arXiv subjects

Romain Pascual

Publications and source records attributed to Romain Pascual.

3 recordsLinked to original sources

arXiv2024

Instantiation of Jerboa Rule Schemes, a Set-based Explanation

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.

cs.CG↗

arXiv2024

How Far Does Los's Theorem Extend To Kripke-Joyal Semantics? Sufficient Conditions, Counterexamples, and a Conjecture

Los's theorem, also known as the fundamental result of ultraproducts, states that the ultraproduct over a family of structures for the same language satisfies a first-order formula if and only if the set of indices for which the structures satisfy the formula belongs to the underlying ultrafilter. The associated notion of satisfaction is the Tarskian one via the elements of the set-theoretic structure that allow interpreting the formula. In the context of topoi, Kripke--Joyal semantics extends Tarski's notion to categorical logic. In this article, we investigate Los's theorem for first-order structures on locally presentable topoi with Kripke--Joyal semantics. More precisely, we identify a set of structural properties on the ambient topos (two-valuedness and projectivity of the terminal object) used to derive a categorical analog of Los's theorem. As an application, we derive compactness results for several fragments of first-order logic. We also show via explicit counterexamples that our structural assumptions are only sufficient. The counterexamples suggest an external approach and we conjecture that Los's property for full first-order logic is equivalent to the existence of a conservative family of logical points compatible with the ultraproduct construction.

cs.LO↗

arXiv2024

Ultraproducts in abstract categorical logic

In a previous publication, we introduced an abstract logic via an abstract notion of quantifier. Drawing upon concepts from categorical logic, this abstract logic interprets formulas from context as subobjects in a specific category, e.g., Cartesian, regular, or coherent categories, Grothendieck, or elementary toposes. We proposed an entailment system formulated as a sequent calculus which we proved complete. Building on this foundation, our current work explores model theory within abstract logic. More precisely, we generalize one of the most important and powerful classical model theory methods, namely the ultraproduct method, and show its fundamental theorem, i.e., Los's theorem. The result is shown as independently as possible of a given quantifier.

cs.LO↗