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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| Vydané v: | Logical methods in computer science Ročník 19, Issue 4 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Logical Methods in Computer Science e.V
01.01.2023
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| Predmet: | |
| ISSN: | 1860-5974, 1860-5974 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | 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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| ISSN: | 1860-5974 1860-5974 |
| DOI: | 10.46298/lmcs-19(4:26)2023 |