First-Order Sketch Conditions and Constraints -- A Category Independent Approach

TL;DR

This paper introduces a category-independent framework for first-order sketch conditions and constraints within the Diagram Predicate Framework, enabling formalization and reasoning about diagrammatic models across various categories.

Contribution

It generalizes existing graph conditions and constraints to arbitrary categories using functors, providing a unified, formal approach for diagrammatic modeling and reasoning.

Findings

Defines general first-order sketch conditions and constraints for arbitrary categories

Demonstrates how sketch constraints can describe and deduce structural properties

Uses Category Theory as an example to illustrate the framework

Abstract

Generalizing different variants of "graph conditions and constraints" as well as "universal constraints" and "negative universal constraints" in the Diagram Predicate Framework (DPF), we introduce for arbitrary categories $\mathbf{Cxt}$ and "statement" functors $\mathtt{Stm}:\mathbf{Cxt}\to\mathbf{Set}$ general first-order sketch conditions and constraints. Sketches are used in DPF to formalize different kinds of diagrammatic software models. We discuss the use of sketch constraints for describing the syntactic structure of sketches. We outline the use of sketch constraints to deduce knowledge implicitly given in a sketch as well as procedures to deduce sketch constraints from given sketch constraints. We use the simple but paradigmatic modeling formalism "Category Theory" as running example.