Monotonicity Constraints for Termination in the Integer Domain

TL;DR

This paper extends the theory of size-change termination to the integer domain using monotonicity constraints, providing decision procedures and methods to construct global ranking functions with optimal complexity.

Contribution

It develops a theoretical foundation and decision procedures for using monotonicity constraints to prove termination in the integer domain, which is not well-founded.

Findings

Established decision procedures for termination using monotonicity constraints in the integer domain.

Presented a method to construct global ranking functions in singly-exponential time.

Demonstrated the applicability of monotonicity constraints beyond well-founded domains.

Abstract

Size-Change Termination (SCT) is a method of proving program termination based on the impossibility of infinite descent. To this end we use a program abstraction in which transitions are described by Monotonicity Constraints over (abstract) variables. When only constraints of the form x>y' and x\geq y' are allowed, we have size-change graphs. In the last decade, both theory and practice have evolved significantly in this restricted framework. The crucial underlying assumption of most of the past work is that the domain of the variables is well-founded. In a recent paper I showed how to extend and adapt some theory from the domain of size-change graphs to general monotonicity constraints, thus complementing previous work, but remaining in the realm of well-founded domains. However, monotonicity constraints are, interestingly, capable of proving termination also in the integer domain, which is not well-founded. The purpose of this paper is to explore the application of monotonicity constraints in this domain. We lay the necessary theoretical foundation, and present precise decision procedures for termination; finally, we provide a procedure to construct explicit global ranking functions from monotonicity constraints in singly-exponential time, and of optimal worst-case size and dimension (ordinal).