Implementation Examples
# BitMatrix Spatial Computing: Implementation Examples
*Developed by Euan Craig (DigitalEuan.com) New Zealand, 2025*
*With assistance from GPT, Gemini, Grok, and Manus*
This document provides practical implementation examples and pseudocode for key components of the BitMatrix Spatial Computing framework. These examples demonstrate how the theoretical concepts and mathematical foundations can be applied in real-world applications.
## 1. Core BitMatrix Implementation
### 1.1 BitField3D Class Implementation
```python
class BitField3D:
"""
Implementation of a 3D bitfield for the BitMatrix architecture.
"""
def __init__(self, x_dim, y_dim, z_dim):
"""Initialize a 3D bitfield with the specified dimensions."""
self.x_dim = x_dim
self.y_dim = y_dim
self.z_dim = z_dim
# Initialize bit storage (using a 3D numpy array for efficiency)
self.bits = np.zeros((x_dim, y_dim, z_dim), dtype=np.uint8)
# Initialize property storage (using a dictionary for flexibility)
self.properties = {}
def getBit(self, x, y, z):
"""Get the bit value at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and 0 <= z < self.z_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}) out of bounds")
return self.bits[x, y, z]
def setBit(self, x, y, z, value):
"""Set the bit value at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and 0 <= z < self.z_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}) out of bounds")
self.bits[x, y, z] = value
def setProperty(self, x, y, z, property_name, property_value):
"""Set a property for the bit at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and 0 <= z < self.z_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}) out of bounds")
# Create position key
pos_key = (x, y, z)
# Initialize property dictionary for this position if it doesn't exist
if pos_key not in self.properties:
self.properties[pos_key] = {}
# Set the property
self.properties[pos_key][property_name] = property_value
def getProperty(self, x, y, z, property_name):
"""Get a property for the bit at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and 0 <= z < self.z_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}) out of bounds")
# Create position key
pos_key = (x, y, z)
# Check if property exists
if pos_key not in self.properties or property_name not in self.properties[pos_key]:
raise KeyError(f"Property '{property_name}' not found at coordinates ({x}, {y}, {z})")
return self.properties[pos_key][property_name]
def getPropertyNames(self, x, y, z):
"""Get all property names for the bit at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and 0 <= z < self.z_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}) out of bounds")
# Create position key
pos_key = (x, y, z)
# Return property names or empty list if none exist
if pos_key in self.properties:
return list(self.properties[pos_key].keys())
else:
return []
def getDimensions(self):
"""Get the dimensions of the bitfield."""
return (self.x_dim, self.y_dim, self.z_dim)
def copy(self):
"""Create a deep copy of this bitfield."""
new_bitfield = BitField3D(self.x_dim, self.y_dim, self.z_dim)
new_bitfield.bits = np.copy(self.bits)
# Deep copy properties
for pos_key, props in self.properties.items():
new_bitfield.properties[pos_key] = props.copy()
return new_bitfield
```
### 1.2 BitField4D Class Implementation
```python
class BitField4D:
"""
Implementation of a 4D bitfield for the BitMatrix architecture.
"""
def __init__(self, x_dim, y_dim, z_dim, t_dim):
"""Initialize a 4D bitfield with the specified dimensions."""
self.x_dim = x_dim
self.y_dim = y_dim
self.z_dim = z_dim
self.t_dim = t_dim
# Initialize bit storage (using a 4D numpy array for efficiency)
self.bits = np.zeros((x_dim, y_dim, z_dim, t_dim), dtype=np.uint8)
# Initialize property storage (using a dictionary for flexibility)
self.properties = {}
def getBit(self, x, y, z, t):
"""Get the bit value at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and
0 <= z < self.z_dim and 0 <= t < self.t_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}, {t}) out of bounds")
return self.bits[x, y, z, t]
def setBit(self, x, y, z, t, value):
"""Set the bit value at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and
0 <= z < self.z_dim and 0 <= t < self.t_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}, {t}) out of bounds")
self.bits[x, y, z, t] = value
def setProperty(self, x, y, z, t, property_name, property_value):
"""Set a property for the bit at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and
0 <= z < self.z_dim and 0 <= t < self.t_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}, {t}) out of bounds")
# Create position key
pos_key = (x, y, z, t)
# Initialize property dictionary for this position if it doesn't exist
if pos_key not in self.properties:
self.properties[pos_key] = {}
# Set the property
self.properties[pos_key][property_name] = property_value
def getProperty(self, x, y, z, t, property_name):
"""Get a property for the bit at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and
0 <= z < self.z_dim and 0 <= t < self.t_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}, {t}) out of bounds")
# Create position key
pos_key = (x, y, z, t)
# Check if property exists
if pos_key not in self.properties or property_name not in self.properties[pos_key]:
raise KeyError(f"Property '{property_name}' not found at coordinates ({x}, {y}, {z}, {t})")
return self.properties[pos_key][property_name]
def getPropertyNames(self, x, y, z, t):
"""Get all property names for the bit at the specified coordinates."""
if not (0 <= x < self.x_dim and 0 <= y < self.y_dim and
0 <= z < self.z_dim and 0 <= t < self.t_dim):
raise IndexError(f"Coordinates ({x}, {y}, {z}, {t}) out of bounds")
# Create position key
pos_key = (x, y, z, t)
# Return property names or empty list if none exist
if pos_key in self.properties:
return list(self.properties[pos_key].keys())
else:
return []
def getDimensions(self):
"""Get the dimensions of the bitfield."""
return (self.x_dim, self.y_dim, self.z_dim, self.t_dim)
def copy(self):
"""Create a deep copy of this bitfield."""
new_bitfield = BitField4D(self.x_dim, self.y_dim, self.z_dim, self.t_dim)
new_bitfield.bits = np.copy(self.bits)
# Deep copy properties
for pos_key, props in self.properties.items():
new_bitfield.properties[pos_key] = props.copy()
return new_bitfield
```
## 2. Spatial Operations Implementation
### 2.1 Rotation Implementation
```python
def rotate3D(bitfield, axis, angle_degrees):
"""
Rotate a 3D bitfield around the specified axis.
