From Termination to Complexity Analysis with Monotonicity Constraints

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 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 bounded in terms of the initial state. We show that if it can be bounded, the bound is polynomial. Depending on how the abstract program represents a concrete program, such a bound may have different implications on the concrete program’s complexity. In fact, we can characterize PTIME, PSPACE and EXPTIME by appropriate abstractions plus the bounded termination criterion. We argue that, augmented with certain heuristics for efficiency and precision, this approach may be useful for practical complexity analysis of programs. Keywords-Cost Analysis; Size-change termination; Implicit Computational Complexity