Modular and Certified Semantic Labeling and Unlabeling

Abstract

Semantic labeling is a powerful transformation technique to prove termination of term rewrite systems. The dual technique is unlabeling. For unlabeling it is essential to drop the so called decreasing rules which sometimes have to be added when applying semantic labeling.  We indicate two problems concerning unlabeling and present our solutions.

The first problem is that currently unlabeling cannot be applied as a modular step, since the decreasing rules are determined by a semantic labeling step which may have taken place much earlier. To this end, we give an implicit definition of decreasing rules that does not depend on any knowledge about preceding labelings.

The second problem is that unlabeling is in general unsound.  To solve this issue, we introduce the notion of extended termination problems.  Moreover, we show how existing termination techniques can be lifted to operate on extended termination problems.

All our proofs have been formalized in Isabelle/HOL as part of the IsaFoR/CeTA project.