  Existential rules, long known as tuple-generating dependencies in database
theory, have been intensively studied in the last decade as a powerful
formalism to represent ontological knowledge in the context of ontology-based
query answering. A knowledge base is then composed of an instance that contains
incomplete data and a set of existential rules, and answers to queries are
logically entailed from the knowledge base. This brought again to light the
fundamental chase tool, and its different variants that have been proposed in
the literature. It is well-known that the problem of determining, given a chase
variant and a set of existential rules, whether the chase will halt on any
instance, is undecidable. Hence, a crucial issue is whether it becomes
decidable for known subclasses of existential rules. In this work, we consider
linear existential rules, a simple yet important subclass of existential rules
that generalizes inclusion dependencies. We show the decidability of the all
instance chase termination problem on linear rules for three main chase
variants, namely semi-oblivious, restricted and core chase. To obtain these
results, we introduce a novel approach based on so-called derivation trees and
a single notion of forbidden pattern. Besides the theoretical interest of a
unified approach and new proofs, we provide the first positive decidability
results concerning the termination of the restricted chase, proving that chase
termination on linear existential rules is decidable for both versions of the
problem: Does every fair chase sequence terminate? Does some fair chase
sequence terminate?
