Update README.md

README.md

@@ -17,14 +17,24 @@ Codes and other materials are provided via [https://github.com/lamm-mit/LifeGPT]
 
 ## Sample results
 
+### '`'Glider gun' pattern, from Life, played out on the surface of a 3D torus
+ 
 To best demonstrate our use of a toroidal grid topology (which is functionally the same as periodic boundary conditions), we include a GIF animation of the famous 'glider gun' pattern, from Life, played out on the surface of a 3D torus (see [toroidal_grid_glider_gun.gif](https://github.com/lamm-mit/LifeGPT/raw/main/toroidal_grid_glider_gun.gif)). In this animations, live cells are represented with blue dots, and dead cells are represented as the absence of a dot. The torus is made to be translucent to better illustrate its unique geometry. We hope that this facilitates a more intuitive understanding of the manner in which periodic boundary conditions function with respect to Life. 
 
 ![Glider Gun GIF](https://github.com/lamm-mit/LifeGPT/raw/main/toroidal_grid_glider_gun.gif)
 
-Other results include comparisons of multi-step regression and comparison to ground truth. 
+### Autoregressive Autoregressor (ARAR)
+
+ARAR is achived through recursive process of inference based on an input token sequence given to LifeGPT, resulting in a new sequence of tokens which are subsequently fed back into the input of LifeGPT. This loop may go on until a desired number of iterations are reached. Our ARAR scripts ([ARAR_9_iterations.ipynb](https://github.com/lamm-mit/LifeGPT/blob/main/ARAR_9_iterations.ipynb) and [ARAR_249_iterations.ipynb](https://github.com/lamm-mit/LifeGPT/blob/main/ARAR_249_iterations.ipynb)) demonstrate the application of this method for running Life in the case of 9 iterations for multiple model temperatures, and 249 iterations for only temperature=0, respectively.
+
+Note: ARAR only utilizes versions of LifeGPT trained on broad-entropy data.
+
+The afformentioned ARAR scripts include code that generates GIF animations and figures (se [Testing_Set_ARAR_Animations_and_Figures](https://github.com/lamm-mit/LifeGPT/tree/main/Testing_Set_ARAR_Animations_and_Figures)) showing the evolution of LifeGPT's recursive NGS predictions, Life (GT), and the discrepancy (error) between the two. GIFs are generated for each sample in the testing set, for differing temperatures and epochs, for 9 iterations of Life. 250 iterations are generated for only one version of LifeGPT (epoch=50, temperature=0) due to time and compute constraints. The following figures give examples of predictions made with ARAR.
 
 ![image/png](https://cdn-uploads.huggingface.co/production/uploads/623ce1c6b66fedf374859fe7/AMKWGJXj4psBwaJ5ZCzs7.png)
 
+
+
 ```bibtex
 @article{berkovich2024lifegpt,
   title={LifeGPT: Topology-Agnostic Generative Pretrained Transformer Model for Cellular Automata},