Abstract
This paper formalises the followership in networks that agents following or unfollowing each other dynamically. The semantics is based on the basic hybrid logic and we extend the logic with a propositional action modality \([{a}\uparrow {\theta }]\) for the changes of followership. The main contribution of this paper is the completeness result. Moreover, we have proved that all pure axiomatic extensions have completeness and discussed some possible future works and extensions, in particular, some features of the extended action modality \([{a}\uparrow {\varphi }]\), like repetition regrets are discussed.
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Notes
- 1.
Our action here is a little bit in difference, agents will follow and only all the agents satisfied by the claimed property. Action itself does not preserving the old relations.
- 2.
In the later section, when we talk about the extended languages, the set of Agt will then be substituted by the extended set as well.
- 3.
The converse of \([\overrightarrow{a}]\) is \([\overleftarrow{a}]\) which is defined by \([{a_n}\uparrow {\theta _n}]\ldots [{a_2}\uparrow {\theta _2}][{a_1}\uparrow {\theta _1}]\).
- 4.
It means that agents are dropping p-property, if they update with q-property.
- 5.
\(@_b\theta \rightarrow @_a[{a}\uparrow {\theta }]\Diamond {b}\) indicates that if b satisfies \(\theta \), then b will be followed by a after the executing \([{a}\uparrow {\theta }]\). \(@_a[{a}\uparrow {\theta }]\Diamond {b}\rightarrow @_b\theta \) says that since agent a is following b after executing \([{a}\uparrow {\theta }]\), then agent b satisfies \(\theta \).
- 6.
As the propositional case is trivial, and it’s also a technical reason why we need to restrict our actions into propositional formulas.
- 7.
Due to the property of \(\Diamond {}\), it is not possible to interpret the action that “agent a chooses to follow all agents that are followed by b”, actions like \([{a}\uparrow {(i\wedge @_b\Diamond {i})}]\) only means that “agent a chooses to follow agent i and b is following i”.
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Acknowledgment
We thank anonymous reviewers of LORI-2019 for many helpful comments, and in particular, for the contribution of related works. The first author is supported by the Fundamental Research Funds for the Central Universities under research no. SWU1809669, and by the Key Research Funds for the Key Liberal Science Research Base of Chongqing under research no. 18SK045.
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Xiong, Z., Guo, M. (2019). A Dynamic Hybrid Logic for Followership. In: Blackburn, P., Lorini, E., Guo, M. (eds) Logic, Rationality, and Interaction. LORI 2019. Lecture Notes in Computer Science(), vol 11813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60292-8_31
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