Founded (Auto)Epistemic Equilibrium Logic Satisfies Epistemic Splitting

Abstract: In a recent line of research, two familiar concepts from logic programming semantics (unfounded sets and splitting) were extrapolated to the case of epistemic logic programs. The property of epistemic splitting provides a natural and modular way to understand programs without epistemic cycles but, surprisingly, was only fulfilled by Gelfond's original semantics (G91), among the many proposals in the literature. On the other hand, G91 may suffer from a kind of self-supported, unfounded derivations when epistemic cycles come into play. Recently, the absence of these derivations was also formalised as a property of epistemic semantics called foundedness. Moreover, a first semantics proved to satisfy foundedness was also proposed, the so-called Founded Autoepistemic Equilibrium Logic (FAEEL). In this paper, we prove that FAEEL also satisfies the epistemic splitting property something that, together with foundedness, was not fulfilled by any other approach up to date. To prove this result, we provide an alternative characterisation of FAEEL as a combination of G91 with a simpler logic we called Founded Epistemic Equilibrium Logic (FEEL), which is somehow an extrapolation of the stable model semantics to the modal logic S5. Under consideration for acceptance in TPLP.

• The paper establishes that FAEEL achieves epistemic splitting by decomposing logic programs into a modular bottom and a dependent top component.
• The methodology employs an intermediate logic, FEEL, to bridge Equilibrium Logic and S5, rigorously comparing FAEEL with G91 semantics.
• This framework enhances modular reasoning in AI systems and prevents self-supporting derivations within complex epistemic logic programs.

Founded (Auto)Epistemic Equilibrium Logic Satisfies Epistemic Splitting

Introduction

The paper presents a novel examination of the Founded Autoepistemic Equilibrium Logic (FAEEL) within epistemic logic programming and explores the property of epistemic splitting. The primary focus is on demonstrating that FAEEL satisfies epistemic splitting, which is crucial for modular and intuitive reasoning in logic programs without epistemic cycles. This property was previously known to be satisfied only by Gelfond's original semantics (G91), despite FAEEL's additional feature of addressing unfounded derivations. This is a noteworthy contribution given that many semantics for epistemic logic programs lack this property.

Theoretical Foundation

Epistemic logic programs extend disjunctive logic programs by including modal operators like $\mathbf{K} l$ (where $l$ is true in every stable model) and $\mathbf{M} l$ (where $l$ is true in some stable model). The semantics of these programs is challenging due to the cyclic interdependence between subjective literals and the stable models they query, leading to multiple alternative semantics proposals.

FAEEL is introduced as a semantics that not only satisfies foundedness, avoiding self-supported derivations, but now is also shown to satisfy epistemic splitting. The paper positions FAEEL as a promising candidate to serve as an epistemic logic semantics by fulfilling both properties, unlike any other pre-existing approach.

Methodology

The property of epistemic splitting allows programs to be decomposed into a "bottom" part that is evaluated first, independently, and a "top" part that depends on the results of the bottom. To demonstrate that FAEEL satisfies this property, the paper introduces the intermediate logic Founded Epistemic Equilibrium Logic (FEEL). FEEL acts as a bridge, characterized by combining Equilibrium Logic with the modal logic S5.

The key result achieved is that FAEEL-world views can be precisely characterized as those G91-world views that are also FEEL-world views. This facilitates analyzing FAEEL properties by exploring them in FEEL and G91 separately.

Results

• Epistemic Splitting: It is rigorously proven that FAEEL satisfies the epistemic splitting property. By introducing and utilizing FEEL, the authors provide alternative characterizations enabling this proof.
• Relation with G91: For epistemically tight programs and those without positive epistemic cycles, FAEEL coincides with G91 semantics, preserving compatibility while extending functionality by addressing unfounded derivations.
• Alternative Semantics: The work draws clear distinctions between FAEEL and similar frameworks, such as G91 and other epistemic logics, reinforcing the robustness of FAEEL in managing self-support and epistemic dependencies.

Practical Implications

By satisfying epistemic splitting, FAEEL allows a natural decomposition of logic programs, facilitating modular reasoning and simplifying the understanding of complex programs. This makes it suitable for applications requiring robust knowledge representation with modular verification, such as formal methods in AI systems, advanced decision support systems, and knowledge-based AI controllers.

Conclusion

The paper successfully establishes FAEEL as a comprehensive semantic framework for epistemic logic programs by fulfilling the important properties of foundedness and epistemic splitting. This positions FAEEL as a viable and effective formalism for future developments in AI, particularly where modularity and the prevention of self-supporting conclusions are critical.