Abstract
We study the expressivity and the complexity of various logics in probabilistic team semantics with the Boolean negation. In particular, we study the extension of probabilistic independence logic with the Boolean negation, and a recently introduced logic FOPT. We give a comprehensive picture of the relative expressivity of these logics together with the most studied logics in probabilistic team semantics setting, as well as relating their expressivity to a numerical variant of second-order logic. In addition, we introduce novel entropy atoms and show that the extension of first-order logic by entropy atoms subsumes probabilistic independence logic. Finally, we obtain some results on the complexity of model checking, validity, and satisfiability of our logics.
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Notes
- 1.
In [21] Li recently introduced first-order theory of random variables with probabilistic independence (FOTPI) whose variables are interpreted by discrete distributions over the unit interval. The paper shows that true arithmetic is interpretable in FOTPI whereas probabilistic independence logic is by our results far less complex.
- 2.
In some sources, the term probabilistic team only refers to teams that are distributions, and the functions \(\mathbb {X}:X\rightarrow \mathbb {R}_{\ge 0}\) that are not distributions are called weighted teams.
- 3.
In [11], two sublogics of \({\textrm{FOPT}(\le _{c}^\delta )}\), called \(\textrm{FOPT}(\le ^\delta )\) and \(\textrm{FOPT}(\le ^{\delta },\perp \!\!\!\perp _{\textrm{c}}^{\delta })\), were also considered. Note that the results of this section also hold for these sublogics.
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Acknowledgements
The first author is supported by the ERC grant 101020762. The second author is supported by Academy of Finland grant 345634. The third author is supported by Academy of Finland grants 338259 and 345634. The fourth author appreciates funding by the European Union’s Horizon Europe research and innovation programme within project ENEXA (101070305). The fifth author appreciates funding by the German Research Foundation (DFG), project ME 4279/3-1. The sixth author is partially funded by the German Research Foundation (DFG), project VI 1045/1-1.
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Hannula, M., Hirvonen, M., Kontinen, J., Mahmood, Y., Meier, A., Virtema, J. (2023). Logics with Probabilistic Team Semantics and the Boolean Negation. In: Gaggl, S., Martinez, M.V., Ortiz, M. (eds) Logics in Artificial Intelligence. JELIA 2023. Lecture Notes in Computer Science(), vol 14281. Springer, Cham. https://doi.org/10.1007/978-3-031-43619-2_45
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