Abstract
We propose a Logic of Abstraction, meant to formalize the act of “abstracting away” the irrelevant features of a model. We give complete axiomatizations for a number of variants of this formalism, and explore their expressivity. As a special case, we consider the “logics of filtration”.
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Notes
- 1.
However, we’ll show that, in combination with applying relational transformers described by regular PDL programs, this lifting can capture other filtrations.
- 2.
In Sect. 3, we will show precisely how filtrations fit into our framework.
- 3.
The finiteness of \(\varSigma \) is in fact irrelevant for the definition of quotient models, however, this will be required in order to be able to provide reduction axioms for our new dynamic modalities introduced later in this section. This is why we keep the setting simple and work only with finite \(\varSigma \)s.
- 4.
Note that two \(\varSigma \)-equivalent worlds may disagree on the propositional variables that are not in the set \(\varSigma \).
- 5.
This definition is known to modal logicians under the name of smallest filtration (see, e.g., [7, Chap. 2.3]).
- 6.
In this section—since the formalism is based on Kripke models with a single relation—we have only one basic program r in our syntax. In Sect. 4, we work with multi-relational Kripke models allowing for more than one basic programs, as standard in \(\mathbf {PDL}\).
- 7.
The filtrations in the aforementioned sources are defined for a language without the universal modality. However, as observed in [13, Sect. 5.2], the universal modality does not cause any problems in the theory of filtrations.
- 8.
Since filtrations are usually only defined for subformula closed sets—the reason being that the Filtration Theorem can only be proved in this case—we add this as an additional condition.
- 9.
Recall that a transitive Kripke model \(\mathfrak {M}\) is called rooted if there is \(s \in W\) such that sRw for all \(w \in \mathfrak {M}\).
- 10.
Similar to the case in Sect. 2, the sets \(\varSigma \) being finite is essential in order to obtain reduction axioms for the corresponding dynamic logic.
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Acknowledgments
A. Özgün acknowledges financial support from European Research Council grant EPS 313360.
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Baltag, A., Bezhanishvili, N., Ilin, J., Özgün, A. (2017). Quotient Dynamics: The Logic of Abstraction. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_13
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