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VerifyThis 2016: Strassen's Matrix Multiplication

This is a complete solution to VerifyThis 2016 challenge 1. The details are described in the paper The Matrix Reproved.

A self-contained solution to task 1 can be found in file naive.mlw Other files solve task 2 and 3:

To replay proofs:

Challenge text:

Challenge 1: Matrix Multiplication

Consider the following pseudocode algorithm, which is naive implementation of matrix multiplication. For simplicity we assume that the matrices are square.

 int[][] matrixMultiply(int[][] A, int[][] B) {
    int n = A.length;

    // initialise C
    int[][] C = new int[n][n];

    for (int i = 0; i < n; i++) {
           for (int k = 0; k < n; k++) {
                   for (int j = 0; j < n; j++) {
                       C[i][j] += A[i][k] * B[k][j];
                   }
           }
    }
    return C;
 }

Tasks.
1. Provide a specification to describe the behaviour of this algorithm, and prove
   that it correctly implements its specification.

2. Show that matrix multiplication is associative, i.e., the order in which
   matrices are multiplied can be disregarded: A(BC) = (AB)C. To show this,
   you should write a program that performs the two different computations,
   and then prove that the result of the two computations is always the same.

3. [Optional, if time permits] In the literature, there exist many proposals
   for more efficient matrix multiplication algorithms. Strassen’s algorithm
   was one of the first. The key idea of the algorithm is to use a recursive
   algorithm that reduces the number of multiplications on submatrices
   (from 8 to 7), see https://en.wikipedia.org/wiki/Strassen_algorithm for an
   explanation. A relatively clean Java implementation (and Python and C++)
   can be found here: https://martin-thoma.com/strassen-algorithm-in-python-java-cpp/.
   Prove that the naive algorithm above has the same behaviour as Strassen’s
   algorithm. Proving it for a restricted case, like a 2x2 matrix should be
   straightforward, the challenge is to prove it for arbitrary matrices with size 2^n.

Authors: Martin Clochard / Léon Gondelman / Mário Pereira

Topics: Matrices / Proofs by reflection

Tools: Why3

References: VerifyThis @ ETAPS 2016

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