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On Topologically Relevant Fragments of the Logic of Linear Flows of Time

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Logic, Language, Information, and Computation (WoLLIC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9160))

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Abstract

Moss and Parikh’s bi-modal logic of subset spaces not only facilitates reasoning about knowledge and topology, but also provides an interesting example of a bi-topological system. This results from the fact that two interrelated S4s are involved in it. In the search for other examples of such kind, the temporal logic of linear flows of time might cross one’s mind. And although the full system itself is not bi-S4, a specific fragment sharing most of the corresponding characteristics can be identified. We here examine, among other things, to what extent the two modalities determining the latter set of formulas are related with regard to the respective canonical topo-model.

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Notes

  1. 1.

    See Chap. 6 of the handbook [2] for a report on much of the early research into this topic; more recent developments include, e.g., the papers [3] and [13].

  2. 2.

    See also [10], Sect. 6. With regard to the following, cf. the discussion on the Diodorean conception of modality there.

  3. 3.

    It should be noted that this is not mandatory for other topological interpretations of the modal operators that have been studied in the literature; see, e.g., [2], Sect. 5.3.1.

  4. 4.

    We do not discuss the adequacy of these structures for the context of tense logic further, as our focus will be on topological matters subsequently.

  5. 5.

    I.e., e.g., \([\mathsf {F}]^+\alpha :\equiv \alpha \wedge [\mathsf {F}]\alpha \); see [6], p. 98. – We do not care about redundancies throughout this paper.

  6. 6.

    This acronym is chosen since the basic temporal logic including the schemata 1 – 4 is usually denoted by \(\mathrm {K}_t\).

  7. 7.

    This modality has already been studied in [10], Sect. 6. Therefore, it is defined this way here (although \(\square _f\alpha \) and \(\square _p\alpha \) could have been used as well, as can be seen easily).

  8. 8.

    Concerning this notion, see, e.g., [2], p. 251.

  9. 9.

    These schemata are designated ‘T’ historically; see [10], p. 22, and for the common names of some other axioms used here, too.

  10. 10.

    Regarding \(\phi \), note the formal analogy between the two contexts.

  11. 11.

    It is meant here that all modalities are interpreted topologically, contrasting, e.g., the logics considered in [5].

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Acknowledgement

I am very grateful to the anonymous referees for their valuable comments, hints, and suggestions.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Hagen, 58084, Hagen, Germany

    Bernhard Heinemann

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  1. Bernhard Heinemann
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Correspondence to Bernhard Heinemann .

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Editors and Affiliations

  1. Nuance Communications, Sunnyvale, California, USA

    Valeria de Paiva

  2. Universidade Federal de Pernambuco, Centro de Informática, Recife, Pernambuco, Brazil

    Ruy de Queiroz

  3. Department of Mathematics, Indiana University Bloomington, Bloomington, Indiana, USA

    Lawrence S. Moss

  4. nDepartment of Computer Science, Indiana University Bloomington, Bloomington, Indiana, USA

    Daniel Leivant

  5. Universidade Federal de Pernambuco, Centro de Informática, Recife, Pernambuco, Brazil

    Anjolina G. de Oliveira

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Heinemann, B. (2015). On Topologically Relevant Fragments of the Logic of Linear Flows of Time. In: de Paiva, V., de Queiroz, R., Moss, L., Leivant, D., de Oliveira, A. (eds) Logic, Language, Information, and Computation. WoLLIC 2015. Lecture Notes in Computer Science(), vol 9160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47709-0_3

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  • DOI: https://doi.org/10.1007/978-3-662-47709-0_3

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