Parameters:
bitfield (BitField3D): Bitfield to rotate
axis (str): Rotation axis ('x', 'y', or 'z')
angle_degrees (float): Rotation angle in degrees
Returns:
BitField3D: Rotated bitfield
"""
# Convert angle to radians
angle_radians = np.radians(angle_degrees)
# Get dimensions
x_dim, y_dim, z_dim = bitfield.getDimensions()
# Create rotation matrix
if axis.lower() == 'x':
rotation_matrix = np.array([
[1, 0, 0],
[0, np.cos(angle_radians), -np.sin(angle_radians)],
[0, np.sin(angle_radians), np.cos(angle_radians)]
])
elif axis.lower() == 'y':
rotation_matrix = np.array([
[np.cos(angle_radians), 0, np.sin(angle_radians)],
[0, 1, 0],
[-np.sin(angle_radians), 0, np.cos(angle_radians)]
])
elif axis.lower() == 'z':
rotation_matrix = np.array([
[np.cos(angle_radians), -np.sin(angle_radians), 0],
[np.sin(angle_radians), np.cos(angle_radians), 0],
[0, 0, 1]
])
else:
raise ValueError(f"Invalid rotation axis: {axis}")
# Create result bitfield
result_bitfield = BitField3D(x_dim, y_dim, z_dim)
# Calculate center of rotation
center_x = (x_dim - 1) / 2
center_y = (y_dim - 1) / 2
center_z = (z_dim - 1) / 2
# Apply rotation to each bit
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Calculate coordinates relative to center
rel_x = x - center_x
rel_y = y - center_y
rel_z = z - center_z
# Apply rotation matrix
rotated_coords = np.dot(rotation_matrix, np.array([rel_x, rel_y, rel_z]))
# Convert back to absolute coordinates
rot_x = int(round(rotated_coords[0] + center_x))
rot_y = int(round(rotated_coords[1] + center_y))
rot_z = int(round(rotated_coords[2] + center_z))
# Check if rotated coordinates are within bounds
if (0 <= rot_x < x_dim and 0 <= rot_y < y_dim and 0 <= rot_z < z_dim):
# Copy bit value from original position to rotated position
result_bitfield.setBit(rot_x, rot_y, rot_z, bitfield.getBit(x, y, z))
# Copy properties
for prop_name in bitfield.getPropertyNames(x, y, z):
prop_value = bitfield.getProperty(x, y, z, prop_name)
result_bitfield.setProperty(rot_x, rot_y, rot_z, prop_name, prop_value)
return result_bitfield
```
### 2.2 Block Operations Implementation
```python
def createBlock(dimensions, value=0):
"""
Create a block (bitfield) with the specified dimensions and initial value.
Parameters:
dimensions (tuple): Dimensions of the block (x_dim, y_dim, z_dim)
value (int): Initial value for all bits (0 or 1)
Returns:
BitFieldND: Created block
"""
if len(dimensions) == 3:
x_dim, y_dim, z_dim = dimensions
block = BitField3D(x_dim, y_dim, z_dim)
# Set all bits to the specified value
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
block.setBit(x, y, z, value)
elif len(dimensions) == 4:
x_dim, y_dim, z_dim, t_dim = dimensions
block = BitField4D(x_dim, y_dim, z_dim, t_dim)
# Set all bits to the specified value
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
block.setBit(x, y, z, t, value)
else:
raise ValueError("Dimensions must specify a 3D or 4D block")
return block
def insertBlock(target_bitfield, block, position):
"""
Insert a block into a target bitfield at the specified position.
Parameters:
target_bitfield (BitFieldND): Target bitfield to insert into
block (BitFieldND): Block to insert
position (tuple): Position to insert the block at
Returns:
BitFieldND: Modified target bitfield
"""
target_dimensions = target_bitfield.getDimensions()
block_dimensions = block.getDimensions()
# Validate dimensionality match
if len(target_dimensions) != len(block_dimensions):
raise ValueError("Target bitfield and block must have the same dimensionality")
# Validate position
if len(position) != len(target_dimensions):
raise ValueError("Position must have the same dimensionality as the target bitfield")
# Create a copy of the target bitfield
result_bitfield = target_bitfield.copy()
# Insert block
if len(target_dimensions) == 3:
target_x_dim, target_y_dim, target_z_dim = target_dimensions
block_x_dim, block_y_dim, block_z_dim = block_dimensions
pos_x, pos_y, pos_z = position
# Check if block fits within target
if (pos_x + block_x_dim > target_x_dim or
pos_y + block_y_dim > target_y_dim or
pos_z + block_z_dim > target_z_dim):
raise ValueError("Block does not fit within target bitfield at the specified position")
# Copy block bits and properties to target
for z in range(block_z_dim):
for y in range(block_y_dim):
for x in range(block_x_dim):
# Calculate target position
target_x = pos_x + x
target_y = pos_y + y
target_z = pos_z + z
# Copy bit
result_bitfield.setBit(target_x, target_y, target_z, block.getBit(x, y, z))
# Copy properties
for prop_name in block.getPropertyNames(x, y, z):
prop_value = block.getProperty(x, y, z, prop_name)
result_bitfield.setProperty(target_x, target_y, target_z, prop_name, prop_value)
elif len(target_dimensions) == 4:
target_x_dim, target_y_dim, target_z_dim, target_t_dim = target_dimensions
block_x_dim, block_y_dim, block_z_dim, block_t_dim = block_dimensions
pos_x, pos_y, pos_z, pos_t = position
# Check if block fits within target
if (pos_x + block_x_dim > target_x_dim or
pos_y + block_y_dim > target_y_dim or
pos_z + block_z_dim > target_z_dim or
pos_t + block_t_dim > target_t_dim):
raise ValueError("Block does not fit within target bitfield at the specified position")
# Copy block bits and properties to target
for t in range(block_t_dim):
for z in range(block_z_dim):
for y in range(block_y_dim):
for x in range(block_x_dim):
# Calculate target position
target_x = pos_x + x
target_y = pos_y + y
target_z = pos_z + z
target_t = pos_t + t
# Copy bit
result_bitfield.setBit(target_x, target_y, target_z, target_t,
block.getBit(x, y, z, t))
# Copy properties
for prop_name in block.getPropertyNames(x, y, z, t):
prop_value = block.getProperty(x, y, z, t, prop_name)
result_bitfield.setProperty(target_x, target_y, target_z, target_t,
prop_name, prop_value)
return result_bitfield
```
## 3. Temporal Operations Implementation
### 3.1 Wave Propagation Implementation
```python
def propagateWave(bitfield, num_steps, speed=0.5, damping=0.05):
"""
Propagate a wave through a 3D bitfield for a specified number of time steps.
