Deciding Conditional Termination

TL;DR

This paper introduces a method to determine initial program configurations that guarantee termination by analyzing the greatest fixpoint of the transition relation, with decidability results for specific relation classes.

Contribution

It defines a dual set for conditional termination, providing a way to compute non-termination preconditions for deterministic and well-founded relations.

Findings

Decidable termination for loops with octagonal relations.

Decidable termination for loops with finite monoid affine relations.

A method to compute the weakest non-termination precondition.

Abstract

We address the problem of conditional termination, which is that of defining the set of initial configurations from which a given program always terminates. First we define the dual set, of initial configurations from which a non-terminating execution exists, as the greatest fixpoint of the function that maps a set of states into its pre-image with respect to the transition relation. This definition allows to compute the weakest non-termination precondition if at least one of the following holds: (i) the transition relation is deterministic, (ii) the descending Kleene sequence overapproximating the greatest fixpoint converges in finitely many steps, or (iii) the transition relation is well founded. We show that this is the case for two classes of relations, namely octagonal and finite monoid affine relations. Moreover, since the closed forms of these relations can be defined in Presburger arithmetic, we obtain the decidability of the termination problem for such loops.