Analysing Temporal Reasoning in Description Logics Using Formal Grammars

We establish a correspondence between (fragments of) $\mathcal{TEL}^\bigcirc$, a temporal extension of the $\mathcal{EL}$ description logic with the LTL operator $\bigcirc^k$, and some specific kinds of formal grammars, in particular, conjunctive grammars (context-free grammars equipped with the operation of intersection). This connection implies that $\mathcal{TEL}^\bigcirc$ does not possess the property of ultimate periodicity of models, and further leads to undecidability of query answering in $\mathcal{TEL}^\bigcirc$, closing a question left open since the introduction of $\mathcal{TEL}^\bigcirc$. Moreover, it also allows to establish decidability of query answering for some new interesting fragments of $\mathcal{TEL}^\bigcirc$, and to reuse for this purpose existing tools and algorithms for conjunctive grammars.