The chapter discusses a theorem by Stephen Warshall on the transformation of boolean matrices. Given a boolean matrix \( M \) of size \( d \times d \), the goal is to transform it into a new matrix \( M' \) such that \( m'_{ij} = 1 \) if and only if either \( m_{ij} = 1 \) or there exists a sequence of indices \( k_1, k_2, \ldots, k_n \) where \( m_{ik_1} = m_{k_1 k_2} = \cdots = m_{k_{n-1} k_n} = m_{k_n j} = 1 \). The transformation is achieved through a series of steps involving boolean products and sums. Warshall proves that the algorithm for this transformation runs in time slightly faster than the square of \( d \). The proof involves showing that the constructed matrix \( M^* \) is equivalent to \( M' \) by demonstrating that if \( m_{ij} = 1 \), then \( m'_{ij} = 1 \), and vice versa. The proof also includes a detailed argument to show that if \( m_{ij} = 0 \) but there exists a sequence of indices satisfying the condition for \( m'_{ij} = 1 \), it leads to a contradiction, thus proving the validity of the transformation.The chapter discusses a theorem by Stephen Warshall on the transformation of boolean matrices. Given a boolean matrix \( M \) of size \( d \times d \), the goal is to transform it into a new matrix \( M' \) such that \( m'_{ij} = 1 \) if and only if either \( m_{ij} = 1 \) or there exists a sequence of indices \( k_1, k_2, \ldots, k_n \) where \( m_{ik_1} = m_{k_1 k_2} = \cdots = m_{k_{n-1} k_n} = m_{k_n j} = 1 \). The transformation is achieved through a series of steps involving boolean products and sums. Warshall proves that the algorithm for this transformation runs in time slightly faster than the square of \( d \). The proof involves showing that the constructed matrix \( M^* \) is equivalent to \( M' \) by demonstrating that if \( m_{ij} = 1 \), then \( m'_{ij} = 1 \), and vice versa. The proof also includes a detailed argument to show that if \( m_{ij} = 0 \) but there exists a sequence of indices satisfying the condition for \( m'_{ij} = 1 \), it leads to a contradiction, thus proving the validity of the transformation.