Abstract
We develop inference procedures for a recently proposed model of probabilistic argumentation called PABA, taking advantages of well-established dialectical proof procedures for Assumption-based Argumentation and Bayesian Network algorithms. We establish the soundness and termination of our inference procedures for a general class of PABA frameworks. We also discuss how to translate other models of probabilistic argumentation into this class of PABA frameworks so that our inference procedures can be used for these models as well.
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Notes
- 1.
Probabilistic values are made up for demonstration.
- 2.
\(ABA\,\mathcal F \vdash _{sem} \pi \) iff wrt this PABA framework, \(Prob_{sem}(\pi ) = 1\).
- 3.
- 4.
Preferred/grounded/ideal semantics.
- 5.
For convenience, define \(head(r) = l_0\) and \(body(r) = \{l_1,\dots l_n\}\).
- 6.
Any PABA framework in [5] is also an PABA framework in our extended definition, but the reverse may not hold.
- 7.
\(\lnot \) is the classical negation operator.
- 8.
In examples, we will not list complementary rules to save space.
- 9.
We will use this framework in running examples from now on.
- 10.
That is, each pair \(\alpha , \lnot \alpha \) of probabilistic assumptions of \(\mathcal P\) corresponds to truth assignments of variable \(\alpha \in V\) and vice versa; and each probabilistic rule in \(\mathcal R_p\) corresponds to one entry of an CPT in \(\mathcal N\) and vice versa.
- 11.
If \(\pi \) does not occur in \(\mathcal P\), then \(Prob_{sem}(\pi )=0\) for any semantics sem.
- 12.
From now on we assume an arbitrary but fixed \(PABA\,\mathcal P = (\mathcal A_p, \mathcal R_p, \mathcal F)\) with
if not explicitly stated otherwise. - 13.
Silence about a component means it remains the same as the previous step. In this case 2.a.i, for example, \(A_{i+1} = A_i\) and \(C_{i+1} = C_i\).
- 14.
That is, neither \(\sigma \) nor its complement are elements of \(\omega \).
- 15.
\(\mathcal A\) is the set of assumptions in \(ABA\,\mathcal F\).
- 16.
That is, \(Prob_{gr}(A) \triangleq \sum \limits _{\omega \in \mathcal W: (AR_{\omega }, Att \cap (AR_{\omega } \times AR_{\omega })) \vdash _{gr} A} P(\omega )\).
if not explicitly stated otherwise.References
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Acknowledgment
This work was funded by SIIT Young Researcher Grant under Contract No SIIT-2014-YRG1.
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Hung, N.D. (2016). Computing Probabilistic Assumption-Based Argumentation. In: Booth, R., Zhang, ML. (eds) PRICAI 2016: Trends in Artificial Intelligence. PRICAI 2016. Lecture Notes in Computer Science(), vol 9810. Springer, Cham. https://doi.org/10.1007/978-3-319-42911-3_13
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