Dynamic Cantor Derivative Logic
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 $...
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| Veröffentlicht in: | Logical methods in computer science Jg. 19, Issue 4 |
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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Logical Methods in Computer Science e.V
01.01.2023
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| ISSN: | 1860-5974, 1860-5974 |
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| 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. |
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| AbstractList | 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. 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. |
| Author | Montacute, Yoàv Fernández-Duque, David |
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| Snippet | Topological semantics for modal logic based on the Cantor derivative operator
gives rise to derivative logics, also referred to as $d$-logics. Unlike logics... Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as $d$-logics. Unlike logics... |
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| Title | Dynamic Cantor Derivative Logic |
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