Prism: Symbolic Superoptimization of Tensor Programs

1. 1
   Understand Symbolic Tensor Program Definition

   Grasp the concept of defining tensor programs using symbolic parameters (e.g., 'B' for batch size, 'D' for dimension) instead of fixed concrete values. This allows for generalized optimization across varying inputs.
2. 2
   Define a Conceptual Tensor Operation

   Outline a fundamental tensor operation, such as matrix multiplication or convolution, using symbolic dimensions. This represents the abstract input that a superoptimizer like Prism would process.
3. 3
   Simulate Superoptimizer Input

   Conceptually feed this symbolic tensor program definition into a hypothetical Prism-like superoptimizer. Imagine the system receiving this high-level, symbolic graph (sGraph).
4. 4
   Envision Two-Level Optimization

   Understand that the optimizer would apply a two-level strategy: first, high-level structural transformations (like operator fusion), then low-level instruction selection and scheduling for a target hardware.
5. 5
   Interpret Optimized Output

   Envision receiving an optimized program representation or highly efficient, hardware-specific code generated from your symbolic input, achieving performance gains without manual tuning.

import numpy as np

# This snippet conceptually defines a tensor operation using symbolic placeholders.
# A superoptimizer like Prism would take such a definition as input to optimize.

# Define symbolic dimensions (Prism would process these abstractly)
BATCH_SIZE = 'B' # Symbolic batch size
INPUT_DIM = 'D_in' # Symbolic input dimension
HIDDEN_DIM = 'D_hidden' # Symbolic hidden dimension

print(f"Conceptual Tensor Program Definition for Superoptimization:")
print(f"  Input Tensor: (Batch={BATCH_SIZE}, Features={INPUT_DIM})")
print(f"  Weight Matrix: (Features={INPUT_DIM}, Hidden_Units={HIDDEN_DIM})")

# Represent a matrix multiplication operation conceptually
# This is the 'program' Prism would analyze and optimize.
conceptual_operation = f"MatMul(Input({BATCH_SIZE}, {INPUT_DIM}), Weights({INPUT_DIM}, {HIDDEN_DIM})) -> Output({BATCH_SIZE}, {HIDDEN_DIM})"

print(f"  Operation: {conceptual_operation}")
print(f"  Followed by a non-linear activation (e.g., ReLU).")
print(f"\nThis symbolic representation is what Prism would 'read' and transform into an optimal execution plan.")

# Example of how concrete values would be used AFTER optimization or for testing
# For Prism, the optimization happens BEFORE concrete values are known.
def run_with_concrete_values(batch, input_d, hidden_d):
    print(f"\n--- Running with Concrete Values ---")
    print(f"  Batch: {batch}, Input Dim: {input_d}, Hidden Dim: {hidden_d}")
    input_data = np.random.rand(batch, input_d)
    weights = np.random.rand(input_d, hidden_d)
    output = np.dot(input_data, weights)
    print(f"  Output shape: {output.shape}")

# Call the conceptual definition
run_with_concrete_values(32, 128, 256)