Dynamic Topological Logic Interpreted over Minimal Systems
Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one uni...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Fernández-Duque, David [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2010 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media B.V. 2010 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of philosophical logic - Springer Netherlands, 1972, 40(2010), 6 vom: 16. Nov., Seite 767-804 |
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| Übergeordnetes Werk: |
volume:40 ; year:2010 ; number:6 ; day:16 ; month:11 ; pages:767-804 |
| Links: |
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| DOI / URN: |
10.1007/s10992-010-9160-4 |
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| Katalog-ID: |
OLC2073633811 |
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| 520 | |a Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within $\mathcal{DTL}$ one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within $\mathcal{DTL}$ translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while $\mathcal{DTL}$s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that $\mathcal{DTL}$ interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of $\mathcal{DTL}$ which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. | ||
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| 650 | 4 | |a Temporal logic | |
| 650 | 4 | |a Multimodal logic | |
| 650 | 4 | |a Topological dynamics | |
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10.1007/s10992-010-9160-4 doi (DE-627)OLC2073633811 (DE-He213)s10992-010-9160-4-p DE-627 ger DE-627 rakwb eng 100 VZ 5,1 17,1 ssgn PHILOS DE-12 fid LING DE-30 fid Fernández-Duque, David verfasserin aut Dynamic Topological Logic Interpreted over Minimal Systems 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2010 Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within $\mathcal{DTL}$ one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within $\mathcal{DTL}$ translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while $\mathcal{DTL}$s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that $\mathcal{DTL}$ interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of $\mathcal{DTL}$ which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. Dynamic topological logic Spatial logic Temporal logic Multimodal logic Topological dynamics Enthalten in Journal of philosophical logic Springer Netherlands, 1972 40(2010), 6 vom: 16. Nov., Seite 767-804 (DE-627)129290696 (DE-600)120391-5 (DE-576)014472031 0022-3611 nnns volume:40 year:2010 number:6 day:16 month:11 pages:767-804 https://doi.org/10.1007/s10992-010-9160-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC FID-PHILOS FID-LING SSG-OLC-PHI SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_69 GBV_ILN_72 GBV_ILN_130 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4082 GBV_ILN_4112 GBV_ILN_4317 AR 40 2010 6 16 11 767-804 |
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10.1007/s10992-010-9160-4 doi (DE-627)OLC2073633811 (DE-He213)s10992-010-9160-4-p DE-627 ger DE-627 rakwb eng 100 VZ 5,1 17,1 ssgn PHILOS DE-12 fid LING DE-30 fid Fernández-Duque, David verfasserin aut Dynamic Topological Logic Interpreted over Minimal Systems 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2010 Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within $\mathcal{DTL}$ one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within $\mathcal{DTL}$ translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while $\mathcal{DTL}$s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that $\mathcal{DTL}$ interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of $\mathcal{DTL}$ which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. Dynamic topological logic Spatial logic Temporal logic Multimodal logic Topological dynamics Enthalten in Journal of philosophical logic Springer Netherlands, 1972 40(2010), 6 vom: 16. Nov., Seite 767-804 (DE-627)129290696 (DE-600)120391-5 (DE-576)014472031 0022-3611 nnns volume:40 year:2010 number:6 day:16 month:11 pages:767-804 https://doi.org/10.1007/s10992-010-9160-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC FID-PHILOS FID-LING SSG-OLC-PHI SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_69 GBV_ILN_72 GBV_ILN_130 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4082 GBV_ILN_4112 GBV_ILN_4317 AR 40 2010 6 16 11 767-804 |
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10.1007/s10992-010-9160-4 doi (DE-627)OLC2073633811 (DE-He213)s10992-010-9160-4-p DE-627 ger DE-627 rakwb eng 100 VZ 5,1 17,1 ssgn PHILOS DE-12 fid LING DE-30 fid Fernández-Duque, David verfasserin aut Dynamic Topological Logic Interpreted over Minimal Systems 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2010 Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within $\mathcal{DTL}$ one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within $\mathcal{DTL}$ translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while $\mathcal{DTL}$s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that $\mathcal{DTL}$ interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of $\mathcal{DTL}$ which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. Dynamic topological logic Spatial logic Temporal logic Multimodal logic Topological dynamics Enthalten in Journal of philosophical logic Springer Netherlands, 1972 40(2010), 6 vom: 16. Nov., Seite 767-804 (DE-627)129290696 (DE-600)120391-5 (DE-576)014472031 0022-3611 nnns volume:40 year:2010 number:6 day:16 month:11 pages:767-804 https://doi.org/10.1007/s10992-010-9160-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC FID-PHILOS FID-LING SSG-OLC-PHI SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_69 GBV_ILN_72 GBV_ILN_130 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4082 GBV_ILN_4112 GBV_ILN_4317 AR 40 2010 6 16 11 767-804 |
