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Dynamic Cantor Derivative Logic

David Fernández-Duque, Yoàv Montacute

TL;DR

This work develops dynamic topological logics based on the Cantor derivative, introducing the $d$-logics $wK4C$, $K4C$, and $GLC$ and proving their finite model property, soundness, and completeness over appropriate classes of dynamic systems. It presents a general proof framework built from a canonical model, a finitary accessibility relation, stories and $oldsymbol{ ext Phi}$-morphisms, to transfer Kripke model results to dynamic topological semantics, including scattered and $T_D$ spaces, and extends the theory to invertible dynamics with $oldsymbol{wK4H}$, $oldsymbol{K4H}$, and $oldsymbol{GLH}$. The key result is that $oldsymbol{GLC}$ precisely captures the $d$-logic of dynamic topological systems on scattered spaces, while $oldsymbol{wK4C}$ and $oldsymbol{K4C}$ align with more general and $T_D$ topologies, respectively, with finite model properties established via a rigorous filtration-like construction. The paper lays a foundation for a finite axiomatisation of the trimodal topo-temporal language in the $d$-semantics and outlines directions for future work, including extending to immersions and Euclidean settings.

Abstract

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as $d$-logics. Unlike logics based on the topological closure operator, $d$-logics have not previously been studied in the framework of dynamical systems, which are pairs $(X,f)$ consisting of a topological space $X$ equipped with a continuous function $f\colon X\to X$. We introduce the logics $\bf{wK4C}$, $\bf{K4C}$ and $\bf{GLC}$ and show that they all have the finite Kripke model property and are sound and complete with respect to the $d$-semantics in this dynamical setting. In particular, we prove that $\bf{wK4C}$ is the $d$-logic of all dynamic topological systems, $\bf{K4C}$ is the $d$-logic of all $T_D$ dynamic topological systems, and $\bf{GLC}$ is the $d$-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where $f$ is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems $\bf{wK4H}$, $\bf{K4H}$ and $\bf{GLH}$. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological $d$-logics. Furthermore, our result for $\bf{GLC}$ constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -- something known to be impossible over the class of all spaces.

Dynamic Cantor Derivative Logic

TL;DR

This work develops dynamic topological logics based on the Cantor derivative, introducing the -logics , , and and proving their finite model property, soundness, and completeness over appropriate classes of dynamic systems. It presents a general proof framework built from a canonical model, a finitary accessibility relation, stories and -morphisms, to transfer Kripke model results to dynamic topological semantics, including scattered and spaces, and extends the theory to invertible dynamics with , , and . The key result is that precisely captures the -logic of dynamic topological systems on scattered spaces, while and align with more general and topologies, respectively, with finite model properties established via a rigorous filtration-like construction. The paper lays a foundation for a finite axiomatisation of the trimodal topo-temporal language in the -semantics and outlines directions for future work, including extending to immersions and Euclidean settings.

Abstract

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as -logics. Unlike logics based on the topological closure operator, -logics have not previously been studied in the framework of dynamical systems, which are pairs consisting of a topological space equipped with a continuous function . We introduce the logics , and and show that they all have the finite Kripke model property and are sound and complete with respect to the -semantics in this dynamical setting. In particular, we prove that is the -logic of all dynamic topological systems, is the -logic of all dynamic topological systems, and is the -logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems , and . The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological -logics. Furthermore, our result for constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -- something known to be impossible over the class of all spaces.

Paper Structure

This paper contains 11 sections, 40 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamic Topological Logics
  4. Prior Work
  5. The tangled closure on scattered spaces
  6. The Canonical Model
  7. A finitary accessibility relation
  8. Stories and $\Phi$-morphisms
  9. Topological $d$-completeness
  10. Invertible Systems
  11. Conclusion

Key Result

Lemma 2.5

If $\langle W,\sqsubset \rangle$ is a derivative frame, then $\langle W,{\downarrow_\sqsubset} \rangle$ is a derivative space.

Figures (1)

  • Figure 1: An example of a $\bold{GL}$-story. The squiggly arrows represent the relation $\sqsubset_{\mathfrak S}$ while the straight arrows represent the function $f_{\mathfrak S}$. Each vertical slice represents a $\bold{GL}$-moment. In the case of other types of stories, we may also have clusters besides singletons.

Theorems & Definitions (88)

  • Definition 2.1: topological space
  • Definition 2.2: Cantor derivative
  • Definition 2.3: derivative space
  • Definition 2.4: derivative frame
  • Lemma 2.5
  • Lemma 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Definition 2.9
  • ...and 78 more