Ranking Functions for Size-Change Termination II

TL;DR

This paper introduces simpler algorithms for deriving global ranking functions in size-change termination proofs, reducing complexity bounds and providing insights into the scope and optimality of the method.

Contribution

It presents more transparent algorithms for size-change termination, lowering the upper bound on ranking expression size and establishing exponential lower bounds for complexity.

Findings

Improved upper bound on ranking expression size from triply exponential to singly exponential.

Introduction of a framework for lower bounds on ranking expression complexity.

Proof of exponential lower bounds demonstrating the optimality of the results.

Abstract

Size-Change Termination is an increasingly-popular technique for verifying program termination. These termination proofs are deduced from an abstract representation of the program in the form of "size-change graphs". We present algorithms that, for certain classes of size-change graphs, deduce a global ranking function: an expression that ranks program states, and decreases on every transition. A ranking function serves as a witness for a termination proof, and is therefore interesting for program certification. The particular form of the ranking expressions that represent SCT termination proofs sheds light on the scope of the proof method. The complexity of the expressions is also interesting, both practicaly and theoretically. While deducing ranking functions from size-change graphs has already been shown possible, the constructions in this paper are simpler and more transparent than previously known. They improve the upper bound on the size of the ranking expression from triply exponential down to singly exponential (for certain classes of instances). We claim that this result is, in some sense, optimal. To this end, we introduce a framework for lower bounds on the complexity of ranking expressions and prove exponential lower bounds.