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10.1007/s10992-010-9160-4 doi (DE-627)OLC2073633811 (DE-He213)s10992-010-9160-4-p DE-627 ger DE-627 rakwb eng 100 VZ 5,1 17,1 ssgn PHILOS DE-12 fid LING DE-30 fid Fernández-Duque, David verfasserin aut Dynamic Topological Logic Interpreted over Minimal Systems 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2010 Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within $\mathcal{DTL}$ one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within $\mathcal{DTL}$ translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while $\mathcal{DTL}$s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that $\mathcal{DTL}$ interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of $\mathcal{DTL}$ which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. Dynamic topological logic Spatial logic Temporal logic Multimodal logic Topological dynamics Enthalten in Journal of philosophical logic Springer Netherlands, 1972 40(2010), 6 vom: 16. Nov., Seite 767-804 (DE-627)129290696 (DE-600)120391-5 (DE-576)014472031 0022-3611 nnns volume:40 year:2010 number:6 day:16 month:11 pages:767-804 https://doi.org/10.1007/s10992-010-9160-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC FID-PHILOS FID-LING SSG-OLC-PHI SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_69 GBV_ILN_72 GBV_ILN_130 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4082 GBV_ILN_4112 GBV_ILN_4317 AR 40 2010 6 16 11 767-804 |
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10.1007/s10992-010-9160-4 doi (DE-627)OLC2073633811 (DE-He213)s10992-010-9160-4-p DE-627 ger DE-627 rakwb eng 100 VZ 5,1 17,1 ssgn PHILOS DE-12 fid LING DE-30 fid Fernández-Duque, David verfasserin aut Dynamic Topological Logic Interpreted over Minimal Systems 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2010 Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within $\mathcal{DTL}$ one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within $\mathcal{DTL}$ translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while $\mathcal{DTL}$s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that $\mathcal{DTL}$ interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of $\mathcal{DTL}$ which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. Dynamic topological logic Spatial logic Temporal logic Multimodal logic Topological dynamics Enthalten in Journal of philosophical logic Springer Netherlands, 1972 40(2010), 6 vom: 16. Nov., Seite 767-804 (DE-627)129290696 (DE-600)120391-5 (DE-576)014472031 0022-3611 nnns volume:40 year:2010 number:6 day:16 month:11 pages:767-804 https://doi.org/10.1007/s10992-010-9160-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC FID-PHILOS FID-LING SSG-OLC-PHI SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_69 GBV_ILN_72 GBV_ILN_130 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4082 GBV_ILN_4112 GBV_ILN_4317 AR 40 2010 6 16 11 767-804 |
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Dynamic Topological Logic Interpreted over Minimal Systems |
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Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within $\mathcal{DTL}$ one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within $\mathcal{DTL}$ translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while $\mathcal{DTL}$s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that $\mathcal{DTL}$ interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of $\mathcal{DTL}$ which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. © Springer Science+Business Media B.V. 2010 |
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Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within $\mathcal{DTL}$ one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within $\mathcal{DTL}$ translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while $\mathcal{DTL}$s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that $\mathcal{DTL}$ interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of $\mathcal{DTL}$ which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. © Springer Science+Business Media B.V. 2010 |
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Abstract Dynamic Topological Logic ($\mathcal{DTL}$) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within $\mathcal{DTL}$ one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within $\mathcal{DTL}$ translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while $\mathcal{DTL}$s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that $\mathcal{DTL}$ interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of $\mathcal{DTL}$ which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. © Springer Science+Business Media B.V. 2010 |
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