Parameters:
bitfield (BitField3D): Initial bitfield state
num_steps (int): Number of time steps to simulate
speed (float): Wave propagation speed (0.0 to 1.0)
damping (float): Damping factor (0.0 to 1.0)
Returns:
BitField4D: 4D bitfield containing the wave propagation over time
"""
# Get dimensions
x_dim, y_dim, z_dim = bitfield.getDimensions()
# Create 4D bitfield to store the wave propagation over time
result_bitfield = BitField4D(x_dim, y_dim, z_dim, num_steps + 1)
# Initialize arrays for current, previous, and next states
current_state = np.zeros((x_dim, y_dim, z_dim))
previous_state = np.zeros((x_dim, y_dim, z_dim))
# Copy initial state from input bitfield
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
current_state[x, y, z] = bitfield.getBit(x, y, z)
result_bitfield.setBit(x, y, z, 0, bitfield.getBit(x, y, z))
# Define discrete Laplacian operator
def laplacian(state):
laplacian_state = np.zeros_like(state)
# Apply 3D discrete Laplacian operator
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Count neighbors (6-connected)
neighbors_sum = 0
neighbor_count = 0
# Check each neighbor
for dx, dy, dz in [(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1)]:
nx, ny, nz = x + dx, y + dy, z + dz
# Check bounds
if 0 <= nx < x_dim and 0 <= ny < y_dim and 0 <= nz < z_dim:
neighbors_sum += state[nx, ny, nz]
neighbor_count += 1
# Calculate Laplacian (average of neighbors minus center)
if neighbor_count > 0:
laplacian_state[x, y, z] = neighbors_sum / neighbor_count - state[x, y, z]
return laplacian_state
# Simulate wave propagation for each time step
for t in range(1, num_steps + 1):
# Calculate next state using discrete wave equation
# next = 2*current - previous + speed^2 * laplacian(current)
next_state = (2 * current_state - previous_state +
speed**2 * laplacian(current_state))
# Apply damping
next_state = next_state * (1 - damping)
# Store result in 4D bitfield (threshold at 0.5)
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Threshold continuous value to binary
bit_value = 1 if next_state[x, y, z] >= 0.5 else 0
result_bitfield.setBit(x, y, z, t, bit_value)
# Store actual wave amplitude as property
result_bitfield.setProperty(x, y, z, t, 'amplitude', float(next_state[x, y, z]))
# Update states for next iteration
previous_state = current_state.copy()
current_state = next_state.copy()
return result_bitfield
```
### 3.2 Temporal Pattern Implementation
```python
def createTemporalPattern(spatial_dimensions, temporal_dimension, pattern_type='sine', frequency=0.1, phase=0):
"""
Create a bitfield with a temporal pattern.
Parameters:
spatial_dimensions (tuple): Spatial dimensions (x_dim, y_dim, z_dim)
temporal_dimension (int): Number of time steps
pattern_type (str): Type of pattern ('sine', 'square', 'sawtooth', 'pulse')
frequency (float): Pattern frequency (cycles per time step)
phase (float): Initial phase (in radians)
Returns:
BitField4D: 4D bitfield with the temporal pattern
"""
x_dim, y_dim, z_dim = spatial_dimensions
t_dim = temporal_dimension
# Create 4D bitfield
result_bitfield = BitField4D(x_dim, y_dim, z_dim, t_dim)
# Generate temporal pattern
time_steps = np.arange(t_dim)
if pattern_type == 'sine':
# Sine wave pattern
pattern = 0.5 + 0.5 * np.sin(2 * np.pi * frequency * time_steps + phase)
elif pattern_type == 'square':
# Square wave pattern
pattern = 0.5 + 0.5 * np.sign(np.sin(2 * np.pi * frequency * time_steps + phase))
elif pattern_type == 'sawtooth':
# Sawtooth wave pattern
pattern = np.mod(frequency * time_steps + phase / (2 * np.pi), 1.0)
elif pattern_type == 'pulse':
# Pulse pattern (periodic impulses)
pattern = np.zeros(t_dim)
pulse_period = int(1.0 / frequency)
pulse_indices = np.arange(0, t_dim, pulse_period)
pattern[pulse_indices] = 1.0
else:
raise ValueError(f"Unsupported pattern type: {pattern_type}")
# Apply pattern to bitfield
for t in range(t_dim):
# Threshold pattern value to get bit value
bit_value = 1 if pattern[t] >= 0.5 else 0
# Set all spatial positions to the same temporal pattern
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
result_bitfield.setBit(x, y, z, t, bit_value)
# Store actual pattern value as property
result_bitfield.setProperty(x, y, z, t, 'amplitude', float(pattern[t]))
return result_bitfield
```
## 4. Neural Network Integration Implementation
### 4.1 Neural Layer Implementation
```python
def createNeuralNetwork(input_dimensions, hidden_dimensions, output_dimensions):
"""
Create a neural network for processing BitMatrix data.
