Develop AI systems to efficiently learn new skills and solve open-ended problems, rather than depend exclusively on AI systems trained with extensive datasets
Before even getting into tackling the reasoning aspect of ARC, I'm thinking about how we encode the data.
I think we'd be handicapped by traditional methods because of the sequential nature of those encoding methods. So much of reasoning relies on being able to see the whole of the input and output at once.
We have a 3D input space, and classical computing methods are going to encode that as 1D or 2D. A 3D matrix in classical terms is just an array of arrays - and arrays are inherently a 1-dimensional data structure. Nesting them only gives the illusion of proper dimensionality. You use indexing tricks to manage the object as if it was 3D.
So we come to the idea of holograms.
A hologram is a three-dimensional image created using light interference patterns. Unlike traditional 2D images, holograms can show different perspectives of an object as you move around them, providing depth and parallax effects.
Holograms record the following properties:
X and Y position, encoded in the spatial distribution of the interference pattern
Phase, crucial for depth information
Amplitude, contributes to the intensity of the reconstructed image.
Parallel processing: Holographic encoding could allow for simultaneous access to all parts of the input, potentially mimicking how human vision processes scenes holistically.
Pattern recognition: The interference patterns in holograms might naturally highlight relationships and patterns within the data.
Dimensionality preservation: Holographic encoding preserves 3D relationships in a way that could be more intuitive for reasoning tasks.
Discretization: ARC inputs are discrete grids, while holograms typically encode continuous fields. We'd need to develop a method to "discretize" the holographic representation.
Computational complexity: Generating and interpreting holographic patterns is computationally intensive.
Abstraction: While holograms are good for spatial relationships, we'd need to develop methods to encode abstract concepts and rules that are crucial for ARC tasks.
Develop a mapping between discrete grid states and holographic interference patterns.
Explore how transformations in the ARC tasks (rotations, color changes, etc.) could be represented as operations on the holographic encoding.
Investigate how machine learning models could be adapted to work with holographic data representations.
Discover assets like this one.
Here's some sample code. We'd want to apply transformations to the holographic version then reconstruct to see how it effects the output. This approach opens up possibilities for parallel processing of the entire input at once, potentially revealing patterns or relationships that might not be immediately apparent in the original representation.
import numpy as np
from scipy.fft import fft2, ifft2
def encode_arc_to_hologram(input_array, reference_amplitude=1.0, reference_phase=0.0):
"""
Encode a 3D ARC input array into a simplified 2D holographic representation.
Args:
input_array (numpy.ndarray): 3D numpy array representing the ARC input.
reference_amplitude (float): Amplitude of the reference beam.
reference_phase (float): Phase of the reference beam.
Returns:
numpy.ndarray: 2D numpy array representing the holographic encoding.
"""
# Ensure input is 3D
if input_array.ndim != 3:
raise ValueError("Input must be a 3D numpy array")
# Normalize input to [0, 1] range
input_normalized = (input_array - input_array.min()) / (input_array.max() - input_array.min())
# Create a complex amplitude representing the object wave
object_wave = np.zeros(input_array.shape[:2], dtype=np.complex128)
for i in range(input_array.shape[2]):
object_wave += input_normalized[:,:,i] * np.exp(1j * 2 * np.pi * i / input_array.shape[2])
# Create reference wave
reference_wave = reference_amplitude * np.exp(1j * reference_phase)
# Compute interference pattern
hologram = np.abs(object_wave + reference_wave) ** 2
# Normalize hologram to [0, 1] range
hologram_normalized = (hologram - hologram.min()) / (hologram.max() - hologram.min())
return hologram_normalized
def reconstruct_hologram(hologram, reference_amplitude=1.0, reference_phase=0.0):
"""
Reconstruct the object wave from the hologram.
Args:
hologram (numpy.ndarray): 2D numpy array representing the hologram.
reference_amplitude (float): Amplitude of the reference beam.
reference_phase (float): Phase of the reference beam.
Returns:
numpy.ndarray: Reconstructed object wave.
"""
# Create reference wave
reference_wave = reference_amplitude * np.exp(1j * reference_phase)
# Simulate illumination of the hologram
reconstructed = hologram * reference_wave.conj()
# Use Fourier filtering to isolate the object wave
ft = fft2(reconstructed)
ft_shifted = np.fft.fftshift(ft)
# Create a filter to isolate the object wave
rows, cols = hologram.shape
crow, ccol = rows // 2, cols // 2
mask = np.zeros((rows, cols), dtype=np.uint8)
mask[crow-30:crow+30, ccol-30:ccol+30] = 1
# Apply the filter
ft_filtered = ft_shifted * mask
ft_filtered_shifted = np.fft.ifftshift(ft_filtered)
# Inverse Fourier transform to get the reconstructed object wave
reconstructed_object = ifft2(ft_filtered_shifted)
return np.abs(reconstructed_object)
# Example usage
if __name__ == "__main__":
# Create a sample 3D ARC input (3x3x3 cube)
arc_input = np.random.rand(3, 3, 3)
# Encode the ARC input into a hologram
hologram = encode_arc_to_hologram(arc_input)
# Reconstruct the object from the hologram
reconstructed = reconstruct_hologram(hologram)
print("Original ARC input shape:", arc_input.shape)
print("Hologram shape:", hologram.shape)
print("Reconstructed shape:", reconstructed.shape)
# You can visualize these using matplotlib if desired
# import matplotlib.pyplot as plt
# plt.imshow(hologram, cmap='gray')
# plt.title("Hologram")
# plt.show()
#
# plt.imshow(reconstructed, cmap='gray')
# plt.title("Reconstructed")
# plt.show()