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Instantiation of Jerboa Rule Schemes, a Set-based Explanation

Romain Pascual

TL;DR

A lightweight, set-theoretic description that exploits the proximity between presheaf topoi and sets to provide an explanation that does not rely on extensive theoretical background is proposed.

Abstract

This report presents a set-theoretic framework for the instantiation of rule schemes in the Jerboa platform, a tool for developing domain-specific geometric modelers. Jerboa enables the design of geometric modeling operations as graph transformation rules generalized to rule schemes for genericity over the topological content of the operations. Current approaches to algebraic graph transformations are typically described within a finitary $\mathcal{M}$-adhesive category (where $\mathcal{M}$ is a suitable system of monomorphisms), employing compositional double-pushout (DPO) semantics for rewriting. In this report, we propose a lightweight, set-theoretic description that exploits the proximity between presheaf topoi and sets to provide an explanation that does not rely on extensive theoretical background. The proposed method simplifies the formal description of modeling operations to bridge the gap between abstract concepts and their practical application in geometric modeling. The framework offers a complementary perspective to categorical approaches at the foundation of Jerboa.

Instantiation of Jerboa Rule Schemes, a Set-based Explanation

TL;DR

A lightweight, set-theoretic description that exploits the proximity between presheaf topoi and sets to provide an explanation that does not rely on extensive theoretical background is proposed.

Abstract

This report presents a set-theoretic framework for the instantiation of rule schemes in the Jerboa platform, a tool for developing domain-specific geometric modelers. Jerboa enables the design of geometric modeling operations as graph transformation rules generalized to rule schemes for genericity over the topological content of the operations. Current approaches to algebraic graph transformations are typically described within a finitary -adhesive category (where is a suitable system of monomorphisms), employing compositional double-pushout (DPO) semantics for rewriting. In this report, we propose a lightweight, set-theoretic description that exploits the proximity between presheaf topoi and sets to provide an explanation that does not rely on extensive theoretical background. The proposed method simplifies the formal description of modeling operations to bridge the gap between abstract concepts and their practical application in geometric modeling. The framework offers a complementary perspective to categorical approaches at the foundation of Jerboa.

Paper Structure

This paper contains 12 sections, 5 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Representing Objects as Generalized Maps
  3. Topological Representation
  4. Topological Cells and Orbits
  5. Geometric Information and Embedding
  6. A Folded Representation of Modeling Operations
  7. Relabeling Function
  8. Instantiation
  9. Nodes
  10. Arcs
  11. Rule Scheme
  12. Conclusion

Figures (8)

  • Figure 1: Variations of topology and geometry: \ref{['fig:topovsgeom:object']} an cube, \ref{['fig:topovsgeom:geom']} same topology as \ref{['fig:topovsgeom:object']} but with a different geometry, and \ref{['fig:topovsgeom:topo']} same geometry as \ref{['fig:topovsgeom:object']} but with a different topology.
  • Figure 2: Gmap construction: \ref{['fig:object:1']} 2D object, \ref{['fig:object:2']} darts ($\bullet$), \ref{['fig:object:3']}$0$-arcs ($\mathbin{\bullet\mkern-2mu{}\mkern-2mu\bullet}$), \ref{['fig:object:4']}$1$-arcs ($\mathbin{\bullet\mkern-2mu{}\mkern-2mu\bullet}$), \ref{['fig:object:5']}$2$-arcs ($\mathbin{\bullet\mkern-2mu{}\mkern-2mu\bullet}$). Cells: \ref{['fig:cells:1']}$\langle \textcolor{alpha1}{ $1$ }\xspace, \textcolor{alpha2}{ $2$ }\xspace \rangle$-orbit (vertices), \ref{['fig:cells:2']}$\langle \textcolor{alpha0}{ $0$\xspace }, \textcolor{alpha2}{ $2$ }\xspace \rangle$-orbit (edges), \ref{['fig:object:4']}$\langle \textcolor{alpha0}{ $0$\xspace }, \textcolor{alpha1}{ $1$ }\xspace \rangle$-orbit (faces).
  • Figure 3: Embeddings: \ref{['fig:ebd:1']} embedded Gmap, \ref{['fig:ebd:2']}$\mathop{\mathrm{\mathtt{position}}}\nolimits {\,\colon\,}\newline \langle \textcolor{alpha1}{ $1$ }\xspace, \textcolor{alpha2}{ $2$ }\xspace \rangle \to \mathop{\mathrm{\mathtt{Point3}}}\nolimits$, \ref{['fig:ebd:3']}$\mathop{\mathrm{\mathtt{color}}}\nolimits {\,\colon\,}\newline \langle \textcolor{alpha0}{ $0$\xspace }, \textcolor{alpha1}{ $1$ }\xspace \rangle \to \mathop{\mathrm{\mathtt{ColorRGB}}}\nolimits$.
  • Figure 4: Graph transformation rules for the vertex insertion. Graph transformation rule for the vertex insertion in a free edge \ref{['fig:ajoutSommet1']} and its application on a $2$-Gmap on an outer edge \ref{['fig:ajoutSommet2']} via the match deduced from $x \mapsto a$. Graph transformation rule for the vertex insertion in a sewn edge \ref{['fig:ajoutSommet3']} and its application on a $2$-Gmap on an inner edge \ref{['fig:ajoutSommet4']} via the match deduced from $x \mapsto e$.
  • Figure 5: rule schemes for the vertex insertion: \ref{['fig:vertexinsertion2folded']} by folding the $2$-links and \ref{['fig:vertexinsertion02folded']} both the $0$- and $2$-links.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Definition 2.1: Generalized map -- adapted from pascual_topological_2022
  • Definition 2.2: Orbit -- adapted from pascual_topological_2022
  • Definition 4.1: Relabeling function -- from pascual_inferring_2022
  • Definition 4.2: Graph scheme, rule scheme -- from pascual_inferring_2022
  • Definition 5.1: Node instantiation -- from pascual_inferring_2022
  • Definition 5.2: Arc instantiation -- from pascual_inferring_2022