*5.1. Experimental Setup*

All problem instances (DeLP3E knowledge bases and queries) were synthetically generated to be able to adequately control the independent variables in our analysis. To obtain an instance, we first randomly generate the AM as a classical DeLP program with a balanced set of facts and rules; rule bodies and heads are generated in such a way as to ensure overlap, in order to yield nontrivial arguments (see [31] for details on such a procedure). The general design of the program generator consists of the following steps:


For the Bayesian networks in the EM, we randomly generated a graph on the basis of the desired number of EM variables (and a random number of edges set to the number of nodes as a maximum) using the networkx library (https://networkx.github.io, accessed on 21 August 2022). To control the entropy of the encoded distribution, we took each node probability table entry and randomly choose between *true* and *false*; then, we randomly assigned a probability to that outcome in the interval [*α*, 1], where *α* is a parameter varied in {0.7, 0.9}.

Annotation functions are lastly randomly generated by assigning to each element in the AM an element randomly chosen from the set of (possibly negated) EM variables plus "*true*" (AM elements annotated with *true* hold in all worlds).

**Quality Metric.**Given a probability interval *i*<sup>1</sup> = [*a*, *b*], we used the following metric to gauge the quality of a sound approximation *i*<sup>2</sup> = [*c*, *d*] (that is [*a*, *b*] ⊆ [*c*, *d*] always holds):

$$Q\_{i1}(i\_2) = \frac{1 - (d - c)}{1 - (b - a)}$$

Intuitively, this metric calculates the probability mass that is *discarded* by one interval in relation to another. The resulting value is always a real number in [0, 1], where a value of zero indicates the poorest possible approximation ([0, 1], which is always a sound approximation for problem instance), and a value of 1 yields the best possible approximation, which corresponds precisely with the exact interval. Thus, we generally apply this metric by using the result of the exact algorithm in the numerator and an approximation in the denominator.
