A Logic of East and West

We propose a logic of east and west (LEW) for points in 1D Euclidean space. It formalises primitive direction relations: east (E), west (W) and indeterminate east/west (I_ew). It has a parameter τ ∈ N_>1, which is referred to as the level of indeterminacy in directions. For every τ ∈ N_>1, we provide a sound and complete axiomatisation of LEW, and prove that its satisfiability problem is NP-complete. In addition, we show that the finite axiomatisability of LEW depends on τ: if τ = 2 or τ = 3, then there exists a finite sound and complete axiomatisation; if τ > 3, then the logic is not finitely axiomatisable. LEW can be easily extended to higher-dimensional Euclidean spaces. Extending LEW to 2D Euclidean space makes it suitable for reasoning about not perfectly aligned representations of the same spatial objects in different datasets, for example, in crowd-sourced digital maps.