Abstract
First-order typed model counting extends first-order model counting by the ability to distinguish between different types of models. In this paper, we exploit this benefit in order to calculate weighted conditional impacts (WCIs) which play a central role in nonmonotonic reasoning based on conditionals. More precisely, WCIs store information about the verification and the falsification of conditionals with respect to a possible worlds semantics, and therefore serve as sufficient statistics for maximum entropy (ME) distributions as models of probabilistic conditional knowledge bases. Formally, we annotate formulas with algebraic types that encode concisely all structural information needed to compute WCIs, while allowing for a systematic and efficient counting of models. In this way, our approach to typed model counting for ME-reasoning integrates both structural and counting aspects in the same framework.
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Notes
- 1.
Non-conditional statements of the form “\(A\in \mathsf {FOL}\) holds with probability p” can be accommodated into \(\mathcal {R}\) by adding the conditional \((A|\top )[p]\) to \(\mathcal {R}\).
- 2.
We usually omit the operation symbol “\(\cdot \)”.
- 3.
The operation \(\circ \) shall be the one that binds weakest.
- 4.
We write edge labels in dashed frames and omit the label “\(\mathbf {1_\mathcal {S}}\)”.
- 5.
Note that \(\mathsf {sd}\text {-}\mathsf {DNNF}^{\mathcal {S}}\)-circuits principally represent grounded formulas, and so are \(A,B,C\in \mathsf {FOL}^\mathcal {S}\) in the following definitions. The power of \(\mathsf {sd}\text {-}\mathsf {DNNF}^{\mathcal {S}}\)-circuits lies in the capability of consolidating isomorphic formulas via set conjunction respectively set disjunction.
- 6.
In \(\mathsf {TMC}(A)\) resp. \(\mathsf {TMC}(B)\), those structured interpretations are considered that are restricted to the ground atoms in A resp. B.
- 7.
Here, \(\forall _{X\in \mathsf {var}(r_i)} X.\phi _i(r_i)=\forall X_1. (\ldots \forall X_m.\phi _i(r_i))\) where \(\mathsf {var}(r_i)=\{X_1,\ldots ,X_m\}\). Note that \(\mathsf {var}(r_i)=\mathsf {var}(\phi _i(r_i))\) for \(i=1,\ldots ,n\), which can be proved easily.
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Acknowledgements
This research was supported by the German National Science Foundation (DFG) Research Unit FOR 1513 on Hybrid Reasoning for Intelligent Systems.
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Wilhelm, M., Finthammer, M., Kern-Isberner, G., Beierle, C. (2017). First-Order Typed Model Counting for Probabilistic Conditional Reasoning at Maximum Entropy. In: Moral, S., Pivert, O., Sánchez, D., Marín, N. (eds) Scalable Uncertainty Management. SUM 2017. Lecture Notes in Computer Science(), vol 10564. Springer, Cham. https://doi.org/10.1007/978-3-319-67582-4_19
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