From Geometry to Inference: Qualitative Reasoning in the QRPC Spatiotemporal Calculus

The Qualitative Rectilinear Projection Calculus (QRPC) is a binary qualitative framework that represents spatiotemporal relations between oriented objects through a set of 48 jointly exhaustive and pairwise disjoint basic relations, each defined by six qualitative components encoding directional and positional aspects of their interaction. Despite its expressive power, QRPC originally lacked an operational framework for inference. This work addresses that gap by formally defining and implementing the converse and composition operations, and by presenting a closed-form geometric classification model that determines the basic relation between two objects in constant time from six scalar quantities derived from their heading vectors and displacement vector — providing the operational bridge from sensor measurements to the QRPC relational vocabulary. Composition is handled through a constructive algorithm that derives any of the 2304 pairwise compositions from a minimal generating subset of canonical representatives, exploiting seven transformation rules based on π-rotational symmetries to avoid exhaustive enumeration. The resulting composition table is shown to satisfy path-consistency conditions by construction, as confirmed computationally across every ordered triplet in the full composition space, while the converse operator is shown to be strong and involutive. Although universal associativity does not hold, as is characteristic of weakly represented calculi, the non-associative cases follow a structured pattern: they are strongly correlated with geometrically degenerate boundary relations, the divergence between alternative groupings is small and bounded, and the two groupings are never entirely disjoint. QRPC's theoretical properties are validated empirically by applying the PC-2 pathconsistency algorithm to randomly generated constraint networks under the A(n,d,l) random instance generation model, characterising inferential behaviour under varying network size, density, and label count. The complete theoretical framework and its verification procedures are supported by an interactive tool designed for exploration and reproducibility, whose implementation is provided as open-source software. Future work includes developing a comprehensive inference engine for QRPC, identifying tractable fragments and exploring hybrid reasoning approaches that combine PC-2, stronger consistency algorithms and machine learning techniques for applications such as autonomous navigation and dynamic event prediction.