  Existential rules form an expressive Datalog-based language to specify
ontological knowledge. The presence of existential quantification in
rule-heads, however, makes the main reasoning tasks undecidable. To overcome
this limitation, in the last two decades, a number of classes of existential
rules guaranteeing the decidability of query answering have been proposed.
Unfortunately, only some of these classes fully encompass Datalog and, often,
this comes at the price of higher computational complexity. Moreover,
expressive classes are typically unable to exploit tools developed for classes
exhibiting lower expressiveness. To mitigate these shortcomings, this paper
introduces a novel general syntactic condition that allows us to define,
systematically and in a uniform way, from any decidable class $\mathcal{C}$ of
existential rules, a new class called Dyadic-$\mathcal{C}$ enjoying the
following properties: $(i)$ it is decidable; $(ii)$ it generalises Datalog;
$(iii)$ it generalises $\mathcal{C}$; $(iv)$ it can effectively exploit any
reasoner for query answering over $\mathcal{C}$; and $(v)$ its computational
complexity does not exceed the highest between the one of $\mathcal{C}$ and the
one of Datalog. Under consideration in Theory and Practice of Logic Programming
(TPLP).
