Counting Paths: Nondeterminism as Linear Algebra

Nondeterminism is considered to be ignorance about the actual state transition sequence performed during a computation. The number of distinct potential paths from state i to j forms a matrix [nij]. The behavior of a nondeterministic program is defined to be this multiplicity matrix of the state transitions. The standard programming constructs have behaviors defined in terms of the behaviors of their constituents using matrix addition and multiplication only. The spectral radius of the matrix assigned to an iterating component characterizes its convergence. The spectral radius is shown to be either 0 or else ¿ 1. The program converges iff the spectral radius is zero, diverges deterministically iff the spectral radius is one, and has a proper nondeterministic divergence iff the spectral radius exceeds one. If the machine has an infinite number of states the characterization of convergence is given graph theoretically. The spectral radii of synchronous and interleaved parallel noncommunicating systems are easily computed in terms of the spectral radii of the components.