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

2025-05-15

AI Summary

This work investigates the fine-grained parameterized complexity of propositional abduction, using the number $n$ of variables in the knowledge base as the natural parameter. It systematically delineates computational boundaries across NP-, coNP-, and $Sigma_2^P$-complete fragments. Methodologically, it introduces the first parameterized algorithm for $Sigma_2^P$-complete abduction that improves upon brute-force enumeration, yielding the first nontrivial upper bound parameterized by $n$; concurrently, it establishes tight lower bounds under the Exponential Time Hypothesis (ETH) across multiple logical fragments, proving optimality for most cases. Integrating techniques from parameterized algorithm design, SAT solving, and fine-grained complexity analysis, the paper bridges a critical gap between monotonic and nonmonotonic reasoning research. Its main contributions are: (i) the first $n$-parameterized upper bound for $Sigma_2^P$-complete abduction; (ii) ETH-based tight lower bounds confirming asymptotic optimality; and (iii) a definitive complexity dichotomy that identifies the core tractability frontier of abductive reasoning.

Technology Category

• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Analyzing complexity of intractable abduction problems
• Bridging gap between monotonic and non-monotonic reasoning
• Beating exhaustive search for Σ^P_2-complete problems

Innovation

• Analyzing abduction complexity using variable count
• Beating exhaustive search for Σ^P_2-complete problems
• Providing lower bounds under exponential-time hypothesis