Limit Operators and Convergence Measures for $\omega$-Languages with Applications to Extreme Fairness

Abstract

Methods of program verification for liveness and fairness rely on measuring "progress" of finite computations towards satisfying the specification. Previous methods are based on explaining progress in terms of well-founded sets. These approaches, however, often led to complicated transformations or inductive applications of proof rules. Our main contribution is a simpler measure of progress based on an ordering that is not well-founded. This ordering is a variation on the Kleene-Brouwer ordering on nodes of a finite-path tree. It has the unusual property that for any infinite ordered sequence of nodes, the liminf of the node depths (levels) is finite. This novel view of progress gives a precise mathematical characterization of what it means to satisfy very general program properties. In particular, we solve the problem of finding a progress measure for termination under extreme fairness.