# Parameterised Complexity of Abduction in Schaefer's Framework

## TL;DR

This paper classifies the parameterized complexity of abduction problems within Schaefer's framework, analyzing various Boolean fragments and size constraints to identify tractable cases and develop new analytical methods.

## Contribution

It provides a comprehensive two-dimensional classification of abduction problems' complexity, extending previous work and introducing an alternative algebraic analysis method.

## Key findings

- Almost complete parameterized complexity classification
- Identification of fixed-parameter tractable cases
- Development of a new algebraic analysis approach

## Abstract

Abductive reasoning is a non-monotonic formalism stemming from the work of Peirce. It describes the process of deriving the most plausible explanations of known facts. Considering the positive version asking for sets of variables as explanations, we study, besides asking for existence of the set of explanations, two explanation size limited variants of this reasoning problem (less than or equal to, and equal to). In this paper, we present a thorough two-dimensional classification of these problems. The first dimension is regarding the parameterised complexity under a wealth of different parameterisations. The second dimension spans through all possible Boolean fragments of these problems in Schaefer's constraint satisfaction framework with co-clones (STOC 1978). Thereby, we almost complete the parameterised picture started by Fellows et al. (AAAI 2012), partially building on results of Nordh and Zanuttini (Artif. Intell. 2008). In this process, we outline a fine-grained analysis of the inherent parameterised intractability of these problems and pinpoint their FPT parts. As the standard algebraic approach is not applicable to our problems, we develop an alternative method that makes the algebraic tools partially available again.