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Advancing Algorithmic Approaches to Probabilistic Argumentation under the Constellation Approach

Andrei Popescu, Johannes P. Wallner

TL;DR

The paper tackles probabilistic argumentation under the constellation approach, addressing the high computational complexity of reasoning tasks. It refines complexity results to show a divergence: the probability that a set is a complete extension is $\#\P$-complete, while the probability that an argument is accepted is $\#\cdot\NP$-complete, even for acyclic attack structures. A dynamic-programming algorithm on nice tree-decompositions computes $P^{\mathit{ext}}$ for complete semantics, supported by preprocessing to simplify instances and an experimental evaluation on PAFs with up to about 750 arguments demonstrating scalability dependent on tree-width. Extensions to dependencies among arguments and other semantics are discussed to broaden applicability. Overall, the work provides both theoretical complexity insights and a scalable DP method that advances practical probabilistic argumentation under the constellation approach.

Abstract

Reasoning with defeasible and conflicting knowledge in an argumentative form is a key research field in computational argumentation. Reasoning under various forms of uncertainty is both a key feature and a challenging barrier for automated argumentative reasoning. It was shown that argumentative reasoning using probabilities faces in general high computational complexity, in particular for the so-called constellation approach. In this paper, we develop an algorithmic approach to overcome this obstacle. We refine existing complexity results and show that two main reasoning tasks, that of computing the probability of a given set being an extension and an argument being acceptable, diverge in their complexity: the former is #P-complete and the latter is #-dot-NP-complete when considering their underlying counting problems. We present an algorithm for the complex task of computing the probability of a set of arguments being a complete extension by using dynamic programming operating on tree-decompositions. An experimental evaluation shows promise of our approach.

Advancing Algorithmic Approaches to Probabilistic Argumentation under the Constellation Approach

TL;DR

The paper tackles probabilistic argumentation under the constellation approach, addressing the high computational complexity of reasoning tasks. It refines complexity results to show a divergence: the probability that a set is a complete extension is -complete, while the probability that an argument is accepted is -complete, even for acyclic attack structures. A dynamic-programming algorithm on nice tree-decompositions computes for complete semantics, supported by preprocessing to simplify instances and an experimental evaluation on PAFs with up to about 750 arguments demonstrating scalability dependent on tree-width. Extensions to dependencies among arguments and other semantics are discussed to broaden applicability. Overall, the work provides both theoretical complexity insights and a scalable DP method that advances practical probabilistic argumentation under the constellation approach.

Abstract

Reasoning with defeasible and conflicting knowledge in an argumentative form is a key research field in computational argumentation. Reasoning under various forms of uncertainty is both a key feature and a challenging barrier for automated argumentative reasoning. It was shown that argumentative reasoning using probabilities faces in general high computational complexity, in particular for the so-called constellation approach. In this paper, we develop an algorithmic approach to overcome this obstacle. We refine existing complexity results and show that two main reasoning tasks, that of computing the probability of a given set being an extension and an argument being acceptable, diverge in their complexity: the former is #P-complete and the latter is #-dot-NP-complete when considering their underlying counting problems. We present an algorithm for the complex task of computing the probability of a set of arguments being a complete extension by using dynamic programming operating on tree-decompositions. An experimental evaluation shows promise of our approach.
Paper Structure (15 sections, 6 theorems, 6 equations, 3 figures, 1 table, 4 algorithms)

This paper contains 15 sections, 6 theorems, 6 equations, 3 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Tree-Decompositions and Tree-Width
  4. Complexity Results for Probabilistic AFs
  5. Preprocessing Probabilistic AFs
  6. Tree-Decomposition-based Algorithm
  7. Introduction Nodes
  8. Forget Nodes
  9. Join Nodes
  10. Experimental Evaluation
  11. Implementations
  12. Instances
  13. Setup and Results
  14. Extensions
  15. Discussion

Key Result

Proposition 1

Let $F=(A,R)$ be an AF and $\sigma \in \{\mathit{com},{\mathit{stb}}\}$.

Figures (3)

  • Figure 1: A PAF with certain (solid lines) and uncertain (dashed lines) arguments and attacks.
  • Figure 2: Preprocessing a PAF
  • Figure 3: Nice tree decomposition of the PAF of Example \ref{['ex:paf']} and corresponding tables (partial) for the computation of $P^{\mathit{ext}}_{com}(\{a, c, e\})$.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1
  • Example 1
  • Definition 6
  • Example 2
  • Definition 7
  • ...and 11 more