A Fine-Grained Complexity View on Propositional Abduction -- Algorithms and Lower Bounds

TL;DR

This paper explores the fine-grained complexity of propositional abduction, providing new algorithms and lower bounds, and bridging the understanding between monotonic and non-monotonic reasoning.

Contribution

It introduces the first complexity analysis of abduction parameterized by variables, offering algorithms and lower bounds for various complexity classes.

Findings

First algorithms for abduction with complexity bounds

Lower bounds ruling out improvements under ETH

Beating exhaustive search for a $ ext{Sigma}_2^P$-complete problem

Abstract

The Boolean satisfiability problem (SAT) is a well-known example of monotonic reasoning, of intense practical interest due to fast solvers, complemented by rigorous fine-grained complexity results. However, for non-monotonic reasoning, e.g., abductive reasoning, comparably little is known outside classic complexity theory. In this paper we take a first step of bridging the gap between monotonic and non-monotonic reasoning by analyzing the complexity of intractable abduction problems under the seemingly overlooked but natural parameter n: the number of variables in the knowledge base. We obtain several positive results for $\Sigma^P_2$- as well as NP- and coNP-complete fragments, which implies the first example of beating exhaustive search for a $\Sigma^P_2$-complete problem (to the best of our knowledge). We complement this with lower bounds and for many fragments rule out improvements under the (strong) exponential-time hypothesis.