Parameters:
input_dimensions (tuple): Dimensions of the input bitfield
hidden_dimensions (list): List of dimensions for hidden layers
output_dimensions (tuple): Dimensions of the output bitfield
Returns:
dict: Neural network object with layers and parameters
"""
# Calculate total sizes
def calc_total_size(dimensions):
total = 1
for dim in dimensions:
total *= dim
return total
input_size = calc_total_size(input_dimensions)
output_size = calc_total_size(output_dimensions)
# Create network structure
network = {
'input_dimensions': input_dimensions,
'hidden_dimensions': hidden_dimensions,
'output_dimensions': output_dimensions,
'layers': []
}
# Create input layer
current_input_size = input_size
# Create hidden layers
for i, hidden_size in enumerate(hidden_dimensions):
# Create layer
layer = {
'weights': np.random.randn(hidden_size, current_input_size) * 0.1,
'biases': np.random.randn(hidden_size) * 0.1,
'activation': 'relu' if i < len(hidden_dimensions) - 1 else 'sigmoid'
}
network['layers'].append(layer)
current_input_size = hidden_size
# Create output layer
output_layer = {
'weights': np.random.randn(output_size, current_input_size) * 0.1,
'biases': np.random.randn(output_size) * 0.1,
'activation': 'sigmoid'
}
network['layers'].append(output_layer)
return network
def applyNeuralNetwork(input_bitfield, network):
"""
Apply a neural network to process a bitfield.
Parameters:
input_bitfield (BitFieldND): Input bitfield to process
network (dict): Neural network object
Returns:
BitFieldND: Output bitfield after neural processing
"""
# Validate input dimensions
input_dimensions = input_bitfield.getDimensions()
if input_dimensions != network['input_dimensions']:
raise ValueError(f"Input bitfield dimensions {input_dimensions} do not match network input dimensions {network['input_dimensions']}")
# Flatten input bitfield to 1D array
input_array = []
if len(input_dimensions) == 3: # 3D input
x_dim, y_dim, z_dim = input_dimensions
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
input_array.append(input_bitfield.getBit(x, y, z))
elif len(input_dimensions) == 4: # 4D input
x_dim, y_dim, z_dim, t_dim = input_dimensions
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
input_array.append(input_bitfield.getBit(x, y, z, t))
# Convert to numpy array
input_array = np.array(input_array)
# Forward pass through the network
current_activation = input_array
for layer in network['layers']:
# Apply weights and biases
z = np.dot(layer['weights'], current_activation) + layer['biases']
# Apply activation function
if layer['activation'] == 'sigmoid':
current_activation = 1 / (1 + np.exp(-z))
elif layer['activation'] == 'relu':
current_activation = np.maximum(0, z)
elif layer['activation'] == 'tanh':
current_activation = np.tanh(z)
else:
raise ValueError(f"Unsupported activation function: {layer['activation']}")
# Output activation is now in current_activation
output_array = current_activation
# Create output bitfield
output_dimensions = network['output_dimensions']
if len(output_dimensions) == 3: # 3D output
x_dim, y_dim, z_dim = output_dimensions
output_bitfield = BitField3D(x_dim, y_dim, z_dim)
# Fill output bitfield
index = 0
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Threshold at 0.5 for binary output
bit_value = 1 if output_array[index] >= 0.5 else 0
output_bitfield.setBit(x, y, z, bit_value)
# Store actual activation value as property
output_bitfield.setProperty(x, y, z, 'activation', float(output_array[index]))
index += 1
elif len(output_dimensions) == 4: # 4D output
x_dim, y_dim, z_dim, t_dim = output_dimensions
output_bitfield = BitField4D(x_dim, y_dim, z_dim, t_dim)
# Fill output bitfield
index = 0
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Threshold at 0.5 for binary output
bit_value = 1 if output_array[index] >= 0.5 else 0
output_bitfield.setBit(x, y, z, t, bit_value)
# Store actual activation value as property
output_bitfield.setProperty(x, y, z, t, 'activation', float(output_array[index]))
index += 1
return output_bitfield
```
## 5. Quantum-Inspired Implementation
### 5.1 Quantum Search Implementation
```python
def quantumSearch(bitfield, target_pattern, num_iterations=None):
"""
Implement a quantum-inspired search algorithm to find a target pattern.
Parameters:
bitfield (BitFieldND): Bitfield to search within
target_pattern (BitFieldND): Pattern to search for
num_iterations (int): Number of iterations (if None, uses optimal value)
Returns:
list: List of positions where the pattern was found
"""
bitfield_dimensions = bitfield.getDimensions()
pattern_dimensions = target_pattern.getDimensions()
# Validate dimensionality match
if len(bitfield_dimensions) != len(pattern_dimensions):
raise ValueError("Bitfield and pattern must have the same dimensionality")
# Check if pattern fits within bitfield
for i in range(len(bitfield_dimensions)):
if pattern_dimensions[i] > bitfield_dimensions[i]:
raise ValueError(f"Pattern dimension {i} exceeds bitfield dimension")
# Find all possible positions where the pattern could fit
possible_positions = []
if len(bitfield_dimensions) == 3:
x_dim, y_dim, z_dim = bitfield_dimensions
x_pat, y_pat, z_pat = pattern_dimensions
for z_start in range(z_dim - z_pat + 1):
for y_start in range(y_dim - y_pat + 1):
for x_start in range(x_dim - x_pat + 1):
possible_positions.append((x_start, y_start, z_start))
elif len(bitfield_dimensions) == 4:
x_dim, y_dim, z_dim, t_dim = bitfield_dimensions
x_pat, y_pat, z_pat, t_pat = pattern_dimensions
for t_start in range(t_dim - t_pat + 1):
for z_start in range(z_dim - z_pat + 1):
for y_start in range(y_dim - y_pat + 1):
for x_start in range(x_dim - x_pat + 1):
possible_positions.append((x_start, y_start, z_start, t_start))
# Calculate optimal number of iterations if not specified
# In Grover's algorithm, optimal is approximately π/4 * sqrt(N)
if num_iterations is None:
num_iterations = int(np.pi / 4 * np.sqrt(len(possible_positions)))
# Initialize uniform superposition
amplitudes = np.ones(len(possible_positions)) / np.sqrt(len(possible_positions))
# Define oracle function to mark matching positions
def oracle(position):
"""Return True if pattern matches at the given position."""
