The signature of the formal language of mereotopology contains two predicates $P$ and $C$, which stand for “being a part of” and “contact,” respectively. This paper will deal with the decidability issue of the mereotopological theories which can be formed by the axioms found in the literature. Three main results to be given are as follows: all axiomatized mereotopological theories are separable; all mereotopological theories up to $\mathbf{ACEMT}$, $\mathbf{SACEMT}$, or $\mathbf{SACEMT}^{\prime}$ are finitely inseparable; all axiomatized mereotopological theories except $\mathbf{SAX}$, $\mathbf{SAX}^{\prime}$, (...) or $\mathbf{S\overline{B}X}^{\prime}$, where $\mathbf{X}$ is strictly stronger than $\mathbf{CEMT}$, are undecidable. Then it can also be easily seen that all axiomatized mereotopological theories proved to be undecidable here are neither essentially undecidable nor strongly undecidable but are hereditarily undecidable. Result will be shown by constructing strongly undecidable mereotopological structures based on two-dimensional Euclidean space, and it will be pointed out that the same construction cannot be carried through if the language is not rich enough. (shrink)

We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that the two systems (...) are dual to each other. The duality is expressed in a formal way which enables us to define a transformation of proofs in one of the systems into the proofs of the other. (shrink)

We present relational proof systems for the four groups of theories of spatial reasoning: contact relation algebras, Boolean algebras with a contact relation, lattice-based spatial theories, spatial theories based on a proximity relation.

We describe a relational framework that uniformly supports formalization and automated reasoning in varied propositional modal logics. The proof system we propose is a relational variant of the classical Rasiowa-Sikorski proof system. We introduce a compact graph-based representation of formulae and proofs supporting an efficient implementation of the basic inference engine, as well as of a number of refinements. Completeness and soundness results are shown and a Prolog implementation is described.