product iamge

Products of Topological Modal Logics [PhD Thesis]

This document was uploaded by one of our users. The uploader already confirmed that they had the permission to publish it. If you are author/publisher or own the copyright of this documents, please report to us by using this DMCA report form.

Download
Search on Amazon

Simply click on the Download Book button.

Yes, Book downloads on Ebookily are 100% Free.

Sometimes the book is free on Amazon As well, so go ahead and hit "Search on Amazon"

Author(s): Darko Sarenac
Series: ILLC Dissertation Series DS-2006-08
Publisher: University of Amsterdam
Year: 2006

Language: English
Pages: 134
City: Amsterdam

Dedication iv
Abstract v
Acknowledgments vi
1 Introduction 1
2 Modal Logics for Products of Topologies 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Topological completeness of S4 . . . . . . . . . . . . . . . . . 8
2.2.2 The fusion S4 " S4 . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 The product S4 × S4 . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Product spaces and product topo-bisimulations . . . . . . . . . . . . 13
2.3.1 Horizontal and vertical topologies . . . . . . . . . . . . . . . . 13
2.3.2 Failure of com and chr on R × R . . . . . . . . . . . . . . . . 16
2.3.3 Product topo-bisimulations . . . . . . . . . . . . . . . . . . . 17
2.4 Correspondence for com and chr . . . . . . . . . . . . . . . . . . . . . 19
2.5 Cardinal Spaces and chr and com . . . . . . . . . . . . . . . . . . . . 23
2.5.1 Bimodal formulae in products of cardinal spaces . . . . . . . . 23
2.6 The logic of product spaces . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Adding the true product interior . . . . . . . . . . . . . . . . . . . . . 29
2.8 Conclusions and further directions . . . . . . . . . . . . . . . . . . . . 33
2.8.1 Special spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.8.2 Enriching the language . . . . . . . . . . . . . . . . . . . . . . 34
2.8.3 Further exploration of the connection with Kripke semantics . 35
3 Combining Order and Topology 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Topo-Directional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Expressive Power . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Minimal Logic of TDL . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.2 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 The logic TDLGO over the class of generalized order topologies . . . 46
3.5 Decidability of TDLGO . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 The logic TDLQ over rational numbers . . . . . . . . . . . . . . . . . 55
3.7 The logic TDLN over natural numbers . . . . . . . . . . . . . . . . . 57
3.7.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.7.2 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.8 The logic TDLO over the class of (standard) order topologies . . . . 59
3.9 Topological Compass Logic . . . . . . . . . . . . . . . . . . . . . . . . 61
3.10 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.10.1 Completeness on abstract frames . . . . . . . . . . . . . . . . 64
3.11 TCQ, the complete logic for Q × Q . . . . . . . . . . . . . . . . . . . 65
3.12 TCN, the complete logic for N × N . . . . . . . . . . . . . . . . . . . 68
3.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 The Geometry of Knowledge 70
4.1 Epistemic logic in its standard guise . . . . . . . . . . . . . . . . . . . 70
4.1.1 Basic epistemic logic . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.2 Group knowledge . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.1.3 Agents as epistemic accessibility relations . . . . . . . . . . . . 73
4.1.4 Alternative views of common knowledge . . . . . . . . . . . . 74
4.1.5 Computing epistemic fixed-points . . . . . . . . . . . . . . . . 74
4.1.6 Merging Information . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Epistemic Models in Topological Semantics . . . . . . . . . . . . . . 78
4.2.1 From graphs to topological spaces. . . . . . . . . . . . . . . . 78
4.2.2 Topology and information . . . . . . . . . . . . . . . . . . . . 79
4.2.3 Common knowledge in product spaces . . . . . . . . . . . . . 80
4.2.4 Complete logic of common knowledge on topo-products . . . . 84
4.2.5 More on epistemic agents as topologies . . . . . . . . . . . . . 86
4.2.6 Operations that are safe for topo-bisimulation . . . . . . . . . 89
4.2.7 Merging information revisited . . . . . . . . . . . . . . . . . . 93
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Conclusions and Further Directions 95
5.1 Products in richer modal languages . . . . . . . . . . . . . . . . . . . 96
5.2 First-order extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Constraints on product operations . . . . . . . . . . . . . . . . . . . . 100
5.4 Decidability and complexity . . . . . . . . . . . . . . . . . . . . . . . 101
5.5 Conclusion once more . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A Plug-And-Play Unravelling and other issues 107