if len(bitfield_dimensions) == 3:
x_start, y_start, z_start = position
x_pat, y_pat, z_pat = pattern_dimensions
# Check if pattern matches at this position
for z in range(z_pat):
for y in range(y_pat):
for x in range(x_pat):
if bitfield.getBit(x_start + x, y_start + y, z_start + z) != target_pattern.getBit(x, y, z):
return False
return True
elif len(bitfield_dimensions) == 4:
x_start, y_start, z_start, t_start = position
x_pat, y_pat, z_pat, t_pat = pattern_dimensions
# Check if pattern matches at this position
for t in range(t_pat):
for z in range(z_pat):
for y in range(y_pat):
for x in range(x_pat):
if bitfield.getBit(x_start + x, y_start + y, z_start + z, t_start + t) != target_pattern.getBit(x, y, z, t):
return False
return True
# Find matching positions
matching_positions = [i for i, pos in enumerate(possible_positions) if oracle(pos)]
# If no matches, return empty list
if not matching_positions:
return []
# Simulate Grover's algorithm iterations
for _ in range(num_iterations):
# Oracle: Flip the sign of matching positions
for i in matching_positions:
amplitudes[i] = -amplitudes[i]
# Diffusion operator: Reflect about the mean
mean_amplitude = np.mean(amplitudes)
amplitudes = 2 * mean_amplitude - amplitudes
# Calculate probabilities
probabilities = np.abs(amplitudes)**2
# Sort positions by probability (highest first)
sorted_indices = np.argsort(-probabilities)
# Return positions with highest probability
result_positions = [possible_positions[i] for i in sorted_indices[:len(matching_positions)]]
return result_positions
```
### 5.2 Quantum Fourier Transform Implementation
```python
def quantumFourierTransform(bitfield):
"""
Implement a quantum-inspired Fourier transform.
Parameters:
bitfield (BitFieldND): Bitfield to transform
Returns:
BitFieldND: Transformed bitfield
"""
dimensions = bitfield.getDimensions()
# Extract bit values into numpy array
if len(dimensions) == 3: # 3D bitfield
x_dim, y_dim, z_dim = dimensions
data_array = np.zeros(dimensions, dtype=complex)
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Get bit value and convert to complex number
bit_value = bitfield.getBit(x, y, z)
data_array[x, y, z] = complex(bit_value)
elif len(dimensions) == 4: # 4D bitfield
x_dim, y_dim, z_dim, t_dim = dimensions
data_array = np.zeros(dimensions, dtype=complex)
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Get bit value and convert to complex number
bit_value = bitfield.getBit(x, y, z, t)
data_array[x, y, z, t] = complex(bit_value)
# Apply Fast Fourier Transform
fft_result = np.fft.fftn(data_array)
# Normalize
fft_result = fft_result / np.sqrt(np.prod(dimensions))
# Create result bitfield
if len(dimensions) == 3:
result_bitfield = BitField3D(*dimensions)
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Calculate magnitude (absolute value)
magnitude = np.abs(fft_result[x, y, z])
phase = np.angle(fft_result[x, y, z])
# Set bit based on threshold
bit_value = 1 if magnitude > 0.5 else 0
result_bitfield.setBit(x, y, z, bit_value)
# Store complex value as properties
result_bitfield.setProperty(x, y, z, 'magnitude', float(magnitude))
result_bitfield.setProperty(x, y, z, 'phase', float(phase))
elif len(dimensions) == 4:
result_bitfield = BitField4D(*dimensions)
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Calculate magnitude (absolute value)
magnitude = np.abs(fft_result[x, y, z, t])
phase = np.angle(fft_result[x, y, z, t])
# Set bit based on threshold
bit_value = 1 if magnitude > 0.5 else 0
result_bitfield.setBit(x, y, z, t, bit_value)
# Store complex value as properties
result_bitfield.setProperty(x, y, z, t, 'magnitude', float(magnitude))
result_bitfield.setProperty(x, y, z, t, 'phase', float(phase))
return result_bitfield
```
## 6. Data Compression Implementation
### 6.1 LZW Compression Implementation
```python
def compressLZW(bitfield):
"""
Compress a bitfield using LZW compression.
Parameters:
bitfield (BitFieldND): Bitfield to compress
Returns:
tuple: (compressed_data, dictionary)
"""
dimensions = bitfield.getDimensions()
# Flatten bitfield to 1D sequence
bit_sequence = []
if len(dimensions) == 3:
x_dim, y_dim, z_dim = dimensions
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
bit_sequence.append(str(bitfield.getBit(x, y, z)))
elif len(dimensions) == 4:
x_dim, y_dim, z_dim, t_dim = dimensions
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
bit_sequence.append(str(bitfield.getBit(x, y, z, t)))
# Convert bit sequence to string
bit_string = ''.join(bit_sequence)
# Initialize dictionary with single characters
dictionary = {str(i): i for i in range(2)} # 0 and 1
next_code = 2
# Initialize variables
current_string = ""
compressed_data = []
# Compress
for bit in bit_string:
# Try to extend current string
new_string = current_string + bit
if new_string in dictionary:
# If extended string exists in dictionary, continue extending
current_string = new_string
else:
# Output code for current string
compressed_data.append(dictionary[current_string])
# Add new string to dictionary
dictionary[new_string] = next_code
next_code += 1
# Reset current string
current_string = bit
# Output code for any remaining string
if current_string:
compressed_data.append(dictionary[current_string])
return compressed_data, dictionary
def decompressLZW(compressed_data, dictionary):
"""
Decompress LZW-compressed data.
