Canonical form and axiomatic soundness for the hypergraph extension

Develop a canonical normal form for the proposed hypergraph algebra obtained by extending the edge graph algebra via numerically labelling edge ends instead of using the pits/tips distinction, and establish that the corresponding axiom system is sound and complete with respect to the intended hypergraph semantics.

Background

The paper introduces a new algebraic representation for edge-indexed graphs and proves soundness and completeness of its axioms with respect to a flow-based semantics. In the conclusion, the authors discuss extending this framework beyond edge graphs.

They propose a concrete route to generalize the algebra to hypergraphs by numerically labelling edge ends, replacing the pits/tips notion. While they have an initial formulation, they explicitly state that two key components are unresolved: producing a canonical form for the extended algebra and proving the axiom system’s soundness and completeness.

References

We are confident that our edge graph algebra can be extended to a hypergraph algebra by numerically labelling the ends of the graph edges instead of using the names pits and tips. We have the beginnings of an elegant formulation of the algebra in this format, but finding a canonical form and validating the axioms are sound and complete remains future work.

— Let a Thousand Flowers Bloom: An Algebraic Representation for Edge Graphs (2403.02273 - Liell-Cock et al., 4 Mar 2024) in Section 7 (Conclusion), page 9:22