🤖 AI Summary
To address limitations in engineering and systems modeling—including insufficient expressiveness of binary relations, implicit semantics, and difficulty representing multilevel structures in diagrammatic notations—this paper proposes a hypernetwork modeling paradigm grounded in *n*-ary relations. Methodologically, it introduces typed hyper-simplices (alpha/beta) to enable explicit semantic binding and defines five semantics-preserving, deterministic structural operators (merge, meet, difference, prune, split), integrated with boundary operators and an ordered role mechanism to form a mechanizable structural algebra. The framework uniformly supports modeling, comparison, decomposition, and reconstruction of both hierarchical and non-hierarchical systems. Under the open-world assumption, it guarantees decidable reasoning and model executability. This work achieves a critical transition from symbolic representation to verifiable, reconstructable system models.
📝 Abstract
Modelling across engineering, systems science, and formal methods remains limited by binary relations, implicit semantics, and diagram-centred notations that obscure multilevel structure and hinder mechanisation. Hypernetwork Theory (HT) addresses these gaps by treating the n-ary relation as the primary modelling construct. Each relation is realised as a typed hypersimplex - alpha (conjunctive, part-whole) or beta (disjunctive, taxonomic) - bound to a relation symbol R that fixes arity and ordered roles. Semantics are embedded directly in the construct, enabling hypernetworks to represent hierarchical and heterarchical systems without reconstruction or tool-specific interpretation. This paper presents the structural kernel of HT. It motivates typed n-ary relational modelling, formalises the notation and axioms (A1-A5) for vertices, simplices, hypersimplices, boundaries, and ordering, and develops a complete algebra of structural composition. Five operators - merge, meet, difference, prune, and split - are defined by deterministic conditions and decision tables that ensure semantics-preserving behaviour and reconcile the Open World Assumption with closure under rules. Their deterministic algorithms show that HT supports reproducible and mechanisable model construction, comparison, decomposition, and restructuring. The resulting framework elevates hypernetworks from symbolic collections to structured, executable system models, providing a rigorous and extensible foundation for mechanisable multilevel modelling.