Position Spaces and Graphs

2026-06-24

AI Summary

This study addresses the formal modeling of relative positional relationships among discrete symbols, balancing logical consistency with computational tractability. To this end, the authors propose a “position graph” framework grounded in positional space theory, employing two strict partial orders to capture horizontal and vertical alignment and precedence relations. Symbol arrangements are constrained by chain conditions and row-column compatibility requirements. The model preserves expressive power while guaranteeing rigorous algebraic consistency. Key contributions include establishing necessary and sufficient conditions for position graph consistency, proving that the induced subgraph isomorphism problem is NP-complete—thereby revealing the intrinsic computational complexity of structural pattern discovery—and constructing a formal logical layer independent of specific data extraction techniques.

Abstract

In this paper, we introduce position graphs, a graph-based reasoning framework based on the formalization of position spaces. This framework utilizes two strict partial orders, representing horizontal and vertical alignment and precedence, to model the relative positions of discrete tokens. Unlike general qualitative spatial calculi, position graphs are constrained by a chain condition and compatibility requirements that focus on rows and columns. We provide a comprehensive theoretical analysis of this representation, beginning with a characterization of graph consistency. Conditions to ensure the consistency of position graphs are established. Furthermore, we investigate the computational complexity of structural pattern discovery, modeled as the induced subgraph isomorphism problem. We demonstrate that this problem remains NP-complete even within the restricted class of position graphs. While initially motivated by document processing, this work focuses on the underlying mathematical properties and algebraic consistency of position-based constraints, providing a formal logical layer that is independent of specific data extraction techniques.

Problem

• position graphs
• qualitative spatial reasoning
• graph consistency
• computational complexity
• spatial constraints

Innovation

• position graphs
• qualitative spatial reasoning
• partial orders
• graph consistency
• induced subgraph isomorphism