Epistemic Logic Programs: Non-Ground and Counting Complexity
Thomas Eiter, Johannes K. Fichte, Markus Hecher, Stefan Woltran
TL;DR
This work analyzes non-ground Epistemic Logic Programs (ELPs), focusing on both qualitative (world-view existence) and quantitative (plausibility via counting) questions. It establishes a detailed complexity landscape: non-ground WV existence is complete for $\text{NEXP}^{\Sigma^P_i}$ for usual fragments (with i=2 for full, i=1 for normal/tight) and becomes $\Sigma^P_{i+2}$-complete under bounded arity, extending up to the fourth level of the polynomial hierarchy; counting results elevate to ${\#EXP}^{\Sigma^P_i}$-complete and ${\#·D^P_{i+1}}$-complete in bounded-arity settings. The authors introduce generalized counting classes and leverage succinct quantified Boolean formulas to obtain tight counting bounds, including SuccQBF-based reductions. They also provide ETH-tight runtime bounds for treewidth-bounded instances, underscoring the practical impact on structure-aware grounding and solver design. Overall, the paper delivers a rigorous, multi-faceted complexity map for non-ground ELPs with broad implications for AI reasoning tasks and epistemic modeling.
Abstract
Answer Set Programming (ASP) is a prominent problem-modeling and solving framework, whose solutions are called answer sets. Epistemic logic programs (ELP) extend ASP to reason about all or some answer sets. Solutions to an ELP can be seen as consequences over multiple collections of answer sets, known as world views. While the complexity of propositional programs is well studied, the non-ground case remains open. This paper establishes the complexity of non-ground ELPs. We provide a comprehensive picture for well-known program fragments, which turns out to be complete for the class NEXPTIME with access to oracles up to Σ^P_2. In the quantitative setting, we establish complexity results for counting complexity beyond #EXP. To mitigate high complexity, we establish results in case of bounded predicate arity, reaching up to the fourth level of the polynomial hierarchy. Finally, we provide ETH-tight runtime results for the parameter treewidth, which has applications in quantitative reasoning, where we reason on (marginal) probabilities of epistemic literals.