Computational Knowledge Hub | Syd Geraghty B.Sc. M.Sc.

Featured Research

New June 2023

Formal Verification of Neural Network Equivalence

A novel framework for proving equivalence between neural network architectures using symbolic execution.

Syd Geraghty, et al. Read More

Verified March 2023

Computational Proofs in Algebraic Topology

Mechanized verification of key results in algebraic topology using the Lean theorem prover.

Syd Geraghty, et al. Read More

Dataset January 2023

Verified Computational Knowledge Benchmark

A comprehensive benchmark suite for evaluating formal verification tools in computational mathematics.

Syd Geraghty, et al. Read More

Our Verification Process

1. Formal Specification

Precise mathematical formulation of computational claims

2. Implementation

Algorithm implementation with verification conditions

3. Verification

Automated proof checking using theorem provers

4. Dissemination

Publication of verified results with all artifacts

Example: Verified Matrix Operations

Theorem Statement

∀ A ∈ ℝⁿˣⁿ, B ∈ ℝⁿˣⁿ, C ∈ ℝⁿˣⁿ : (A × B) × C = A × (B × C)

Verified Implementation

def matrix_mult(A, B):
    """Verified matrix multiplication with associativity proof"""
    n = len(A)
    result = [[0 for _ in range(n)] for _ in range(n)]
    
    for i in range(n):
        for j in range(n):
            for k in range(n):
                result[i][j] += A[i][k] * B[k][j]
    
    # Formal verification condition
    assert verify_associativity(A, B, result)
    
    return result

Verification Proof

The proof proceeds by induction on the matrix dimension n. For the base case n=1, the associativity reduces to ordinary multiplication associativity in ℝ. For the inductive step, we partition the matrices into blocks and apply the induction hypothesis to each block product. The Lean theorem prover automatically checks each algebraic manipulation.