Parameters:
compressed_data (list): Compressed data codes
dictionary (dict): Compression dictionary
Returns:
str: Decompressed bit string
"""
# Invert dictionary for decompression
reverse_dict = {v: k for k, v in dictionary.items()}
# Initialize variables
decompressed_string = ""
# Decompress
for code in compressed_data:
if code in reverse_dict:
decompressed_string += reverse_dict[code]
else:
# Special case for codes not in dictionary
# (can happen in LZW when a pattern is repeated)
previous_string = reverse_dict[compressed_data[-1]]
decompressed_string += previous_string + previous_string[0]
return decompressed_string
```
### 6.2 Run-Length Encoding Implementation
```python
def compressRLE(bitfield):
"""
Compress a bitfield using Run-Length Encoding.
Parameters:
bitfield (BitFieldND): Bitfield to compress
Returns:
list: List of (value, length) tuples
"""
dimensions = bitfield.getDimensions()
# Flatten bitfield to 1D sequence
bit_sequence = []
if len(dimensions) == 3:
x_dim, y_dim, z_dim = dimensions
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
bit_sequence.append(bitfield.getBit(x, y, z))
elif len(dimensions) == 4:
x_dim, y_dim, z_dim, t_dim = dimensions
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
bit_sequence.append(bitfield.getBit(x, y, z, t))
# Compress using RLE
compressed_data = []
current_value = bit_sequence[0]
current_length = 1
for bit in bit_sequence[1:]:
if bit == current_value:
# Continue current run
current_length += 1
else:
# End current run
compressed_data.append((current_value, current_length))
# Start new run
current_value = bit
current_length = 1
# Add final run
compressed_data.append((current_value, current_length))
return compressed_data
def decompressRLE(compressed_data):
"""
Decompress RLE-compressed data.
Parameters:
compressed_data (list): List of (value, length) tuples
Returns:
list: Decompressed bit sequence
"""
decompressed_sequence = []
for value, length in compressed_data:
decompressed_sequence.extend([value] * length)
return decompressed_sequence
```
## 7. Error Correction Implementation
### 7.1 Hamming Code Implementation
```python
def encodeHamming(bitfield):
"""
Encode a bitfield using Hamming code for error correction.
Parameters:
bitfield (BitFieldND): Bitfield to encode
Returns:
BitFieldND: Encoded bitfield with error correction
"""
dimensions = bitfield.getDimensions()
# Flatten bitfield to 1D sequence
data_bits = []
if len(dimensions) == 3:
x_dim, y_dim, z_dim = dimensions
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
data_bits.append(bitfield.getBit(x, y, z))
elif len(dimensions) == 4:
x_dim, y_dim, z_dim, t_dim = dimensions
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
data_bits.append(bitfield.getBit(x, y, z, t))
# Process data in blocks (for simplicity, we'll use Hamming(7,4) code)
encoded_bits = []
for i in range(0, len(data_bits), 4):
# Get 4 data bits (pad with 0 if needed)
block = data_bits[i:i+4]
while len(block) < 4:
block.append(0)
# Calculate parity bits
p1 = (block[0] ^ block[1] ^ block[3]) % 2
p2 = (block[0] ^ block[2] ^ block[3]) % 2
p3 = (block[1] ^ block[2] ^ block[3]) % 2
# Create encoded block (7 bits)
encoded_block = [p1, p2, block[0], p3, block[1], block[2], block[3]]
encoded_bits.extend(encoded_block)
# Create encoded bitfield
# For simplicity, we'll create a 1D bitfield
encoded_bitfield = BitField3D(len(encoded_bits), 1, 1)
for i, bit in enumerate(encoded_bits):
encoded_bitfield.setBit(i, 0, 0, bit)
return encoded_bitfield
def decodeHamming(encoded_bitfield):
"""
Decode a Hamming-encoded bitfield, correcting single-bit errors.
Parameters:
encoded_bitfield (BitFieldND): Encoded bitfield
Returns:
BitFieldND: Decoded bitfield with errors corrected
"""
# Extract encoded bits
dimensions = encoded_bitfield.getDimensions()
encoded_bits = []
if dimensions[0] > 1 and dimensions[1] == 1 and dimensions[2] == 1:
# 1D encoded bitfield
for i in range(dimensions[0]):
encoded_bits.append(encoded_bitfield.getBit(i, 0, 0))
else:
# Flatten general bitfield
if len(dimensions) == 3:
x_dim, y_dim, z_dim = dimensions
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
encoded_bits.append(encoded_bitfield.getBit(x, y, z))
elif len(dimensions) == 4:
x_dim, y_dim, z_dim, t_dim = dimensions
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
encoded_bits.append(encoded_bitfield.getBit(x, y, z, t))
# Process in blocks of 7 bits (Hamming(7,4) code)
decoded_bits = []
for i in range(0, len(encoded_bits), 7):
# Get 7 encoded bits (pad with 0 if needed)
block = encoded_bits[i:i+7]
while len(block) < 7:
block.append(0)
# Extract parity and data bits
p1, p2, d1, p3, d2, d3, d4 = block
# Check parity bits
check1 = (p1 ^ d1 ^ d2 ^ d4) % 2
check2 = (p2 ^ d1 ^ d3 ^ d4) % 2
check3 = (p3 ^ d2 ^ d3 ^ d4) % 2
# Calculate error position
error_pos = check1 + 2*check2 + 4*check3
# Correct error if detected
if error_pos > 0:
if error_pos == 3:
d1 = 1 - d1
elif error_pos == 5:
d2 = 1 - d2
elif error_pos == 6:
d3 = 1 - d3
elif error_pos == 7:
d4 = 1 - d4
# Add corrected data bits to result
decoded_bits.extend([d1, d2, d3, d4])
# Create decoded bitfield
# For simplicity, we'll create a 1D bitfield
decoded_bitfield = BitField3D(len(decoded_bits), 1, 1)
for i, bit in enumerate(decoded_bits):
decoded_bitfield.setBit(i, 0, 0, bit)
return decoded_bitfield
```
## 8. 5D KTA Implementation
### 8.1 KTA Dimensional Transform Implementation
```python
def ktaDimensionShift(bitfield, source_dim, target_dim):
"""
Implement a KTA dimension shift operation.
