The expressive power of first-order topological languages

Abstract

We consider the expressive power of the first-order structure 〈Ω,C 〉 where Ω is either of two of different domains of extended regions in Euclidean space, and C(x,y) is the topological relation ‘Region x is in contact with region y.’ We prove two main theorems: Let P[ℚ] be the domain of bounded, non-empty, rational polyhedra in two- or three-dimensional Euclidean space. A relation Γ over P[ℚ] is definable in the structure 〈P[ℚ], C〉 if and only if Γ is arithmetic and invariant under rational PL-homeomorphisms of the space to itself. We also extend this result to a number of other domains, including the domain of all polyhedra and the domain of semi-algebraic regions. Let R be the space of bounded, non-empty, closed regular regions in n-dimensional Euclidean space. Any analytical relation over lower dimensional (i.e. empty interior) compact point sets that is invariant under homeomorphism is implicitly definable in the structure 〈R⁠,C〉.