Bounded Termination of Monotonicity-Constraint Transition Systems

Intuitively, if we can prove that a program terminates, we expect some conclusion regarding its complexity. But the passage from termination proofs to complexity bounds is not always clear. In this work we consider Monotonicity Constraint Transition Systems, a program abstraction where termination is decidable (based on the size-change termination principle). We show that these programs also have a decidable complexity property: one can determine whether the length of all transition sequences can be bounded in terms of the initial state. This is the bounded termination problem. Interestingly, if a bound exists, it must be polynomial. We prove that the bounded termination problem is PSPACE-complete and, moreover, if a bound exists then a symbolic bound which is constant-factor tight (in the univariate case) can be computed in PSPACE. We present this computation in the form of computing a reachability bound, a bound on the number of visits to a given program location. This presentation is inspired by the practical usefulness of this problem formulation. We also discuss, theoretically, the use of bounds on the abstract program to infer conclusions on a concrete program that has been abstracted. The conclusion maybe a polynomial time bound, or in other cases polynomial space or exponential time. We argue that the monotonicity-constraint abstraction promises to be useful for practical complexity analysis of programs.