Parameters:
bitfield (BitFieldND): Input bitfield
source_dim (int): Source dimension index (0=x, 1=y, 2=z, 3=t)
target_dim (int): Target dimension index (0=x, 1=y, 2=z, 3=t)
Returns:
BitFieldND: Transformed bitfield
"""
dimensions = bitfield.getDimensions()
if len(dimensions) < max(source_dim, target_dim) + 1:
raise ValueError("Bitfield dimensionality too low for specified dimensions")
# Create new dimensions with source and target dimensions swapped
new_dimensions = list(dimensions)
new_dimensions[source_dim], new_dimensions[target_dim] = new_dimensions[target_dim], new_dimensions[source_dim]
# Create result bitfield
if len(dimensions) == 3:
result_bitfield = BitField3D(*new_dimensions)
elif len(dimensions) == 4:
result_bitfield = BitField4D(*new_dimensions)
else:
raise ValueError("Unsupported bitfield dimensionality")
# Copy bits with dimensions swapped
if len(dimensions) == 3:
x_dim, y_dim, z_dim = dimensions
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Create coordinate array
coords = [x, y, z]
# Swap source and target dimensions
coords[source_dim], coords[target_dim] = coords[target_dim], coords[source_dim]
# Get bit value from original position
bit_value = bitfield.getBit(x, y, z)
# Set bit value at transformed position
result_bitfield.setBit(*coords, bit_value)
# Copy properties
for prop_name in bitfield.getPropertyNames(x, y, z):
prop_value = bitfield.getProperty(x, y, z, prop_name)
result_bitfield.setProperty(*coords, prop_name, prop_value)
elif len(dimensions) == 4:
x_dim, y_dim, z_dim, t_dim = dimensions
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Create coordinate array
coords = [x, y, z, t]
# Swap source and target dimensions
coords[source_dim], coords[target_dim] = coords[target_dim], coords[source_dim]
# Get bit value from original position
bit_value = bitfield.getBit(x, y, z, t)
# Set bit value at transformed position
result_bitfield.setBit(*coords, bit_value)
# Copy properties
for prop_name in bitfield.getPropertyNames(x, y, z, t):
prop_value = bitfield.getProperty(x, y, z, t, prop_name)
result_bitfield.setProperty(*coords, prop_name, prop_value)
return result_bitfield
```
### 8.2 KTA Kinetic Operation Implementation
```python
def ktaKineticTransform(bitfield, energy_function):
"""
Implement a KTA kinetic transformation.
Parameters:
bitfield (BitFieldND): Input bitfield
energy_function (function): Function that calculates kinetic energy for each position
Returns:
BitFieldND: Transformed bitfield
"""
dimensions = bitfield.getDimensions()
# Create result bitfield
if len(dimensions) == 3:
result_bitfield = BitField3D(*dimensions)
elif len(dimensions) == 4:
result_bitfield = BitField4D(*dimensions)
else:
raise ValueError("Unsupported bitfield dimensionality")
# Apply kinetic transformation
if len(dimensions) == 3:
x_dim, y_dim, z_dim = dimensions
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Get original bit value
bit_value = bitfield.getBit(x, y, z)
# Calculate kinetic energy
energy = energy_function(x, y, z)
# Apply transformation based on energy
# For example, flip bit if energy exceeds threshold
if energy > 0.5:
transformed_value = 1 - bit_value
else:
transformed_value = bit_value
# Set transformed bit
result_bitfield.setBit(x, y, z, transformed_value)
# Store energy as property
result_bitfield.setProperty(x, y, z, 'kinetic_energy', float(energy))
elif len(dimensions) == 4:
x_dim, y_dim, z_dim, t_dim = dimensions
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
# Get original bit value
bit_value = bitfield.getBit(x, y, z, t)
# Calculate kinetic energy
energy = energy_function(x, y, z, t)
# Apply transformation based on energy
# For example, flip bit if energy exceeds threshold
if energy > 0.5:
transformed_value = 1 - bit_value
else:
transformed_value = bit_value
# Set transformed bit
result_bitfield.setBit(x, y, z, t, transformed_value)
# Store energy as property
result_bitfield.setProperty(x, y, z, t, 'kinetic_energy', float(energy))
return result_bitfield
```
## 9. Application Examples
### 9.1 Enhanced Data Compression Example
```python
def enhancedCompression(bitfield):
"""
Implement enhanced data compression using BitMatrix techniques.
Parameters:
bitfield (BitFieldND): Input bitfield to compress
Returns:
tuple: (compressed_data, metadata)
"""
# Step 1: Apply KTA transformation to increase compressibility
def energy_function(*coords):
# Simple energy function based on coordinate sum
return sum(coords) / (len(coords) * 10)
transformed_bitfield = ktaKineticTransform(bitfield, energy_function)
# Step 2: Apply quantum-inspired Fourier transform to identify patterns
fourier_bitfield = quantumFourierTransform(transformed_bitfield)
# Step 3: Extract high-magnitude frequency components
dimensions = fourier_bitfield.getDimensions()
high_magnitude_components = []
if len(dimensions) == 3:
x_dim, y_dim, z_dim = dimensions
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
magnitude = fourier_bitfield.getProperty(x, y, z, 'magnitude')
if magnitude > 0.1: # Threshold
high_magnitude_components.append((x, y, z, magnitude,
fourier_bitfield.getProperty(x, y, z, 'phase')))
elif len(dimensions) == 4:
x_dim, y_dim, z_dim, t_dim = dimensions
for t in range(t_dim):
for z in range(z_dim):
for y in range(y_dim):
for x in range(x_dim):
magnitude = fourier_bitfield.getProperty(x, y, z, t, 'magnitude')
if magnitude > 0.1: # Threshold
high_magnitude_components.append((x, y, z, t, magnitude,
fourier_bitfield.getProperty(x, y, z, t, 'phase')))
# Step 4: Apply run-length encoding to the transformed bitfield
rle_data = compressRLE(transformed_bitfield)
# Step 5: Apply LZW compression to the RLE data
lzw_data, dictionary = compressLZW(str(rle_data))
# Create metadata for decompression
metadata = {
'dimensions': dimensions,
'high_magnitude_components': high_magnitude_components,
'dictionary': dictionary
}
return lzw_data, metadata
def enhancedDecompression(compressed_data, metadata):
"""
Decompress data using BitMatrix enhanced decompression.
