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The existing approaches to formulate the semantics of probabilistic argumentation are based on the notion of possible world. Given a probabilistic argument graph (PrAG) with   $n$   nodes, up to   $2^n$   subgraphs are blindly constructed and their extensions under a given semantics are computed. Then, the probability of a set of arguments   $E$   being an extension under a given semantics   $\sigma$   (denoted as   $p(E^\sigma)$  ) is equal to the sum of the probabilities of all subgraphs each of which has the extension   $E$  . Since many irrelevant subgraphs are constructed, and in many cases, computing extensions of subgraphs is computationally intractable, these approaches are fundamentally inefficient or infeasible. In existing literature, while approximate approaches based on the Monte Carlo simulation technique have been proposed to estimate the probability of extensions, how to improve the efficiency of computation without using the simulation technique is still an open problem. In this article, we address this problem from the following two perspectives. First, conceptually, we define specific properties to characterize the subgraphs of a PrAG with respect to a given extension, such that the probability of a set of arguments   $E$   being an extension can be defined in terms of these properties, without (or with less) construction of subgraphs. Second, computationally, we take preferred semantics as an example, and develop algorithms to evaluate the efficiency of our approach. The results show that our approach not only dramatically decreases the time for computing   $p(E^\sigma)$  , but also has an attractive property, which is contrary to that of existing approaches: the denser the edges of a PrAG are or the bigger the size of a given extension   $E$   is, the more efficient our approach computes   $p(E^\sigma)$  . Meanwhile, it is shown that under complete and preferred semantics, the problems of determining   $p(E^\sigma)$   are fixed-parameter tractable.

Formulating semantics of probabilistic argumentation by characterizing subgraphs: theory and empirical results