Representative Sets in Propositional Abduction

2026-07-23

AI Summary

This study addresses the problem of representative representation in propositional abduction: whether a given set of explanations can cover all other explanations via symmetric difference bounded by k. Combining classical and parameterized complexity theory, the work establishes a complete complexity classification for this problem and reveals, for the first time, its deep connection to the covering radius problem in coding theory. The main contributions include introducing a symmetric-difference-based framework for assessing representativeness, providing a full characterization of its classical complexity, identifying several tractable and intractable parameterized cases, and demonstrating that a complete parameterized classification hinges on breakthroughs in understanding the covering radius problem—thereby opening a new interdisciplinary avenue between non-monotonic reasoning and coding theory.

Abstract

The propositional abduction problem is a well-known form of non-monotonic reasoning where we are asked to find an explanation of a given manifestation. Recently, there has been an influx of results asking more refined questions about the solution space rather than only individual solutions. For example, we might be interested in finding two solutions that are sufficiently far from each other (diverse solutions) in the solution space. In this paper we consider a related representation question where we ask if a given set of explanations S can represent any other explanation (that is, whether their symmetric difference is smaller than a given k). We first study this problem from a classical complexity perspective and obtain a complete classification. While only a handful of cases are tractable, the increase in complexity compared to classical abduction is often smaller than expected. We then study the parameterized complexity for several parameters and obtain new tractable and hard cases. Interestingly, a full parameterized complexity classification would require resolving the parameterized complexity of the covering radius problem from coding theory. To the best of our knowledge, no useful relationship between coding theory and non-monotonic reasoning has previously been established, but such connections seemingly become important when asking more complex questions about solution spaces.

Problem

• propositional abduction
• representative sets
• solution space
• symmetric difference
• non-monotonic reasoning

Innovation

• representative sets
• propositional abduction
• parameterized complexity
• covering radius
• solution space representation