Products of Topological Modal Logics [PhD Thesis]

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