Parameters:
compressed_data (list): Compressed data
metadata (dict): Compression metadata
Returns:
BitFieldND: Decompressed bitfield
"""
# Extract metadata
dimensions = metadata['dimensions']
high_magnitude_components = metadata['high_magnitude_components']
dictionary = metadata['dictionary']
# Step 1: Apply LZW decompression
decompressed_lzw = decompressLZW(compressed_data, dictionary)
# Step 2: Parse RLE data and apply RLE decompression
# (This is a simplified version, actual implementation would need to parse the string representation)
rle_data = eval(decompressed_lzw) # Warning: eval is used for simplicity, not recommended in production
decompressed_rle = decompressRLE(rle_data)
# Step 3: Reconstruct bitfield from decompressed data
if len(dimensions) == 3:
reconstructed_bitfield = BitField3D(*dimensions)
# Fill with decompressed data
index = 0
for z in range(dimensions[2]):
for y in range(dimensions[1]):
for x in range(dimensions[0]):
if index < len(decompressed_rle):
reconstructed_bitfield.setBit(x, y, z, decompressed_rle[index])
index += 1
elif len(dimensions) == 4:
reconstructed_bitfield = BitField4D(*dimensions)
# Fill with decompressed data
index = 0
for t in range(dimensions[3]):
for z in range(dimensions[2]):
for y in range(dimensions[1]):
for x in range(dimensions[0]):
if index < len(decompressed_rle):
reconstructed_bitfield.setBit(x, y, z, t, decompressed_rle[index])
index += 1
# Step 4: Apply inverse KTA transformation
def inverse_energy_function(*coords):
# Inverse of the energy function used in compression
return sum(coords) / (len(coords) * 10)
decompressed_bitfield = ktaKineticTransform(reconstructed_bitfield, inverse_energy_function)
return decompressed_bitfield
```
### 9.2 Neural Network Pattern Recognition Example
```python
def neuralPatternRecognition(input_bitfield, pattern_library):
"""
Implement neural pattern recognition using BitMatrix techniques.
Parameters:
input_bitfield (BitFieldND): Input bitfield to analyze
pattern_library (dict): Library of known patterns
Returns:
dict: Recognition results
"""
# Step 1: Extract features from input bitfield
feature_size = (2, 2, 2)
features = extractFeatures(input_bitfield, feature_size)
# Step 2: Create neural network for pattern recognition
input_dimensions = input_bitfield.getDimensions()
hidden_dimensions = [64, 32]
output_dimensions = (len(pattern_library),)
network = createNeuralNetwork(input_dimensions, hidden_dimensions, output_dimensions)
# Step 3: Train network with pattern library
# (In a real implementation, this would be done beforehand)
# Here we'll simulate a pre-trained network
# Step 4: Apply neural network to input
output_bitfield = applyNeuralNetwork(input_bitfield, network)
# Step 5: Interpret results
results = {}
# Get activation values for each pattern
if len(output_dimensions) == 1:
for i in range(output_dimensions[0]):
pattern_name = list(pattern_library.keys())[i]
activation = output_bitfield.getProperty(i, 0, 0, 'activation')
results[pattern_name] = activation
# Sort patterns by activation (highest first)
sorted_results = sorted(results.items(), key=lambda x: x[1], reverse=True)
# Return top matches
return {
'top_matches': sorted_results[:3],
'features': len(features),
'confidence': sorted_results[0][1] if sorted_results else 0
}
```
### 9.3 Quantum-Inspired Computing Example
```python
def quantumInspiredAlgorithm(bitfield, iterations=10):
"""
Implement a quantum-inspired algorithm using BitMatrix techniques.
Parameters:
bitfield (BitFieldND): Input bitfield
iterations (int): Number of iterations
Returns:
BitFieldND: Processed bitfield
"""
dimensions = bitfield.getDimensions()
# Step 1: Create superposition of states
states, probabilities = createSuperposition(dimensions, num_states=4)
# Step 2: Apply quantum gates to each state
processed_states = []
for state in states:
# Apply X gate to specific bits
target_bits = [(0, 0, 0), (1, 1, 1)]
processed_state = applyQuantumGate(state, 'X', target_bits)
# Apply Hadamard gate to create superpositions
target_bits = [(2, 2, 2)]
processed_state = applyQuantumGate(processed_state, 'H', target_bits)
# Apply CNOT gate for entanglement
control_bits = [(0, 0, 0)]
target_bits = [(1, 1, 1)]
processed_state = applyQuantumGate(processed_state, 'CNOT', target_bits, control_bits)
processed_states.append(processed_state)
# Step 3: Apply quantum Fourier transform
fourier_states = [quantumFourierTransform(state) for state in processed_states]
# Step 4: Perform quantum walk
walked_states = []
for state in fourier_states:
walked_state = performQuantumWalk(state, steps=iterations)
walked_states.append(walked_state)
# Step 5: Measure superposition to get final state
final_state, selected_index = measureSuperposition(walked_states, probabilities)
return final_state
```
These implementation examples demonstrate how the BitMatrix Spatial Computing framework can be applied to solve various computational problems using its unique multidimensional approach and quantum-inspired techniques.