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The Complexity of Non-Iterated Probabilistic Justification Logic

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Foundations of Information and Knowledge Systems (FoIKS 2016)

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Abstract

The logic \(\mathsf {PJ}\) is a probabilistic logic defined by adding (non-iterated) probability operators to the basic justification logic \(\mathsf {J}\). In this paper we establish upper and lower bounds for the complexity of the derivability problem in the logic \(\mathsf {PJ}\). The main result of the paper is that the complexity of the derivability problem in \(\mathsf {PJ}\) remains the same as the complexity of the derivability problem in the underlying logic \(\mathsf {J}\), which is \(\varPi _2^p\)-complete. This implies hat the probability operators do not increase the complexity of the logic, although they arguably enrich the expressiveness of the language.

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Notes

  1. 1.

    \(\mathsf {J}\) stands for justification, whereas \(\mathsf {PJ}\) stands for probabilistic justification.

  2. 2.

    The two P’s stand for iterations of the probability operator.

  3. 3.

    We agree to the convention that the formula \({!^{n-1}} c : {!^{n-2}} c : \cdots : {!c} : c : \alpha \) represents the formula \(\alpha \) for \(n=0\).

  4. 4.

    Observe that the satisfiability relation of a \(\mathsf {J}_{\mathsf {CS}}\)-evaluation is represented with \(\Vdash \) whereas the satisfiability relation of a model is represented with \(\models \).

  5. 5.

    We will always use bold font for vectors.

  6. 6.

    In the proof of Theorem 4 all vectors have n entries. The entries of the vectors are assumed to be in one to one correspondence with the variables that appear in the original system \(\mathcal {S}\).

    Let \(\varvec{y} \) be a solution of a linear system \(\mathcal {T}\). If \(\varvec{y} \) has more entries than the variables of \(\mathcal {T}\) we imply that entries of \(\varvec{y} \) that correspond to variables that appear in \(\mathcal {T}\) compose a solution of \(\mathcal {T}\).

  7. 7.

    Assume that system \(\mathcal {T}\) has less variables than system \(\mathcal {T}'\). When we say that any solution of \(\mathcal {T}\) is a solution of \(\mathcal {T}'\) we imply that the missing variables are set to 0.

  8. 8.

    A reader unfamiliar with notions of computational complexity theory may consult a textbook on the field, like [24].

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Acknowledgements

The author is grateful to Antonis Achilleos, Thomas Studer and the anonymous referees for valuable comments and remarks that helped him improve the quality of the paper substantially.

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  1. Institute of Computer Science, University of Bern, Bern, Switzerland

    Ioannis Kokkinis

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  1. Ioannis Kokkinis
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  1. Faculteit Wetenschappen, Universiteit Hasselt, Hasselt, Belgium

    Marc Gyssens

  2. Dept. Ciencias Ingeniería Computación, Universidad Nacional del Sur, Bahía Blanca, Argentina

    Guillermo Simari

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Kokkinis, I. (2016). The Complexity of Non-Iterated Probabilistic Justification Logic. In: Gyssens, M., Simari, G. (eds) Foundations of Information and Knowledge Systems. FoIKS 2016. Lecture Notes in Computer Science(), vol 9616. Springer, Cham. https://doi.org/10.1007/978-3-319-30024-5_16

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