In this paper we present a computability and a complexity result on Relational Speciate Reentrant Logic (RSRL). RSRL is a description logic designed to formalise the linguistic framework and theory Head-Driven Phrase Structure Grammar. We show here that given an RSRL-formula and a finite RSRL-interpretation it is in general not decidable if the formula is true in the given interpretation by reduction to Post Correspondence Problems. For so-called chainless RSRL, a semantically weaker version in which the expressive power of RSRL is significantly reduced, we show that if a class of finite structures is definable in chainless RSRL it is decidable by a Turing machine polynomially time bounded in the size of the input structures.
Kepser, S. (2004). On the complexity of RSRL. In Electronic Notes in Theoretical Computer Science (Vol. 53, pp. 146–162). https://doi.org/10.1016/S1571-0661(05)82580-0