Boolean matrix multiplication and transitive closure

Arithmetic operations on matrices are applied to the problem of finding the transitive closure of a Boolean matrix. The best transitive closure algorithm known, due to Munro, is based on the matrix multiplication method of Strassen. We show that his method requires at most O(n α · P(n)) bitwise operations, where α = log 2 7 and P(n) bounds the number of bitwise operations needed for arithmetic modulo n+1. The problems of computing the transitive closure and of computing the "and-or" product of Boolean matrices are shown to be of the same order of difficulty. A transitive closure method based on matrix inverse is presented which can be used to derive Munro's method.