very simple patterns which are often symmetric.

Animation of first 200 steps of Langton's ant

Chaos. After a few hundred moves, a large, irregular pattern of black and white squares appears. The ant traces a pseudo-random path until around 10,000 steps.

Emergent order. Finally the ant starts building a recurrent "highway" pattern of 104 steps that repeats indefinitely.

Squares on a plane are colored variously either black or white. We arbitrarily identify one square as the "ant". The ant can travel in any of the four cardinal directions at each step it takes. The "ant" moves according to the rules below:

• At a white square, turn 90° clockwise, flip the color of the square, move forward one unit

All finite initial configurations tested eventually converge to the same repetitive pattern, suggesting that the "highway" is an attractor of Langton's ant, but no one has been able to prove that this is true for all such initial configurations. It is only known that the ant's trajectory is always unbounded regardless of the initial configuration – this is known as the Cohen–Kong theorem.

• At a black square, turn 90° counter-clockwise, flip the color of the square, move forward one unit

Langton's ant can also be described as a cellular automaton, where the grid is colored black or white and the "ant" square has one of eight different colors assigned to encode the combination of black/white state and the current direction of motion of the ant.[2]

In 2000, Gajardo et al. showed a construction that calculates any boolean circuit using the trajectory of a single instance of Langton's ant. Additionally, it would be possible to simulate an arbitrary Turing machine using the ant's trajectory for computation. This means that the ant is capable of universal computation.

2000年，Gajardo et al. 。展示了一个结构，计算任何布尔电路使用的轨迹，一个单一的实例朗顿的蚂蚁。此外，还可以使用蚂蚁的轨迹来模拟任意的图灵机进行计算。这意味着蚂蚁能够进行通用计算。

Modes of behavior

These simple rules lead to complex behavior. Three distinct modes of behavior are apparent,[3] when starting on a completely white grid.

1. Simplicity. During the first few hundred moves it creates very simple patterns which are often symmetric.

Greg Turk and Jim Propp considered a simple extension to Langton's ant where instead of just two colors, more colors are used. The colors are modified in a cyclic fashion. A simple naming scheme is used: for each of the successive colors, a letter "L" or "R" is used to indicate whether a left or right turn should be taken. Langton's ant has the name "RL" in this naming scheme.

1. Chaos. After a few hundred moves, a large, irregular pattern of black and white squares appears. The ant traces a pseudo-random path until around 10,000 steps.
1. Emergent order. Finally the ant starts building a recurrent "highway" pattern of 104 steps that repeats indefinitely.

Some of these extended Langton's ants produce patterns that become symmetric over and over again. One of the simplest examples is the ant "RLLR". One sufficient condition for this to happen is that the ant's name, seen as a cyclic list, consists of consecutive pairs of identical letters "LL" or "RR". (the term "cyclic list" indicates that the last letter may pair with the first one) The proof involves Truchet tiles.

All finite initial configurations tested eventually converge to the same repetitive pattern, suggesting that the "highway" is an attractor of Langton's ant, but no one has been able to prove that this is true for all such initial configurations. It is only known that the ant's trajectory is always unbounded regardless of the initial configuration[4] – this is known as the Cohen–Kong theorem.[5]

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Some of these extended Langton's ants produce patterns that become symmetric over and over again. One of the simplest examples is the ant "RLLR". One sufficient condition for this to happen is that the ant's name, seen as a cyclic list, consists of consecutive pairs of identical letters "LL" or "RR". (the term "cyclic list" indicates that the last letter may pair with the first one) The proof involves Truchet tiles.

The hexagonal grid permits up to six different rotations, which are notated here as N (no change), R1 (60° clockwise), R2 (120° clockwise), U (180°), L2 (120° counter-clockwise), L1 (60° counter-clockwise).

File:Turmite-121181121020-65932.png|Chaotic growth with a distinctive texture.

File:Turmite-180121020081-223577.png|Growth with a distinctive texture inside an expanding frame.

The hexagonal grid permits up to six different rotations, which are notated here as N (no change), R1 (60° clockwise), R2 (120° clockwise), U (180°), L2 (120° counter-clockwise), L1 (60° counter-clockwise).

File:Turmite-181181121010-10211.png|Constructing a Fibonacci spiral.

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Extension to multiple states

A further extension of Langton's ants is to consider multiple states of the Turing machine – as if the ant itself has a color that can change. These ants are called turmites, a contraction of "Turing machine termites". Common behaviours include the production of highways, chaotic growth and spiral growth.[6]

Multiple Langton's ants can co-exist on the 2D plane, and their interactions give rise to complex, higher-order automata that collectively build a wide variety of organized structures. There is no need for conflict resolution, as every ant sitting on the same square wants to make the same change to the tape. There is a YouTube video showing these multiple ant interactions.

Extension to multiple ants

Multiple Langton's ants can co-exist on the 2D plane, and their interactions give rise to complex, higher-order automata that collectively build a wide variety of organized structures. There is no need for conflict resolution, as every ant sitting on the same square wants to make the same change to the tape. There is a YouTube video showing these multiple ant interactions.

Multiple turmites can co-exist on the 2D plane as long as there is a rule for what happens when they meet. Ed Pegg, Jr. considered ants that can turn for example both left and right, splitting in two and annihilating each other when they meet.[7]

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